Dissertation Alvaro Leitao Rodriguez

Delft University of Technology
Hybrid Monte Carlo methods in computational finance
Leitao Rodriguez, Alvaro
DOI
10.4233/uuid:fc4d5fc5-cee9-44ef-bca1-e69250c1480f
Publication date
2017
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Final published version
Citation (APA)
Leitao Rodriguez, A. (2017). Hybrid Monte Carlo methods in computational finance. [Dissertation (TU Delft),
Delft University of Technology]. https://doi.org/10.4233/uuid:fc4d5fc5-cee9-44ef-bca1-e69250c1480f
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H YBRID M ONTE C ARLO METHODS IN
COMPUTATIONAL FINANCE
H YBRID M ONTE C ARLO METHODS IN
COMPUTATIONAL FINANCE
Proefschrift
ter verkrijging van de graad van doctor

aan de Technische Universiteit Delft,
op gezag van de Rector Magnificus prof. ir. K. C. A. M. Luyben,
voorzitter van het College voor Promoties,
in het openbaar te verdedigen op dinsdag 27 juni 2017 om 12:30 uur.
door
Álvaro L EITAO R ODRÍGUEZ
Master of Science in Mathematical Engineering

Universiteit van Vigo, Spanje
geboren te Xove, Spanje.
Dit proefschrift is goedgekeurd door de promotor:
Prof. dr. ir. C. W. Oosterlee
Samenstelling promotiecommissie:
Rector Magnificus voorzitter
Prof. dr. ir. C. W. Oosterlee Technische Universiteit Delft, promotor
Onafhankelijke leden:
Prof. dr. A. Pascucci Università di Bologna, Italië
Prof. dr. K. In’t Hout Universiteit Antwerpen, België
Prof. dr. B. D. Kandhai Universiteit van Amsterdam
Prof. dr. ir. A. W. Heemink Technische Universiteit Delft
Prof. dr. C. Witteveen Technische Universiteit Delft
Overig lid:
Dr. ir. L. A. Grzelak Technische Universiteit Delft
This research was partially supported by the EU in the FP7-PEOPLE-2012-ITN Program

under Grant Agreement Number 304617 (FP7 Marie Curie Actions, Project Multi-ITN
STRIKE – Novel Methods in Computational Finance).
Hybrid Monte Carlo methods in computational finance

Dissertation at Delft University of Technology
Copyright © 2017 by Á. Leitao Rodríguez
Printed by: IPSKAMP printing
ISBN 978-94-028-0681-6
An electronic version of this dissertation is available at
http://repository.tudelft.nl/.
To Mela;
To my parents;
Two things are infinite:

the universe and human stupidity;
and I’m not sure about the universe.
Albert Einstein
Summary
Monte Carlo methods are highly appreciated and intensively employed in computa-
tional finance in the context of financial derivatives valuation or risk management. The
method offers valuable advantages like flexibility, easy interpretation and straightfor-
ward implementation. Furthermore, the dimensionality of the financial problem can
be increased without reducing the efficiency significantly. The latter feature of Monte
Carlo methods is important since it represents a clear advantage over other competing
numerical methods. Furthermore, in the case of option valuation problems in multiple
dimensions (typically more than five), the Monte Carlo method and its variants become
the only possible choices. Basically, the Monte Carlo method is based on the simulation
of possible scenarios of an underlying process and by then aggregating their values for a
final solution. Pricing derivatives on equity and interest rates, risk assessment or port-
folio valuation are some of the representative examples in finance, where Monte Carlo
methods perform very satisfactorily. The main drawback attributed to these methods is
the rather poor balance between computational cost and accuracy, according to the the-
oretical rate of Monte Carlo convergence. Based on the central limit theorem, the Monte
Carlo method requires hundred³ times ´ more scenarios to reduce the error by one order,
1
i.e. the rate of convergence is O n − 2 .
However, the application of basic Monte Carlo methods, usually called plain Monte
Carlo methods, is often not sufficient to solve modern problems appearing in the finan-
cial sector. Problems like the accurate computation of sensitivities and early-exercise fi-
nancial derivatives in multiple dimensions are particularly interesting examples. Except
for the simplest asset price stochastic models, the generation of the scenarios is not al-
ways a trivial task. Although the so-called Taylor-based discretization schemes, like the
Euler-Maruyama scheme, are heavily used, more involved mechanisms are required to
achieve the goal of accuracy and efficiency in a number of cases. For the computation
of the sensitivities and for early-exercise valuation, the use of plain Monte Carlo is either
not sufficient or computationally unaffordable.
Therefore, hybrid solution methods, i.e., combining Monte Carlo methods with some
other methodologies, can be developed in order to overcome the issues and provide fast,
accurate and consistent computational results. This can be done by relying on mathe-
matical, computational approaches, or combinations of both. Hybrid solution methods
are becoming essential tools to deal with the increasing complexity in quantitative fi-
nance.
In this thesis, we propose Monte Carlo-based hybrid solution methods for some cut-
ting edge problems in computational finance. From the mathematical side, we employ
Fourier-based techniques that, in combination with the Monte Carlo methods, result
in efficient methodologies to simulate and evaluate sensitivities under stochastic local
volatility models, that are nowadays used for foreign-exchange (FX) rate options. We also
employ some recent scientific computing paradigms advances, like high-performance
computing on GPUs, to extend the applicability of a Monte Carlo method for pricing
vii
viii S UMMARY
multi-dimensional early-exercise options and thus enable the computation of very high-
dimensional problems.
In Chapter 1, a general overview of the methodologies that commonly appear in
computational finance is presented. By considering the solution of partial differential
equations (PDEs) provided by the Feynman-Kac theorem and its variants, mathematical
techniques like Fourier inversion, Monte Carlo methods and PDE approaches have been
successfully connected to solve financial problems like option pricing and risk manage-
ment. We provide a brief explanation of the techniques. Other recent developments like
data-driven approaches, where a pricing model is derived solely based on the available
financial data, or parallel GPU computing are also briefly described, as they are consid-
ered in the second part of this thesis.
In Chapter 2, we propose an efficient technique to simulate exactly a representative
stochastic local volatility model, i.e. the Stochastic Alpha Beta Rho (SABR) model, con-
sidered well-established in FX and interest rate markets. We aim to employ only one
time-step in a Monte Carlo procedure. Particularly, we focus on the derivation of the
conditional distribution of the appearing time-integrated variance, an element which is
required in the simulation of the SABR asset dynamics. For that purpose, we compute
the marginal distribution by means of Fourier inversion and the use a copula to obtain a
multivariate distribution which resembles the conditional time-integrated variance dis-
tribution. Being a one time-step method, the method is suitable for the valuation of
European options and, due to the presented improvements in terms of performance,
can be particularly employed for calibration purposes.
The method introduced in Chapter 2 is generalized towards a multiple time-step ver-
sion for the simulation of the SABR model in Chapter 3. Although the multiple time-
step procedure is conceptually similar, this version entails a new challenge regarding the
computational cost. We deal with this challenge by introducing an advanced numeri-
cal interpolation which is based on a stochastic collocation technique. The proposed
method, which we call the mSABR method, allows for an efficient simulation with large
time-steps, and it can be employed to accurately price (exotic), non-standard, options
under the SABR dynamics. This simulation scheme also performs highly satisfactory
even in the context of negative interest rates, that are presently observed in the financial
world.
In Chapter 4, we develop a data-driven Fourier-based option valuation method that
appears beneficial when calculating the sensitivities of financial derivatives in a gen-
eral framework. The method is based on the assumption that only asset data samples
are available. The resulting method is a data-based generalization of the COS method,
which is a Fourier inversion method that employs the characteristic function of the rel-
evant stochastic asset variables. In our method, which we call the data-driven COS (dd-
COS) method, the use of the characteristic function is avoided. Instead, we recover the
probability density function from the available data samples, by using concepts from the
statistical learning theory. Since the ddCOS method is a data-driven procedure it is, thus,
generally applicable and can be employed, for example, in combination with the above
mentioned SABR simulation method, as described in Chapters 2 and 3.
In Chapter 5, we consider another component in our work, the efficient compu-
tation on Graphics Processing Units (GPUs). We present a parallel version of an effi-
S UMMARY ix
cient Monte Carlo-based method for pricing multi-dimensional early-exercise deriva-

tives. Parallelization and GPU computation give an extraordinary gain in terms of com-
putational cost reduction, which allows us to explore new applications of the method.
Very high-dimensional problems become tractable and generalizations regarding the
early-exercise policy are developed thanks to our GPU implementation.
The work developed in this PhD Thesis is based on journal articles, that have either
been published or been submitted during the doctoral research period.
Samenvatting
Monte-Carlomethoden worden zeer gewaardeerd en intensief gebruikt binnen de finan-

ciële wiskunde voor het waarderen van financiële derivaten en voor het beheersen van
risico’s. Monte-Carlosimulatie biedt waardevolle voordelen zoals flexibiliteit, makkelijke
interpretatie en eenvoudige implementatie. Bovendien kan de dimensie van het finan-
ciële vraagstuk worden verhoogd zonder de efficiëntie significant te verminderen. Dit
laatste kenmerk van Monte-Carlomethoden is belangrijk, omdat het een duidelijk voor-
deel is vergeleken met andere, concurrerende numerieke methoden. Voor het waarderen
van opties in meerdere dimensies (typisch meer dan vijf) zijn Monte-Carlosimulatie en
zijn varianten zelfs de enige mogelijke keuzes. In principe is Monte-Carlosimulatie ge-
baseerd op de simulatie van mogelijke scenario’s van een onderliggend proces, waarbij
vervolgens de verkregen scenariowaarden worden gecombineerd om tot de uiteindelijke
oplossing te komen. Het prijzen van derivaten met als onderliggende aandeel- en rente-
koersen, het beoordelen van risico’s en het waarderen van de portefeuille zijn represen-
tatieve financiële voorbeelden waarbij Monte-Carlosimulatie zeer goed presteert. Het
grootste nadeel dat aan Monte-Carlo wordt toegeschreven is de theoretische snelheid
van de convergentie, oftewel het vrij slechte evenwicht tussen de kosten van de bereke-
ning en de bijbehorende nauwkeurigheid. Op basis van de centrale limietstelling vereist
Monte-Carlosimulatie honderd keer meer scenario’s
³ 1´ om de fout met één orde te vermin-
deren, d.w.z. de convergentiesnelheid is O n − 2 .
Echter, de toepassing van basis Monte-Carlomethoden is vaak niet voldoende om
moderne vraagstukken op te lossen die in de financiële sector voorkomen. Problemen
zoals de nauwkeurige berekening van de gevoeligheid en al dan niet vervroegd uitoefenen
van financiële derivaten in meerdere dimensies zijn interessante voorbeelden. Alleen bij
de eenvoudigste stochastische modellen voor financiële producten is het genereren van
de scenario’s altijd triviaal. Hoewel discretisatieschema’s die op Taylorontwikkeling ge-
baseerd zijn, zoals het Euler-Maruyama-schema, veel gebruikt worden, zijn er in een
aantal gevallen ingewikkeldere technieken vereist om een bepaalde nauwkeurigheid en
efficiëntie te bereiken. Voor het berekenen van de gevoeligheid en in het geval van ver-
vroegd uitoefenen is het gebruik van basis Monte-Carlomethoden ofwel niet toereikend
of de rekentijden zijn te hoog.
Daarom kunnen hybride oplosmethoden, d.w.z. het combineren van Monte-Carlo-
methoden met een aantal andere technieken, worden ontwikkeld om deze moeilijkhe-
den te overwinnen en snelle, nauwkeurige en consistente resultaten te behalen. Dit
kan gedaan worden door te vertrouwen op analytische en/of numerieke benaderingen.
Hybride oplosmethoden zullen in de toekomst essentieel zijn om de toenemende com-
plexiteit in kwantitatieve financiële wiskunde te hanteren.
In dit proefschrift dragen we op Monte-Carlo gebaseerde hybride oplosmethoden
aan voor sommige moderne vraagstukken in financiële wiskunde. Vanuit de wiskundige
kant gebruiken we Fouriertechnieken die, gecombineerd met de Monte-Carlomethoden,
resulteren in efficiënte technieken voor het simuleren en schatten van gevoeligheden
xi
xii S AMENVATTING
onder stochastische lokale volatiliteitsmodellen, die op dit ogenblik veel gebruikt worden
voor opties in de valutamarkt (FX). We gebruiken ook enkele recente technologische ont-
wikkelingen, zoals het snel uitvoeren van complexe berekeningen op de grafische kaart,
om zo de toepasbaarheid van Monte-Carlosimulatie uit te breiden voor het berekenen
van multidimensionale opties waarbij vervroegd uitoefenen toegestaan is. Dit maakt het
numeriek oplossen van zeer hoog-dimensionale problemen mogelijk.
In Hoofdstuk 1 wordt een algemeen overzicht gegeven van methoden die vaak in de
financiële wiskunde voorkomen. Door het beschouwen van de oplossing van partiële
differentiaalvergelijkingen (PDV’s) die door de Feynman-Kac-stelling en zijn varianten
worden gegeven, zijn wiskundige technieken zoals Fourier inversie, Monte-Carlosimu-
latie en PDV-benaderingen succesvol gecombineerd om financiële vraagstukken, zoals
het prijzen van opties en risicobeheer, op te lossen. We geven een korte toelichting op
deze technieken. Andere recente ontwikkelingen, die we beschouwen in het tweede deel
van dit proefschrift, worden ook kort beschreven in dit hoofdstuk. Door data gegene-
reerde benaderingen, waarbij een prijsmodel alleen gevormd wordt door de beschikbare
financiële data, en het uitvoeren van parallelle berekeningen op een grafische kaart zijn
voorbeelden van zulke ontwikkelingen.
In Hoofdstuk 2 stellen we een efficiënte techniek voor om een representatief sto-
chastisch lokale volatiliteitsmodel exact te simuleren, namelijk het Stochastic Alpha Beta
Rho-model (SABR), dat in de FX- en rentemarkten vaak gebruikt wordt. Wij streven er-
naar om slechts één stap in een Monte-Carloprocedure te gebruiken. In het bijzonder
richten we ons op de afleiding van de voorwaardelijke verdeling van de naar de tijd ge-
ïntegreerde variantie, een element dat nodig is bij de simulatie van het onderliggende
SABR-proces. Daarvoor berekenen we de marginale verdeling door middel van Fourier-
technieken en het gebruik van een copula om een multivariate verdeling te verkrijgen die
lijkt op de voorwaardelijke verdeling van de naar de tijd geïntegreerde variantie. Doordat
de methode slechts één tijdstap nodig heeft, is het een methode die geschikt is voor het
prijzen van Europese opties. Daarnaast hebben we verbeteringen voorgesteld, waardoor
de methode ook kan worden gebruikt voor kalibratiedoeleinden.
De methode die in Hoofdstuk 2 wordt geïntroduceerd wordt in Hoofdstuk 3 gege-
neraliseerd naar een versie met meerdere tijdstappen voor de simulatie van het SABR-
model. Hoewel het gebruik van meerdere tijdstappen qua concept gelijk is, zorgt deze
versie voor een nieuwe uitdaging met betrekking tot de rekentijd. Hiervoor introduce-
ren we een geavanceerde numerieke interpolatie die gebaseerd is op stochastische col-
locatie. De voorgestelde methode, die we de mSABR-methode noemen, zorgt voor een
efficiënte simulatie met grote tijdstappen en kan worden gebruikt om nauwkeurig niet-
standaard (exotische) opties onder het SABR-model te prijzen. Zelfs onder negatieve ren-
tetarieven, welke momenteel in de financiële wereld worden waargenomen, presteert dit
simulatieschema zeer goed.
In Hoofdstuk 4 ontwikkelen we een data-gestuurde en op Fouriertechnieken geba-
seerde prijsmethode voor opties. Deze methode blijkt in een algemene setting gunstig
bij het berekenen van de gevoeligheden van financiële derivaten. De methode is geba-
seerd op de veronderstelling dat er alleen gegevens van de onderliggende waarde be-
schikbaar zijn. De resulterende methode is een op data gebaseerde generalisatie van de
COS-methode. Dit is een Fouriermethode die de karakteristieke functie van het onderlig-
S AMENVATTING xiii
gende proces gebruikt. In onze methode, die we de data-driven COS (ddCOS) methode
noemen, wordt het gebruik van de karakteristieke functie vermeden. In plaats daarvan
benaderen we de kansdichtheidsfunctie uit de beschikbare gegevens, door gebruik te
maken van concepten uit de statistische leertheorie. Aangezien de ddCOS-methode een
data-gestuurde techniek is, is het dus algemeen toepasbaar en kan het bijvoorbeeld wor-
den gebruikt in combinatie met de SABR-simulatiemethode zoals beschreven in Hoofd-
stukken 2 en 3.
In Hoofdstuk 5 beschouwen we een ander onderdeel van ons werk, efficiënte bereke-
ningen op de grafische kaart, ook wel Graphics Processing Unit (GPU) genoemd. We pre-
senteren een parallelle versie van een efficiënte en op Monte-Carlo gebaseerde methode
voor het prijzen van multidimensionale derivaten waarbij vervroegd uitoefenen toege-
staan is. De parallellisatie en het gebruik van de GPU resulteren in een uitzonderlijke
besparing in de rekentijd, waardoor we nieuwe toepassingen van de methode kunnen
onderzoeken. Zeer hoog-dimensionale problemen worden nu hanteerbaar en genera-
lisaties over het al dan niet vroegtijdig uitoefenen kunnen worden ontwikkeld dankzij
onze GPU-implementatie.
Het werk in dit proefschrift is gebaseerd op artikelen die zijn gepubliceerd of inge-
diend tijdens het promotietraject.
Contents
Summary vii
Samenvatting xi
1 Introduction 1
1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Stochastic processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2.1 The SABR model . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.3 Pricing techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.3.1 Monte Carlo methods . . . . . . . . . . . . . . . . . . . . . . . . 6
1.3.2 Fourier inversion methods . . . . . . . . . . . . . . . . . . . . . . 7
1.3.3 PDE methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.3.4 Hybrid solution techniques . . . . . . . . . . . . . . . . . . . . . 9
1.3.5 Data in computational finance. . . . . . . . . . . . . . . . . . . . 10
1.4 GPU computing. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.5 Multi-ITN-STRIKE project . . . . . . . . . . . . . . . . . . . . . . . . . 11
1.6 Outline of the thesis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2 One time-step Monte Carlo simulation of the SABR model 15
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.2 The SABR model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.2.1 SABR Monte Carlo simulation . . . . . . . . . . . . . . . . . . . . 17
2.3 CDF of SABR’s time-integrated variance . . . . . . . . . . . . . . . . . . 19
2.3.1 ChF of Y M . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.3.2 Treatment of integral I M . . . . . . . . . . . . . . . . . . . . . . . 22
2.3.3 Error and performance analysis . . . . . . . . . . . . . . . . . . . 26
2.4 Simulation of Y (T )|σ(T ): copula approach . . . . . . . . . . . . . . . . . 28
2.4.1 Families of copulas. . . . . . . . . . . . . . . . . . . . . . . . . . 29
2.4.2 Correlation: log Y (T ) and log σ(T ) . . . . . . . . . . . . . . . . . . 30
2.4.3 Sampling Y (T )|σ(T ) . . . . . . . . . . . . . . . . . . . . . . . . . 32
RT
2.4.4 Simulation of S(T ) given S(0), σ(T ) and 0 σ2 (s)ds . . . . . . . . . 33
2.5 Numerical experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
2.5.1 Copula approach analysis . . . . . . . . . . . . . . . . . . . . . . 36
2.5.2 Pricing European options by the one-step SABR method . . . . . . 38
2.6 Conclusions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
3 Multiple time-step Monte Carlo simulation of the SABR model 43
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
3.2 ‘Almost exact’ SABR simulation . . . . . . . . . . . . . . . . . . . . . . . 45
3.2.1 SABR Monte Carlo simulation . . . . . . . . . . . . . . . . . . . . 46
3.2.2 Stochastic Collocation Monte Carlo sampler . . . . . . . . . . . . . 47
xv
xvi C ONTENTS
3.3 Components of the mSABR method. . . . . . . . . . . . . . . . . . . . . 48

Rt
3.3.1 CDF of s σ2 (z)dz|σ(s) using the COS method . . . . . . . . . . . . 48
3.3.2 Efficient sampling of log Ỹ | log σ(s). . . . . . . . . . . . . . . . . . 51
3.3.3 Error analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
Rt
3.3.4 Copula-based simulation of s σ2 (z)dz|σ(t ), σ(s) . . . . . . . . . . 54
Rt
3.3.5 Simulation of S(t ) given S(s), σ(s), σ(t ) and s σ2 (z)dz . . . . . . . 57
3.4 Numerical experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
3.4.1 Convergence test. . . . . . . . . . . . . . . . . . . . . . . . . . . 59
3.4.2 Performance test . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
3.4.3 ‘Almost exact’ SABR simulation by varying ρ . . . . . . . . . . . . . 61
3.4.4 Pricing barrier options . . . . . . . . . . . . . . . . . . . . . . . . 61
3.4.5 Negative interest rates . . . . . . . . . . . . . . . . . . . . . . . . 62
3.5 Conclusions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
4 The data-driven COS method 67
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
4.2 The data-driven COS method . . . . . . . . . . . . . . . . . . . . . . . . 69
4.2.1 The COS method. . . . . . . . . . . . . . . . . . . . . . . . . . . 69
4.2.2 Statistical learning theory for density estimation. . . . . . . . . . . 70
4.2.3 Regularization and Fourier-based density estimators . . . . . . . . 71
4.2.4 The ddCOS method . . . . . . . . . . . . . . . . . . . . . . . . . 73
4.3 Choice of Parameters in ddCOS Method . . . . . . . . . . . . . . . . . . 75
4.3.1 Regularization parameter γn . . . . . . . . . . . . . . . . . . . . . 75
4.4 Applications of the ddCOS method . . . . . . . . . . . . . . . . . . . . . 80
4.4.1 Option valuation and Greeks. . . . . . . . . . . . . . . . . . . . . 80
4.4.2 The SABR model . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
4.4.3 VaR, ES and the Delta-Gamma approach. . . . . . . . . . . . . . . 84
4.5 Conclusions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
5 GPU Acceleration of the Stochastic Grid Bundling Method 91
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
5.2 Bermudan options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
5.3 Stochastic Grid Bundling Method . . . . . . . . . . . . . . . . . . . . . . 94
5.3.1 Bundling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
5.3.2 Parameterizing the option values . . . . . . . . . . . . . . . . . . 96
5.3.3 Estimating the option value . . . . . . . . . . . . . . . . . . . . . 99
5.4 Continuation value computation: new approach . . . . . . . . . . . . . . 101
5.4.1 Discretization, joint ChF and joint moments. . . . . . . . . . . . . 101
5.5 Implementation details . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
5.5.1 GPGPU: CUDA . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
5.5.2 Parallel SGBM . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
5.6 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
5.6.1 Equal-partitioning: convergence test . . . . . . . . . . . . . . . . 108
5.6.2 Bundling techniques: k-means vs. equal-partitioning . . . . . . . . 109
5.6.3 High-dimensional problems . . . . . . . . . . . . . . . . . . . . . 110
5.6.4 Experiments with more general approach . . . . . . . . . . . . . . 112
C ONTENTS xvii
5.7 Conclusions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

6 Conclusions and Outlook 115
6.1 Conclusions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
6.2 Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
References 119
Curriculum Vitæ 129
List of Publications 131
List of Attended Conferences with Presentation 133
Acknowledgement 135
List of most used abbreviations
AV Antithetic Variates
bp Basis points
CDF Cumulative Distribution Function
CEV Constant Elasticity of Variance
ChF Characteristic function
CPU Central Processing Unit
CUDA Compute Unified Device Architecture
ES Expected Shortfall
FX Foreign-eXchange
GBM Geometric Brownian Motion
GOF Goodness-of-fit
GPGPU General-Purpose computing on Graphics Processing Units
GPU Graphics Processing Unit
MCFD Monte Carlo Finite Difference
MISE Mean Integrated Squared Error
MSE Mean Squared Error
PDE Partial Differential Equation
PDF Probability Density Function
PIDE Partial Integro-Differential Equation
RE Relative Error
SABR Stochastic Alpha Beta Rho
SCMC Stochastic Collocation Monte Carlo
SCvM Smirnov-Cramér-von Mises
SDE Stochastic Differential Equation
SGBM Stochastic Grid Bundling Method
SV Stochastic Volatility
SLV Stochastic Local Volatility
VaR Value-at-Risk
xix
CHAPTER 1
Introduction
1.1. I NTRODUCTION
Risk management and valuation problems as they appear in quantitative finance can be
very challenging from both a mathematical and a computational point of view.
The underlying financial assets are typically governed by random and uncertain mo-
vements, that are mathematically modelled by means of stochastic processes and corre-
sponding Stochastic Differential Equations (SDEs). Except for the most basic and classi-
cal models, closed-form expressions of the probability densities, governing these stochas-
tic quantities are not available. For many relevant asset price processes, however, we
know the characteristic function (ChF), which is the Fourier transform of the probabil-
ity density function (PDF) of interest. By Fourier inversion we can thus approximate the
governing PDFs.
Pricing financial derivatives like options is one of the most common research areas
in computational finance. A derivative is a financial product whose valuation depends
on another underlying asset such as stocks, commodities or interest rates. In this thesis,
we focus on option valuation. Except for particular and simple cases, pricing an option
is not a trivial task, depending on the type of the option and/or the stochastic model for
the asset movements.
For any financial institution, risk management forms an essential part of their busi-
ness. Some of the risks appearing in finance include credit risk, market risk and coun-
terparty risk. Important issues in risk management are described in the Basel Accords,
whose requirements must be satisfied by the banks and other financial institutions. Once
the risk is identified, quantification is necessary. The estimation of accurate risk mea-
sures facilitates the interpretation of particular risk factors.
1.2. S TOCHASTIC PROCESSES

A basic technique to deal with stochastic processes and SDEs from a mathematical point
of view is the well-established Itô’s calculus. A central concept within the Itô’s calculus is
the Itô’s integral, which defines a stochastic integral with respect to a Brownian motion.
Based on this, the Itô’s stochastic process, in integral form, is defined as
Z t Z t
X (t ) − X (s) = a(z, X )dz + b(z, X )dW (z)
s s
with W a Brownian motion and a(·) and b(·) predictable and integrable functions.
The stochastic processes appearing in computational finance are often Itô’s pro-
cesses. The most basic and extensively used example is the geometric Brownian mo-
1
2 1. I NTRODUCTION
tion (GBM) where the functions a(·) and b(·) are defined as a(t , X ) = µX (t ) and b(t , X ) =
1 σX (t ), with µ and σ constants. GBM is used to model the underlying asset prices in the
classical Black-Scholes model, leading to log-normal asset dynamics. However, the as-
sumed simplifications (like the constant parameters among others) have often shown to
be inconsistent with the financial market behaviour.
An important indicator of the market inconsistency of GBM and the Black-Scholes
equation is found in the implied volatility smile, which means that the volatility deduced
from option quotes in the market is not constant, but appears as a function of the op-
tion strike prices and of time. A first attempt was found in the local volatility models.
The model reproduces the stochastic behaviour of the volatility observed in the market
already within the parameter calibration process. Parametric and non-parametric local
volatility processes exist, where the parametric version is governed by a functional form
with a few free parameters to be calibrated, and the non-parametric version is solely
based on option market data. One of the first parametric volatility processes was the con-
stant elasticity of variance (CEV) model [31]. The model aims to reproduce the stochastic
behaviour of the volatility observed in the market by including an elasticity parameter.
Another particularly interesting asset processes are formed by exponential Lévy pro-
cesses. Lévy processes are governed by the properties of independent and stationary
increments and continuity in probabilistic sense. A useful formula in the context of Lévy
processes is the celebrated Lévy-Khintchine formula that gives us a representation of the
process by its ChF. The ChFs of Lévy processes and of regular affine diffusion processes
are known, or can be computed, and by these, many relevant asset price processes can
be represented, like jump processes, jump-diffusion or stochastic volatility processes.
With these processes, the market observed volatility smile can be modelled.
The adoption of multiple-dimensional stochastic processes forms another success-
fully attempt in the context of modelling implied volatility smiles, especially by consider-
ing the volatility as a stochastic process. This leads to the stochastic volatility (SV) mod-
els1 or even stochastic local volatility (SLV) models. One of the representative examples
is the Stochastic Alpha Beta Rho (SABR) model in the Foreign-exchange (FX) and interest
rates markets. The SABR model will be intensively studied throughout this thesis, for
that, in the following section we briefly describe its main features.
It is important to notice that particularly these SLV models do not immediately result
in a ChF or PDF, and as such it is not trivial to perform a rapid calibration of the model
for a wide range of model and pricing problem parameters. By means of perturbation
theory, closed-form solutions for implied volatilities can be obtained in the form of an
analytic expression for a range of parameter values, but, as for many solution originating
from asymptotic theory, this expression is not general. Often in practice, contracts are
defined whose parameters lie outside the range of validity of the theory. In that case, it is
really difficult to perform a fast and accurate calibration.
1.2.1. T HE SABR MODEL

As mentioned, the SABR model belongs to the SLV model class. The model was intro-
duced by Hagan et al. [67], aiming to capture the volatility smile and manage the po-
tential risk originating from unexpected future volatility movements. The SABR model is
1 Some SV models indeed fall in the class of Lévy processes, enabling the use of the Lévy-Khintchine formula.
1.2. S TOCHASTIC PROCESSES 3
based on, besides stochasticity in the volatility, a parametric local volatility component
in terms of a model parameter, β as a power in S β . The definition of the SABR model is 1
according to the following system of SDEs,
dS(t ) = σ(t )S β (t )dWS (t ), S(0) = S 0 exp (r T ) ,

dσ(t ) = ασ(t )dWσ (t ), σ(0) = σ0 .
with S(t ) = S̄(t ) exp (r (T − t )) the forward value of the underlying asset S̄(t ), r the interest
rate, S 0 the spot price at t 0 = 0, and T the contract’s expiry time. The second stochastic
process σ(t ) represents the stochastic volatility, with σ(0) = σ0 . The two correlated Brow-
nian motion WS (t ) and Wσ (t ) are correlated with constant correlation coefficient ρ (i.e.
WS Wσ = ρt ). The model parameters are α > 0 (the volatility of the volatility), 0 ≤ β ≤ 1
(the elasticity) and ρ (the correlation coefficient).
In the original SABR model paper, the authors provided a closed-form formula for
the implied volatility under the SABR dynamics based on asymptotic expansion. After a
correction made by Obloj [100], the expression of the so-called Hagan formula reads
³ ´
S0

1 α log K
σi mp (K , S 0 , T ) = · 2 ¸ · ·
(1 − β) 2 S0 (1 − β)4 4 S 0 x(z)
µ ¶ µ ¶
1+ ln + ln +···
24 K 1920 K
σ20
" #
(1 − β)2 1 ρβασ0 2 − 3ρ 2 2
1+ + + α ·T +··· ,
24 (K S 0 )1−β 4 (K S 0 )(1−β)/2 24
with ³ ´
1−β
α S 0 − K 1−β
Ãp !
1 − 2ρz + z 2 + z − ρ
z= , x(z) = log .
σ0 (1 − β) 1−ρ
Some terms are very small and can be then neglected, resulting the following implied
volatility formula,
K K
µ µ ¶ µ ¶ ¶
1
σi mp (K , S 0 , T ) = 1 + B 1 log + B 2 ln2 + B2T ,
ω S0 S0
where the term B 1 , B 2 , B 3 and ω are given by
1¡
1 − β − ραω ,
¢
B1 = −
2
1 ¡
(1 − β)2 + 3 (1 − β) − ραω + 2 − 3ρ 2 α2 ω2 ,
¡ ¢ ¡ ¢ ¢
B2 =
12
¢2 1−β
1−β 1 βρα 1 2 − 3ρ 2 2
¡
S0
B3 = + + α , ω = .
24 ω2 4 ω 24 σ0
The Hagan formula is highly appreciated since it facilitates the model calibration,
i.e., the determination of the model parameters such that the implied volatility given by
the model to the market implied volatility.
4 1. I NTRODUCTION
1 0.3
Hagan
20
Hagan density
Monte Carlo
15
Implied Volatility
0.25
10
f (x)
5
0.2
0
0.15 -5
0.03 0.04 0.05 0.06 0.07 0.08 -0.05 0 0.05 0.1 0.15 0.2
K x
(a) Hagan’s implied volatility. (b) Implied density.
Figure 1.1: SABR parameter configuration: S 0 = 0.05, σ0 = 0.05, α = 0.4, β = 0.5, ρ = −0.7 and T = 7,
However, the use of an asymptotic approximation makes the Hagan formula for the
implied volatility not generally applicable. SABR model parameter configurations exist
for which the formula gives inaccurate or even incorrect values. It is well-known that
Hagan formula gives rise to problems when the asset quotes are close to zero, where a
so-called absorption boundary value must be set. This issue can be seen by constructing
the so-called implied density, i.e. the PDF associated to the implied volatility given by the
Hagan formula. In Figure 1.1, an example of such an incorrect density is shown. In Figure
1.1(a) the implied volatilities given the Hagan formula and also numerically computed
values by means of a Monte Carlo method are depicted for different strike prices K . We
can see that the Hagan formula is not accurate for very low strikes. The problem become
even more clear when representing the implied density, as in Figure 1.1(b), where the
PDF exhibits negative values close to the boundary at zero. This cannot be, and implies
arbitrage opportunities.
The valuation of European and exotic options under the SABR dynamics is generally
not a trivial task. When reliable, the implied volatility given by the Hagan formula can
be used in the Black formula2 to obtain European option prices. However, a problem
remains in the situations where the Hagan formula is not applicable. Then, Monte Carlo
methods are sometimes employed even though an "exact Monte Carlo simulation" of
the SABR model, which allows for simulation with only a few large time-steps, is not
straightforward. The use of Fourier inversion techniques is limited since the ChF is not
available. A PDE formulation for the option price under SABR dynamics has also been
developed, enabling the use of PDE numerical techniques. The two-dimensional SABR
PDE reads
1 2 2β 2−2β ∂2 V 2
β 1−β 2 ∂ V 1 2 2 ∂2 V ∂V ∂V
σ S̄ D + ραS̄ D σ + α σ + + r S̄ − r V = 0,
2 ∂S̄ 2 ∂S̄∂σ 2 ∂σ2 ∂t ∂S̄
where V (t , S̄, σ) is the option price and D(t , T ) := exp(−r (T − t )).

2 The Black-Scholes variant for underlying forwards.
1.3. P RICING TECHNIQUES 5
SLV models in general and the SABR model in particular are very well-suited to model
so-called forward starting options. Forward starting options can be seen as European op- 1
tions, but starting at a predefined time in the future (at t > t 0 ). In that case, the initial
asset value is not known, but still the contract premium and hedge strategy must be set
advance (at today’s date). The SLV models are accurate for modelling the so-called for-
ward volatility, which helps to determine accurate option prices and hedge strategies for
forward starting options. In contrast, plain local volatility models are typically not able to
accurately model the forward volatility, as volatility is flattening out in time under these
models, and, thus, the volatility smile "flattens".
1.3. P RICING TECHNIQUES

Given a SDE governing the asset movements, the financial derivative price can be rep-
resented by the solution of a partial differential equation (PDE) via Itô’s lemma. By the
payoff function, a final condition can be prescribed. Due to the celebrated Feynman-Kac
theorem (and its generalizations towards non-linear equations), the solution of many
PDEs appearing in derivative valuation can be written in terms of a probabilistic repre-
sentation by means of an expectation. The associated risk neutral asset price PDF plays
an important role as it forms the basis of the computation of the expectation. In The-
orem 1.3.1, the Feynman-Kac version for geometric Brownian motion and the Black-
Scholes PDE is given.
Theorem 1.3.1 (Feynman-Kac). Given the money-savings account, modelled by dB (t ) =

r B (t )dt , with constant interest rate, r , let V (t , S) be a sufficiently differentiable function
of time t and stock price S(t ). Suppose that V (t , S) satisfies the following PDE, with drift
term, µ(t , S), and volatility term, σ(t , S):
∂V ∂V 1 2 ∂2 V
+ µ(t , S) + σ (t , S) 2 − r V = 0, (1.1)
∂t ∂S 2 ∂S
with final condition h(T, S). The solution for V (t , S) at time t 0 < T is then given by:
V (t 0 , S) = EQ e−r (T −t0 ) h(T, S)|F (t 0 ) ,

£ ¤
where the expectation is taken under measure Q, with respect to a process S, defined by:
dS(t ) = µQ (t , S)dt + σ(t , S)dW Q (t ), for t > t 0 .
The expectation in Theorem 1.3.1 can be written in integral form, resulting in the
risk-neutral valuation formula,
Z
V (t 0 , S) = e−r (T −t0 ) h(T, y) f (y|F (t 0 ))dy, (1.2)
R
with r the risk-free rate, T the maturity time, and f (·) the PDF of the underlying process.
Many numerical techniques are based on the solution provided by the Feynman-Kac
theorem and, particularly, by the risk-neutral valuation formula.
6 1. I NTRODUCTION
The Feynman-Kac formula can be generalized to more involved types of problems.

1 Suppose, for example, a one-dimensional semi-linear parabolic PDE of the form
∂V ∂V 1 2 ∂2 V ∂V
µ ¶
+ µ(t , S) + σ (t , S) 2 − g t , S,V, σ, = 0,
∂t ∂S 2 ∂S ∂S (1.3)
V (T, S) = h(T, S),
where g can be a function of V and its first derivative. This PDE also can also be ex-
pressed in a probabilistic representation by means of the following forward and back-
ward SDEs,
dS(t ) = µ(t , S)dt + σ(t , S)dW (t ), S(0) = S 0
−dY (t ) = g (t , S, Y , Z )dt + Z dW (t ), Y (T ) = g (S(T )),
with a final condition for the backward SDE. The solution consist of a pair of processes
(Y , Z ) as
∂V
Y (t ) = V (t , S), Z (t ) = σ(t , S) (t , S).
∂S
Many generalizations, to towards multi-dimensional linear PDEs, partial integro-
differential equations (PIDEs), and also towards fully non-linear PDEs have been given
in the literature.
1.3.1. M ONTE C ARLO METHODS

As the integral in Equation 1.2 is typically not solvable in analytic form, numerical-based
approaches can be developed. Monte Carlo methods are well-known numerical tech-
niques to evaluate integrals. They are based on the analogy between probability and
volume. Suppose we need to compute an integral
Z
I := f (x)dx,
C
and we have a technique to draw n independent and identically distributed samples in

C , X 1 , X 2 , . . . , X n . We then define a Monte Carlo estimator as
1 X n
I¯n := f (X j ).
n j =1
If f is integrable over C then, by the strong law of the large numbers,
I¯n → I as n → ∞,
with probability one.

Furthermore, if f is square integrable, we can define
sZ
¡ ¢2
s f := f (x) − I dx,
C
the error of the Monte Carlo estimate I − I¯ is, by means of the central limit theorem, which
p
is assumed normally distributed with mean 0 and standard deviation s f / n.
p
The order of convergence of the plain Monte Carlo method is O (1/ n), which is con-
sidered slow for many applications. 1
However, Monte Carlo methods are highly appreciated in computational finance due
to their simplicity, flexibility and immediate implementation. One of the most impor-
tant advantages is that the methods are easily extended to multi-dimensional problems,
while keeping the order of convergence. Moreover, the method’s simplicity allows to ex-
plore different convergence improvement techniques, like variance reduction techniques
[59] or multi-level Monte Carlo acceleration [57].
In the context of the Monte Carlo approximation of an expectation, like in the so-
lution of the PDE given by the Feynman-Kac theorem, an essential part of the method
is to generate sample paths. Random number generators (RNG) form the basis of these
Monte Carlo paths, and they have been studied for many years. Broadly, they can be
subdivided into “true”, pseudo- and quasi-random generators, and usually generate uni-
formly distributed samples. This is key, because when uniform samples between 0 and
1 are available, samples from any distribution can be obtained as long as the quantile
function, i.e., the inverse of the cumulative distribution function (CDF), is known. The
procedure is then as follows,
d
F Z (Z ) = U thus Zi = F Z−1 (Ui ),
d
where F Z is the CDF, = means equality in the distributional sense, U ∼ U ([0, 1]) and Ui
is a sample from U ([0, 1]). The computational efficiency highly depends on the cost of
calculating F Z−1 . We will require this distribution inversion approach in the Chapters 2
and 3 to simulate the SABR model by means of exact Monte Carlo simulation.
When the “exact” sample generation is not possible, either because the distribution
is not available or the inversion is computationally unaffordable, intermediate steps in
the numerical scenario simulation can be introduced by means of a time discretization
of the associated SDE. The Taylor-based discretization schemes are widely employed in
quantitative finance. The most representative example is the Euler-Maruyama method,
the generalization of the Euler method to SDEs available in the context of the Itô’s calcu-
lus. Another well-known Taylor-based simulation scheme is the Milstein method, which
presents a superior order of convergence to the Euler-Maruyama method.
The application of Monte Carlo methods to European-style options is rather straight-
forward and well-established. However, the use of Monte Carlo methods for other fi-
nancial contracts is usually non-trivial. Some representative examples are the efficient
Monte Carlo computation of the option sensitivities, the Greeks, [56] and the valuation
of early-exercise option contracts [94].
1.3.2. F OURIER INVERSION METHODS

Within the context of the risk-neutral valuation formula, Fourier techniques can be ap-
plied, when the integral is solved in Fourier space. Fourier techniques rely on the avail-
ability of the ChF, i.e., the Fourier transform of the PDF associated to the asset stochastic
8 1. I NTRODUCTION
process. They form the so-called Fourier pair, as

1 Z
fˆ(u) = exp(i xu) f (x)dx,
R
1
Z
f (x) = exp(−i xu) fˆ(u)du,
2π R
where fˆ denotes a Fourier transformed function, i.e., here, the ChF.

Contrary to the PDF, the ChF can be derived for many models in finance, particu-
larly those that are driven by Lévy processes. Therefore, the PDF can be recovered from
the ChF via Fourier inversion. Thus, the PDF can be directly employed within the risk-
neutral valuation formula in Equation (1.2). However, more efficient Fourier techniques
are also available. The starting point is the representation of the unknown PDF in the
form of an orthonormal series expansion on a finite interval [a, b], z ∈ [a, b], as
Ã !
1 ∞
A k · ψk (z) ,
X
f (z) = 1+2
b−a k=1
where ψk (·) are orthonormal functions and the expansion coefficients A k can be ob-
tained by
Z b
A k :=< f , ψk >= f (z)ψk (z)dz.
a
By using Parseval’s identity, i.e., < f , ψk >=< fˆ, ψ̂k >, where fˆ and ψ̂k denote the
Fourier transformations of f and ψk , respectively, the expansion coefficients can be
computed based on the ChF.
Fourier inversion methods usually provide a high accuracy at limited computational
cost. However, their applicability strongly relies on the availability of the ChF. In some
cases when a ChF is not directly available, the ChF can be approximated, but this may
hamper the efficiency of the methodology. The curse of dimensionality in the case of
multi-dimensional problems will also be problematic when Fourier techniques are em-
ployed.
Several pricing techniques have been developed based on a Fourier approach, like
the Carr and Madan method [19], Boyarchenko and Levendorskii [13], Lewis [92], Fang
and Oosterlee [49] and Ortiz-Gracia and Oosterlee [102].
In this thesis, we mainly employ the COS method [49], which is based on a Fourier
cosine series expansion of the PDF and the efficient computation of the expansion co-
efficients. An important advantage of this method is that closed-form solutions can be
derived for many types of option contracts.
1.3.3. PDE METHODS

The financial derivatives valuation problem can be written in PDE form, as in Equation
1.1 for the case of the Black-Scholes model, or as in Equation 1.3. The main advantage of
the PDE-based techniques is that the option Greeks can be obtained without substantial
additional computational effort, and also exotic options can be cast relatively easily in
the PDE framework. In general, the PDEs appearing in finance cannot be solved analyt-
ically and numerical methods should be applied. The computational cost and also the
reduced applicability in higher problem dimensions due to the curse of dimensionality

are drawbacks to methodologies relying on the PDE approach. 1
Obtaining the PDE formulation of a general option pricing problem can be chal-
lenging depending on the assumed underlying model and the type of option itself. An-
other important application of PDEs is given by the so-called Fokker-Planck equation,
also called forward Kolmogorov equation, whose solution is the PDF (and its evolution
in time) associated to a particular SDE. When involved underlying processes (including
jumps, for example) are employed, a generalized version of the Fokker-Planck equation
in the form of a PIDE is also available.
Several extensions have been derived for the Black-Scholes equation, where some of
the basic model assumptions have been relaxed to introduce, for example, transaction
costs or the market’s illiquidity. The latter extensions can be included by adding some
non-linear terms in the PDE. Other well-known example of non-linear PDEs appearing
in finance is given by the Hamilton-Jacobi-Bellman equation, which models, for exam-
ple, optimal asset allocation of portfolio investment problems under constraints. Some
recent representative works are [96] or [80].
Regarding numerical techniques for PDEs, the finite difference method is the most
commonly employed approach. Several finite difference discretization schemes in both
time and space have been studied and developed. In [66] the authors employed al-
ternating direction implicit schemes for a hybrid asset price model with applications
for European and also exotic options. In [125] a new PDE numerical scheme based on
Crank-Nicolson and finite elements for interest rate derivatives was presented. To deal
with non-linearity, Newton-like methods or penalty methods are state-of-the-art itera-
tion techniques.
A summary of the numerical schemes commonly used in finance is provided in [44,
119, 133]. And there are very many other relevant works in the context of the efficient
numerical solution of the PDEs appearing in finance.
1.3.4. H YBRID SOLUTION TECHNIQUES

In practical valuation and risk management applications, we need to deal with rather
involved problems where computing the solution in an efficient way is a complex task.
The techniques explained so far all have their pros and cons, depending on the problems
at hand. Our goal in this thesis is to develop methods that keep the advantages and
circumvent the drawbacks of the individual approaches (Fourier, Monte Carlo and PDE)
for some specific valuation problems, leading to robust, fast and accurate solutions to
these problems. To achieve this, a combination of different numerical approaches may
be considered.
In the literature we can find many examples of hybrid solution methods, combin-
ing two of the techniques presented in the previous sections or employing advanced
methods from other mathematical disciplines. Mentioning a few examples: Gobet et al.
explore the use of Least Squares Monte Carlo methods on PDEs and on backward SDEs
[62]. In [63] Grzelak and Oosterlee proposed a projection method to approximate a ChF
of hybrid asset models. Dang et al. presented in [40] a dimension reduction Monte Carlo
method, combining PDE methods with Monte Carlo simulation.
The techniques and implementations in this thesis are all Monte Carlo-based meth-
10 1. I NTRODUCTION
ods. However, as we are dealing with involved problems, the application of the basic
1 version of the Monte Carlo method is not sufficient to provide a satisfactory balance
between accuracy and efficiency. We therefore combine the Monte Carlo method com-
ponents with other techniques like with copulas, stochastic collocation interpolation,
Fourier inversion, dynamic programming recursion or regression techniques. An inter-
esting aspect of Monte Carlo methods is the independency of the different computa-
tions, which makes the method highly parallelizable which facilitates the use of high-
performance computing. In this thesis we have considered GPU computing as well.
1.3.5. D ATA IN COMPUTATIONAL FINANCE

The amount of data generated in the financial and insurance sectors have been drasti-
cally increasing during the last decades. How to take advantage of big financial data is a
great challenge, due to the "three V’s", volume, variety and velocity.
A common application of financial data is the analysis of the retail business of banks
and insurance companies. Some examples are client knowledge, fraud and anomaly de-
tection, mortgage risk prediction or customer evaluation. As the data available is typi-
cally heterogeneous and unstructured, machine learning and the tools in classification,
clustering and regression may be employed highly successfully.
Other interesting applications of big financial data are found in algorithmic trading,
where investment decisions are carried out by algorithms and thus computers, based on
the combination of mathematical models and big data analytics. This leads to a partly or
fully automated process, where the profit and velocity are maximized and the human in-
fluence (manual errors, emotion, bias, etc.) is minimized. Nowadays, the most popular
example is found in high frequency trading.
When the data available is structured, mathematical techniques can be explored, like
methods for market prediction or risk assessment. However, data can be also interpreted
as just samples from an unknown distribution. When the data available is a set of asset
values, an approach would be to approximate either the PDF or the CDF solely based on
the data, which can then be used within Equation (1.2).
In this thesis, we consider data-based density estimation for the risk neutral asset dy-
namics. By exploring the connection of the density estimation problem and cosine series
expansions, we will develop a data-driven technique that generalizes the COS method.
1.4. GPU COMPUTING

Next to the precision in the financial results, it is also of importance, in many situations,
that solutions are obtained in a very short time. Often the calculations appearing in fi-
nance can be further accelerated by means of advanced scientific computing tools. In
last decades, high performance computing in general and GPU computing in particular
have gained popularity in the quantitative finance context. Several large financial insti-
tutions have GPUs at their disposal.
The GPU computing, or more specifically, GPGPU (General-Purpose computing on
Graphics Processing Units) is a relatively novel approach in the parallel computing. The
GPU is used as a co-processor where computationally heavy tasks can be executed with
high efficiency. Particular computing architectures with hundreds (and even thousands
1.5. M ULTI -ITN-STRIKE PROJECT 11
in the latest generations) of small cores and a multi-level memory hierarchy give an ex-
traordinary performance on embarrassingly parallel computations. 1
Different technologies and platforms have appeared in the framework of GPU com-
puting. The Compute Unified Device Architecture (CUDA) [35] from NVIDIA is probably
the best known and extended technology. Its main features are the GPU drivers, the
programming languages (C, C++ and Fortran) and a complete and mature ecosystem in-
cluding scientific libraries (cuBLAS, cuSOLVER, cuRAND, etc.), editors (Nsight) and de-
bugging and analysis tools (CUDA-DBG, CUDA-MEMCHECK, Visual profiler, Tau). Fur-
thermore, extensions of CUDA to use it in combination with other languages (Python,
Java, .NET, etc.) and scientific platforms (Matlab, Mathematica) have been developed.
OpenCL represents the non-proprietary alternative to CUDA, in which NVIDIA is also
involved. OpenACC is a directive-based implementation tool that allows non program-
ming experts to achieve significant acceleration of their scientific research with reduced
programming effort.
GPU computing appears appropriate for financial calculations since these are typi-
cally compute-bound and not memory-bound. GPUs have an extraordinary computation
power, but a limited amount of memory with respect to other high-performance sys-
tems (multi-core, multi-node, etc.). A particularly interesting application is formed by
the Monte Carlo method. Due to characteristics as the availability of a huge number of
independent scenarios, the Monte Carlo estimator fits perfectly in the GPU architecture
philosophy. However, when hybrid solution methods are considered, like in this thesis,
the corresponding parallelization, and therefore their implementation on GPUs may not
be a trivial task. Some techniques are intrinsically sequential and must be adapted or re-
formulated to take advantage of the GPU architecture. In certain financial applications,
the amount of memory is the limiting factor and an efficient management of the mem-
ory resources becomes essential to obtain a gain in terms of performance. The use of the
highly optimized libraries available on GPUs is strongly recommended when possible.
During the period of the thesis work, we have had access to the Dutch national su-
percomputer Cartesius [20]. In terms of GPU computing the facilities form one of the
most powerful architectures in the world. Thanks to that, we were able to take our im-
plementations to the limit, because of the memory resources and the speed provided by
the latest and most powerful generations of GPUs installed in the Cartesius system. For
prototyping purposes, we also employed the Little Green Machine, an low-consuming
and environmental-friendly system equipped with fast and compact GPUs.
1.5. M ULTI -ITN-STRIKE PROJECT

The research described in this thesis has been partially carried out in the framework of
the EU-funded Multi-ITN-STRIKE Marie Curie project Novel Methods in Computational
Finance. The consortium was formed by several universities from around Europe: B.U.
Wuppertal, C.U. Bratislava, U.P. Valencia, Rousse, I.S.E.G. Lisboa, U.A. Zittau, T.U. Wien,
T.U. Delft, Greenwich, Würzburg and Antwerp as a participants and several other uni-
versities, research institutions and companies as project associate collaborators.
The goal of this international training network was to deepen the understanding of
complex and advanced financial models and to develop effective and robust numeri-
cal schemes for linear and non-linear problems arising from the mathematical theory of
12 1. I NTRODUCTION
pricing financial derivatives and related financial products. This has been accomplished
1 by means of financial modelling, mathematical analysis and numerical simulation, op-
timal control techniques and validation of models. Twelve early-stage researchers (PhD
students) and five experienced researchers (post-doctoral students) have successfully fol-
lowed the high level education provided within the project, building up strong collabo-
rations with the network partners.
In the problems studied, the challenge has been expressed by the non-linearity in
the financial problem formulation. On the one hand, non-linear generalizations of the
Black-Scholes equation (Theorem 1.3.1) have been introduced by considering non-linear
payoffs and market frictions. On the other hand, more complicated asset models have
been studied in ITN-STRIKE, like the already mentioned SV or SLV models.
The modelling research in the ITN-STRIKE project included transaction costs into
the Black-Scholes model, Lévy models [16], Lie Group approaches [11], optimal control
and Fokker-Planck equations [54]. The numerical methods investigated in ITN-STRIKE
included finite difference schemes for PDEs [28, 46, 71, 73] and also hybrid Monte Carlo
methods [88, 89].
Besides the research performed in the project, many scientific events have taken
place, providing a complete educational training program which ranged from advanced
mathematical concepts (Newton methods, optimal control, Lie Group methods, etc.)
to transferable skills (project management, effective writing, cultural awareness, plagia-
rism) and to computational skills (high-performance computing, GPU programming).
Furthermore, the project has provided many networking opportunities in both academic
(ECMI 2014, SCF2015, ICCF 2015, ALGORITMY 2016) and industrial (Postbank, MathFi-
nance) environments.
1.6. O UTLINE OF THE THESIS

In this thesis, we develop hybrid solution methods for involved problems in option pric-
ing and risk management.
In Chapter 2, a one time-step Monte Carlo method for the exact simulation of the
SABR model is introduced. It relies on an accurate approximation of the conditional
distribution of the time-integrated variance, for which Fourier inversion and copulas are
combined. Resulting is a fast simulation algorithm for the SABR model. As a one time-
step method, it is suitable for European options which can be employed for calibration
purposes. We numerically show that our technique overcomes some known issues of the
SABR approximated asymptotic formula.
We extend the method presented in Chapter 2 to the multiple time-step case in Chap-
ter 3. To reduce the method’s complexity, new components need to be added, like an ap-
proximation of the sum of log-normals, stochastic collocation interpolation and a corre-
lation approximation. The resulting method, which we call the mSABR method, is highly
efficient and competitive, as it requires only a few time-steps to achieve high accuracy.
The method is particularly interesting for long-term and exotic options. Furthermore,
the current market situation of negative interest rates is also successfully addressed by
the mSABR scheme.
In Chapter 4, a data-driven generalization of the well-known COS method is pre-
sented, which we call the ddCOS method. The ddCOS method relies on the assumption
1.6. O UTLINE OF THE THESIS 13
that only asset data samples are available and the ChF is not available and no longer
required. Whereas the order of convergence provided by the method is according the 1
convergence of Monte Carlo methods, the method is beneficial for the accurate and ef-
ficient computation of some option sensitivities. Therefore, the method appears suit-
able for risk management. We use the ddCOS method for the Greeks computation with
the SABR simulation scheme presented in Chapters 2 and 3 in the context of the Delta-
Gamma approach.
In Chapter 5, we develop a parallel version of the Stochastic Grid Bundling Method
(SGBM), a method to price multi-dimensional early-exercise option contracts. The par-
allel implementation follows the GPGPU paradigm. A new grid point bundling scheme
is proposed, which is highly suitable for parallel systems, increasing the efficiency in the
use of memory and workload distribution. This parallel version of SGBM generalizes its
applicability, allowing us to solve very high-dimensional problems (up to fifty dimen-
sions). Thanks to the performance of the parallel SGBM, a generally applicable way of
computing the early-exercise policy is proposed.
In Chapter 6, the conclusions of the presented work are summarized. We also include
an outlook for possible future research lines.
CHAPTER 2
One time-step Monte Carlo simulation of the SABR model
In this chapter, we propose a one time-step Monte Carlo method for the SABR model. We
base our approach on an accurate approximation of the CDF for the time-integrated vari-
ance (conditional on the SABR volatility), using Fourier techniques and a copula. Result-
ing is a fast simulation algorithm which can be employed to price European options under
the SABR dynamics. Our approach can thus be seen as an alternative to Hagan analytic
formula for short maturities that may be employed for model calibration purposes.
2.1. I NTRODUCTION
The SABR model [67] is an established SDE system which is often used for interest rates
and FX modelling in practice. The model belongs to the SLV models. The idea behind
SLV models is that the modelling of volatility is partly done by a local volatility and partly
by a stochastic volatility contribution, aiming to preserve the advantages and minimize
the disadvantages of the individual models.
In the original paper [67], the authors have provided a closed-form approximation
formula for the implied volatility under SABR dynamics. This is important for practi-
tioners, as it can be used highly efficiently within the calibration exercise. However, the
closed-form expression is derived by perturbation theory and its applicability is thus not
general. The formula is for example not always accurate for small strike values or for
long time to maturity options. In [100], Oblój provided a correction to the Hagan for-
mula but the same drawbacks remained. In order to improve the approximation for the
SABR implied volatility, different techniques have been proposed in the literature, based
on either perturbation theory [69, 136], heat kernel expansions [72, 104] or model map
techniques [4].
The objective of a bank is typically to employ the SABR model for pricing exotic
derivatives. Before this pricing can take place, the model needs to be calibrated to Euro-
pean options, and, if possible, to some exotic options as well. It is market practice to use
Hagan formula to calibrate the SABR model, however, since it is only an approximation
it will not always result in a sufficient fit to the market. In other words, a SABR Monte
Carlo simulation may give different implied volatilities than the Hagan formula.
In this chapter, we aim to develop a one time-step Monte Carlo method, which is ac-
curate for European derivative contracts up to two years. This kind of contract is often
traded in FX markets. In particular, we focus on the efficient simulation of SABR’s time-
integrated variance, conditional on the volatility dynamics. An analytical expression of
This chapter is based on the article “On a one time-step Monte Carlo simulation approach of the SABR model:
Application to European options”. Published in Applied Mathematics and Computation, 293:461–479, 2017
[88].
15
16 2. O NE TIME - STEP M ONTE C ARLO SIMULATION OF THE SABR MODEL
the conditional distribution of the time-integrated variance is available in differential

form [107]. However, this solution is not tractable in our context, due to the instabil-
ities (related to the Hartman-Watson distribution) appearing when numerical integra-
tion techniques are applied [74]. Furthermore, the use of numerical methods also im-
plies that sampling the distribution becomes computationally unaffordable. Here, we
2 propose to approximate the distribution by the two marginal distributions involved, i.e.
the volatility and time-integrated variance distributions, by using a copula. The copula
methodology has been used intensively in the financial world, see, for example, [26], in
risk management and option pricing. In the field of risk management, some relevant
works are Li [93], Embrechts et al [47], Cherubini and Luciano [24] or Rosenberg and
Schuermann [110]. In the case of pricing multi-asset options, some examples of the use
of copulas are Cherubini and Luciano [25] or Rosenberg [109].
The approximation of the CDF of the time-integrated variance is obtained by ap-
plying a recursive procedure, as described in [137], which was originally developed to
price arithmetic Asian options. The derivation is based on the definition of the ChF of
the time-integrated variance and the use of Fourier inversion techniques. We adapt the
procedure to our problem and improve it in terms of computational costs. Once both
marginal distributions are available, we employ the copula technique to get a multivari-
ate distribution which approximates the conditional distribution of the time-integrated
variance given the volatility. Several copulas are considered and an analysis of their per-
formance is carried out. The conditional time-integrated variance approximation will
be employed in the asset simulation for the complete SABR model.
The chapter is organized as follows. In Section 2.2, the SABR model and the sim-
ulation steps are briefly introduced. Section 2.3 describes the procedure to derive the
CDF of SABR’s time-integrated variance. The copula approach to approximate the con-
ditional time-integrated variance distribution is explained in Section 2.4. In Section 2.5,
some numerical experiments are presented. We conclude in Section 2.6.
The one-step methodology restricts the use of the proposed method to option matu-
rities up to two years and European-type options. In the next chapter, we will generalize
the methodology to the multiple time-step case.
2.2. T HE SABR MODEL

The SABR model is based on a parametric local volatility component in terms of a model
parameter, β. The formal definition of the SABR model reads

(2.1)
dσ(t ) = ασ(t )dWσ (t ), σ(0) = σ0 ,
where S(t ) = S̄(t ) exp (r (T − t )) denotes the forward value of the underlying asset S̄(t ),
with r the interest rate, S 0 the spot price and T the contract’s final time. Quantity σ(t ) de-
notes the stochastic volatility, with σ(0) = σ0 , WS (t ) and Wσ (t ) are two correlated Brow-
nian motions with constant correlation coefficient ρ (i.e. WS Wσ = ρt ). The open param-
eters of the model are α > 0 (the volatility of the volatility), 0 ≤ β ≤ 1 (the elasticity) and
ρ (the correlation coefficient).
A useful aspect for practitioners is that each model parameter can be assigned to a
2.2. T HE SABR MODEL 17
specific feature of the market observed implied volatility when it is represented against
the strike prices, commonly known as a volatility smile or skew. The α parameter mainly
affects the smile curvature. Exponent β is connected to the forward process distribution,
being normal for β = 0 and log-normal for β = 1. In terms of the volatility smile, β also
affects the curvature. Correlation parameter ρ rotates the implied volatility curve around
the at-the-money point, obtaining “more pronounced smile” or “more pronounced skew” 2
shapes.
2.2.1. SABR M ONTE C ARLO SIMULATION

In the first equation of the SABR model (2.1), the forward asset dynamics are defined
as a CEV process [31]. Based on the works of Schroder [117] and Islah [75], an analytic
expression for the CDF of the SABR conditional process has been obtained. For some
S(0) > 0, the conditional CDF of S(t ) with an absorbing boundaryR at S(t ) = 0, and given
t
the volatility, σ(t ), and the conditional time-integrated variance, 0 σ2 (s)ds|σ(t ), reads
µ Z t ¶
Pr S(t ) ≤ K |S(0) > 0, σ(t ), σ2 (s)ds = 1 − χ2 (a; b, c), (2.2)
0
where Ã !2
1 S(0)1−β ρ K 2(1−β)
a= + (σ(t ) − σ(0)) , c= ,
ν(t ) (1 − β) α (1 − β)2 ν(t )
1 − 2β − ρ 2 (1 − β)
Z t
b = 2− , ν(t ) = (1 − ρ 2 ) σ2 (s)ds,
(1 − β)(1 − ρ 2 ) 0
2
and χ (x; δ, λ) is the non-central chi-square CDF.
It should be noted that this formula is exact for the case ρ = 0 and results in an ap-
proximation otherwise. A shifted process with an approximated initial condition is em-
ployed in the derivation for ρ 6= 0. Based on Equation (2.2), an “exact” Monte Carlo sim-
ulation scheme for the SABR model can be defined based on inverting the conditional
SABR CDF, when the conditional time-integrated variance and the volatility are already
simulated. This forms the basis of the one time-step SABR approach, where exact sim-
ulation is required for accuracy reasons when using only one large time-step (meaning
no time discretization). Furthermore, it is well known that the convergence ratio of the
p
Monte Carlo method is 1/ n, with n the number of paths or scenarios.
According to Equation (2.2), in order to apply an “exact” one time-step Monte Carlo
simulation for the SABR dynamics, we need to perform the following steps (with the
terminal time T ):
• Simulation of SABR’s volatility. From Equation (2.1), we observe that the volatility
process of the SABR model is governed by a log-normal distribution. As is well-
known, the solution follows a GBM, i.e. the exact simulation of the volatility at
terminal time, σ(T ), reads
1
σ(T ) ∼ σ(0) exp(αW̃σ (T ) − α2 T ), (2.3)
2
where W̃σ (T ) is an independent Brownian motion.

• Simulation of SABR’s R Ttime-integrated variance, conditional on the terminal value

of the volatility, i.e., 0 σ2 (s)ds|σ(T ). Exact simulation of the time-integrated vari-
ance in the SABR model is not practically possible since the expression of the con-
ditional distribution given the volatility cannot be efficiently treated. One pos-
sibility is to simulate it by a Monte Carlo method, but this implies the use of a
2 nested simulation which can be very expensive or even unaffordable. Another
possibility is to find an approximation for the conditional
RT distribution of SABR’s
time-integrated variance given σ(T ). The integral 0 σ2 (s)ds can also be approxi-
mated by some quadrature rule once σ(0) and σ(T ) (or intermediate values of the
volatility process) are available.
• Simulation of SABR’s forward asset process. We can simulate the forward dynam-
ics by inverting the CDF in Equation (2.2). For this, the conditional SABR’s time-
integrated variance (besides the volatility) is required. Concerning the forward
asset simulation, there is no analytic expression for the inverse SABR distribu-
tion so the inversion has to be calculated by means of some numerical approxi-
mation. Another possibility is to employ SDE discretization schemes, like Taylor-
based schemes (Euler-Maruyama, log-Euler, Milstein or full-truncation) or more
involved ones (low-bias or QE schemes). However, the use of such schemes gives
rise to several issues that have to be managed, like the discretization error, nega-
tivity in the asset values and computational cost.
In Algorithm 1, a schematic representation of the complete one time-step Monte
Carlo simulation procedure for the SABR model is presented. We include references to
the sections of this chapter where the respective parts of the SABR simulation are dis-
cussed.
Algorithm 1: SABR’s one time-step Monte Carlo simulation.

Data: S 0 , σ0 , α, β, ρ, T, n.
for Path p = 1 . . . n do
p
Simulation of W̃σ (t ) ∼ N (0, T ).
Simulation of σ(T )|σ(0) (see Equation (2.3)).
RT
Simulation of 0 σ2 (s)ds|σ(T ) (see Subsection 2.4.3).
RT
Simulation of S(T )|S(0), σ(T ), 0 σ2 (s)ds (see Subsection 2.4.4).
In the above steps of the SABR model exactR T simulation, the challenging part is the
simulation of the time-integrated variance, 0 σ2 (s)ds, conditional on σ(T ) (and σ(0)).
Sometimes moment-matching techniques can be used. Here, however, we propose a
computationally efficient approximation, based
R T on a copula multi-variate distribution,
to simulate the conditional distribution of 0 σ2 (s)ds given the volatility σ(T ). Simula-
tion by means of a copula technique requires the CDF of the involved marginal
RT distribu-
tions. Although the distribution of σ(T ) is known, the distribution of 0 σ2 (s)ds needs
to be approximated. We propose a method based on a Fourier technique, which was al-
ready employed for Asian options in [137]. In Section 2.3, this derivation is presented in
detail.
2.3. CDF OF SABR’ S TIME - INTEGRATED VARIANCE 19
RT
Hereafter, for notational convenience, we will use Y (T ) := 0 σ2 (s)ds.
2.3. CDF OF SABR’ S TIME - INTEGRATED VARIANCE

In this section, we present a procedure to approximate the CDF of the time-integrated
variance, Y (T ). Since we will work in the log-space, an approximation of the CDF of 2
log Y (T ), F log Y (T ) , will be derived. First of all, we approximate Y (T ) by its discrete analog,
i.e. Z T M
σ2 (s)ds ≈ σ2 (t j )∆t =: Ỹ (T ),
X
(2.4)
0 j =1
T
where M is the number of discrete time points1 , i.e., t j = j ∆t , j = 1, . . . , M and ∆t = M .
Ỹ (T ) is subsequently transformed to the logarithmic domain, where we aim to find an
approximation of F log Ỹ (T ) , i.e.
Z x
F log Ỹ (T ) (x) = f log Ỹ (T ) (y)dy, (2.5)
−∞
with f log Ỹ (T ) the PDF of log Ỹ (T ). f log Ỹ (T ) is, in turn, found by approximating the associ-
ated ChF, fˆ , and applying a Fourier inversion procedure. The ChF and the desired
log Ỹ (T )
PDF of log Ỹ (T ) form a so-called Fourier pair. Based on the work in [8] and [137], we
develop a recursive procedure to recover the ChF of f log Ỹ (T ) . We start by defining the
sequence,
σ2 (t j )
Ã !
= log σ2 (t j ) − log σ2 (t j −1 ) ,
¡ ¢ ¡ ¢
R j = log 2 (2.6)
σ (t j −1 )
where R j is the logarithmic increment of σ2 (t ) between t j and t j −1 , j = 1, . . . , M . As

mentioned, in the SABR model context the volatility process follows log-normal dynam-
ics. Since increments of Brownian motion are independent and identically distributed,
d
the R j are also independent and identically distributed, i.e. R j = R. In addition, the ChF
of R j , as a normal increment, is well known and reads, ∀u, j ,
fˆR j (u) = fˆR (u) = exp(−i uα2 ∆t − 2u 2 α2 ∆t ), (2.7)
with α as in (2.1) and i the imaginary unit.

By Equation (2.6), we can write σ2 (t j ) as
σ2 (t j ) = σ2 (t 0 ) exp(R 1 + R 2 + · · · + R j ), (2.8)
because
σ2 (t j )
Ã Ã !!
σ2 (t 1 )
µ 2
σ (t 2 )
µ ¶ ¶
2 2
σ (t j ) = σ (t 0 ) exp log 2 + log 2 + · · · + log 2
σ (t 0 ) σ (t 1 ) σ (t j −1 )
2
σ (t j )
Ã Ã !!
= σ2 (t 0 ) exp log 2 .
σ (t 0 )
1 These time points are not to be confused with the Monte Carlo time-steps. We will have only one Monte Carlo
time-step. M is the number of points for the discrete approximation of Y (T ).
At this point, a backward recursion procedure in terms of R j will be set up by which

we will recover fˆlog Ỹ (T ) . We define
Y1 = R M ,
(2.9)
2 Y j = R M +1− j + Z j −1 , j = 2, . . . , M ,
with Z j = log(1 + exp(Y j )).

Using the definitions in Equations (2.8) and (2.9), the approximated time-integrated
variance can be written as
M
σ2 (t i )∆t = ∆t σ20 exp(Y M ).
X
Ỹ (T ) = (2.10)
i =1
From Equation (2.10), we determine fˆlog Ỹ (T ) , as follows
fˆlog Ỹ (T ) (u) = E[exp(i u log Ỹ (T ))]

= E[exp i u log ∆t σ20 + i uY M ]
¡ ¡ ¢ ¢
(2.11)
= exp i u log ∆t σ20 fˆYM (u).
¡ ¡ ¢¢
We have reduced the computation of fˆlog Ỹ (T ) to the computation of fˆYM . As Y M is

defined recursively, its ChF, fˆYM , can be obtained recursively as well. In Subsection 2.3.1,
a procedure for obtaining an approximation for fˆY , i.e. fˆ∗ , is described. According to
M YM
the definition of the (backward) sequence Y j in Equation (2.9), the required initial and
recursive ChFs are given by the following expressions,
fˆY1 (u) = fˆR M (u) = exp(−i uα2 ∆t − 2u 2 α2 ∆t ),

(2.12)
fˆY j (u) = fˆR M +1− j (u) fˆZ j −1 (u) = fˆR (u) fˆZ j −1 (u),
where Equation (2.7) is used in both expressions and the independence of R M +1− j and
Z j −1 is also employed.
After the computation of fˆYM , and thus of fˆlog Ỹ (T ) , the next step is to compute the
PDF of log Ỹ (T ) by means of the COS method [49]. For that, we need to determine the
support of log Ỹ (T ), which is denoted by interval [ã, b̃]. It can be calculated by employing
the relation
log(Ỹ (T )) = log ∆t σ20 exp(Y M ) = log ∆t σ20 + Y M .
¡ ¢ ¡ ¢
An alternative to the COS method is the so-called SWIFT method [102], based on
Shannon wavelets. This technique is efficient especially when long time horizons are
considered. For the present application, however, the COS method appears superior in
terms of computational time. The use of the one-step SABR simulation will be restricted
to exercise times up to two years (details in Subsection 2.4.2).
We recover f log Ỹ (T ) by employing the COS expression, as follows
NX
−1
kπ
µ ¶
2 0
f log Ỹ (T ) (x) ≈ C k cos (x − ã) , (2.13)
b̃ − ã k=0 b̃ − ã
with
kπ ãkπ
µ µ ¶ µ ¶¶
C k = ℜ fˆlog Ỹ (T ) exp −i ,
b̃ − ã b̃ − ã
and
kπ kπ kπ
µ ¶ µ ¶ µ ¶
fˆlog Ỹ (T ) log ∆t σ20 fˆYM 2
¡ ¢
= exp i
b̃ − ã b̃ − ã b̃ − ã
kπ kπ
µ ¶ µ ¶
log ∆t σ20 fˆY∗M
¡ ¢
≈ exp i ,
b̃ − ã b̃ − ã
where N is the number of COS terms and the prime 0 and ℜ symbols in (2.13) indicate
division of the first term in the summation by two and taking the real part of the complex-
valued expressions in the brackets, respectively.
The CDF F log Ỹ (T ) , as in Equation (2.5), is obtained by integrating the corresponding
approximated PDF from Equation (2.13), as follows
Z x
F log Ỹ (T ) (x) = f log Ỹ (T ) (y)dy
−∞
x NX
−1 (2.14)
¢ kπ
µ ¶
2
Z
0 ¡
≈ C k cos y − ã dy.
ã b̃ − ã k=0 b̃ − ã
2.3.1. C H F OF Y M
In the previous section, we defined a procedure to compute an approximation of the
CDF of Y (T ). This computation relies on the efficient calculation of the ChF of Y M . In
this section, we explain the recursive procedure for approximating fˆYM based on the ini-
tial ChF, fˆY1 , the ChF of the recursive process, fˆR , and the equality fˆY j (u) = fˆR (u) fˆZ j −1 (u)
(see Equations (2.6) and (2.12)). Note that this iterative computation needs to be done
only once. By definition, the ChF of Z j −1 reads
Z ∞
fˆZ j −1 (u) := (exp(x) + 1)i u f Y j −1 (x)dx.
−∞
PDF f Y j −1 is not known. To approximate it, the Fourier cosine series expansion on
f Y j −1 is applied. We suppose a bounded the integration range [a, b]. The calculation of
integration boundaries a and b follows the cumulant-based approach as described in
[137] and [49]. After truncation of the integral, we have
Z b
fˆZ∗j −1 (u) = (exp(x) + 1)i u f Y j −1 (x)dx. (2.15)
a
Now we can apply a cosine series expansion to f Y j −1 , so that fˆZ∗ becomes

j −1
2 NX −1
lπ
Z b µ ¶
0
fˆZ∗j −1 (u) ≈ Bl (exp(x) + 1)i u cos (x − a) dx, (2.16)
b − a l =0 a b−a
with
lπ lπ
µ µ ¶ µ ¶¶
B l = ℜ fˆY∗j −1 exp −i a ,
b−a b−a
where N is the number of expansion terms, fˆY∗1 (u) = fˆR (u) and fˆY∗ (u) = fˆR (u) fˆZ∗ (u).
j j −1
We wish to compute fˆY∗M ( b−akπ

), for k = 0, . . . , N − 1. Considering Equation (2.16), we
rewrite fˆ (u) = fˆR (u) fˆ
∗
Yj
∗
(u) in matrix-vector form, as follows
Z j −1
2 Φ j −1 = M A j −1 , (2.17)
where
¤N −1
Φ j −1 = Φ j −1 (k) k=0 , Φ j −1 (k) = fˆZ∗j −1 (u k ),
£
Z b
N −1
M = [M (k, l )]k=0 , M (k, l ) = (exp(x) + 1)i uk cos ((x − a)u l ) dx, (2.18)
a
2 £ ¤N −1 ³ ´
Aj = A j (l ) l =0 , A j (l ) = ℜ fˆY∗j −1 (u l ) exp (−i au l ) ,
b−a
with
zπ
uz = . (2.19)
b−a
By the recursion procedure in Equation (2.17), we obtain the approximation fˆY∗M .
Before using this approximation, we must compute matrix M in Equation (2.18) highly
efficiently. This matrix M has to be computed only once. Its elements are given by
Z b
I M := (exp(x) + 1)i uk cos ((x − a)u l ) dx. (2.20)
a
The efficient computation of the integral in Equation (2.20) is a key aspect for the
performance of the whole procedure. The number of elements in matrix M can be large,
as it corresponds to the number of elements in the cosine series expansion i.e. N 2 . The
efficiency also depends on the required accuracy. We discuss the numerical treatment in
Subsection 2.3.2.
2.3.2. T REATMENT OF INTEGRAL I M

The efficient calculation of the integral in Equation (2.20) is crucial for an efficient com-
putation of fˆY∗M (and thus of F log Ỹ (T ) ). In this section, we propose several ways to deal
with I M . In Subsection 2.3.3, an analysis in terms of accuracy and efficiency for each of
the approximations is carried out.
Analytic solution based on 2 F 1 . Integral I M in (2.20) can be rewritten as

Z b
IM = ei uk g (x) cos ((x − a)u l ) dx, g (x) = log((exp(x) + 1)), (2.21)
a
with u k and u l as in (2.19).

By the Euler formula, integral I M becomes
1¡ +
Z b
I + I− , I± =
¢
IM = exp(i (u k g (x) ± (x − a)u l ))dx.
2 a
Integrals I + and I − can be solved analytically in terms of the hypergeometric func-

tion 2 F 1 . The solution reads
i e±i ul (x−a)
I± = ∓ 2 F 1 (−i u k , ±i u l ; 1 ± i u l ; − exp(x)).
ul
2
Unfortunately, multiple evaluations of the hypergeometric function make the com-
putation rather time-consuming.
Clenshaw-Curtis quadrature. Regarding numerical approximation methods, quadra-

ture rules can be applied to solve integral I M . In our case, the elements in Equation (2.20)
can, for example, be approximated numerically by the Clenshaw-Curtis quadrature, i.e.,
Z b
(exp(x) + 1)i uk cos ((x − a)u l ) dx ≈ (D T v)T w, (2.22)
a
where the matrix element D(k, m) and the vectors v and w can be computed by the
expressions
¶ (
1/2 if m = {1, n q /2 + 1},
µ
2 (m − 1)(k − 1)π
D(k, m) = cos ·
nq n q /2 1 otherwise,
Ã !T
2 2 2 2
v := 1, , ,··· , , ,
1 − 4 1 − 16 1 − (n q − 2)2 1 − n q2
µ µ ¶¶ µ µ ¶¶
(m − 1)π (m − 1)π
w m := h cos + h − cos ,
nq nq
¶i uk
b−a b−a a +b b−a a +b
µµ ¶ ¶µ µ ¶
h(x) = cos x+ − a u l exp x+ +1 ,
2 2 2 2 2
with k, m = 1, . . . , (n q /2+1) and with n q the number of terms in the quadrature. This was
the approach taken in [137], with computational complexity O(n q N 2 ).
Piecewise linear approximation. In a second numerical approximation, we start with

I M in (2.21) and notice that although function g (x) (the blue curve in Figure 2.1) is not
linear, it is smooth and monotonic. Hence, the numerical technique proposed is to de-
fine sub-intervals of the integration range [a, b] in which g (x) is constant or almost lin-
ear, i.e.,
XL Z bj
IM = exp(i u k g (x)) cos ((x − a)u l ) dx,
j =1 a j
so that we can perform a first-order expansion of function g (x) in each sub-interval

[a j , b j ], j = 1, . . . , L. In order to guarantee continuity of the approximation, g (x) is ap-
proximated by a linear function in each interval as
g (x) ≈ c 1, j + c 2, j x, x ∈ [a j , b j ].
This gives us an approximation I˜M , as follows
L Z bj
I˜M =
X
exp(i u k c 1, j ) exp(i u k c 2, j x) cos ((x − a)u l ) dx . (2.23)
j =1 aj
2
| {z }
IL
In each sub-interval [a j , b j ], we need to determine a simple integral. I L in Equation

(2.23) is known analytically and is given by
¡ ¢
exp(i a j c 2, j u k ) −i c 2, j u k cos((a − a j )u l ) + u l sin((a − a j )u l )
IL =
(u l − c 2, j u k )(u l + c 2, j u k )
¡ ¢
i exp(i b j c 2, j u k ) c 2, j u k cos((a − b j )u l ) + i u l sin((a − b j )u l )
+ .
(u l − c 2, j u k )(u l + c 2, j u k )
The optimal integration points a j and b j in (2.23) are found by differentiation of g (x)
w.r.t. x, i.e.
∂g (x) exp(x) 1
G(x) := = = ,
∂x exp(x) + 1 1 + exp(−x)
giving a relation between the derivative of g (x) and the logistic distribution. This repre-
sentation indicates that G(x) is the CDF of the logistic distribution with a null location
parameter and a scale parameter s = 1. In order to determine the optimal integration
points so that their positions correspond to the logistic distribution, we need to com-
pute the quantiles. The quantiles of the logistic distribution are analytically available
and given by
p
µ ¶
q(p) = log .
1−p
The algorithm for the optimal integration points, given integration interval [a, b] and
the number of intervals L, is as follows:
• Determine an interval in the probability space: P g = [G(a),G(b)].
j
• Divide the interval P g equally in L parts: P g = [p j , p j +1 ], j = 0 . . . L − 1.
• Determine a j and b j by calculating the quantiles, a j = q(p j ) and b j = q(p j +1 ),

j = 0 . . . L − 1.
The algorithm above ensures that integration points are optimally distributed. In
Figures 2.1(a) and 2.1(b), we present the optimal integration points for L = 10 and L = 25,
respectively. The points are well-positioned in the region of interest, as expected.
However, for our specific problem, the integration interval can be rather large. As the
integration points are densely concentrated in the function’s curved part, the errors in
the outer sub-intervals dominate the overall error and more points at the beginning and
at the end of the interval are required to reduce this error. A more balanced distribution
of the integration points can be achieved by introducing a scale parameter, s 6= 1, which
10 10
8 8
6 6
g(x)
4
g(x) 4
2 2
0 0
-10 -5 0 5 10 -10 -5 0 5 10
x x
(a) L = 10 and s = 1 (b) L = 25 and s = 1
10 10
8 8
6 6
g(x)
g(x)
4 4
2 2
0 0
-10 -5 0 5 10 -10 -5 0 5 10
x x
(c) L = 10 and s = 3 (d) L = 25 and s = 3
Figure 2.1: Optimal integration points.

L = 25 L = 50 L = 75 L = 100
Equidistant 2.4075 × 10−3 2.6743 × 10−4 3.1064 × 10−4 2.9547 × 10−6
Optimal (s = 1) 1.3273 × 10−2 2.5343 × 10−3 9.5959 × 10−4 4.7953 × 10−4
Optimal (s = 3) 3.4837 × 10−5 1.1445 × 10−6 2.1506 × 10−7 6.7114 × 10−8
2 Table 2.1: MSE vs. L for I˜M in Equation (2.23).
implies that the integration points correspond to a CDF with increased variance. In Fig-
ures 2.1(c) and 2.1(d), the distributions of the points for s = 3 are shown. In the case of
s 6= 1, the expression for the quantiles reads,
p
µ ¶
q(p) = s log .
1−p
The impact of the optimal point redistribution can be seen in Table 2.1. The mean
squared error (MSE), in absolute value, of the piecewise linear approximation is pre-
sented for a fixed interval [−10, 30] and a fixed number of elements in the cosine expan-
sion, N = 150, considering different numbers of sub-intervals and different integration
point locations (equidistant, optimal with s = 1 and optimal with s = 3). We observe that,
for the problem at hand, the optimal distribution with s > 1 gives better accuracy.
2.3.3. E RROR AND PERFORMANCE ANALYSIS

In the sections above, we have proposed a method to compute the approximated CDF
for SABR’s time-integrated variance. In this section, we perform an error analysis, indi-
cating how to reduce or bound the errors. As usual, there is a relation between the error
(or accuracy) and the performance and we cannot discuss them separately. Therefore,
we will take into account the computational effort here. The sources of error in our ap-
proximation include ²D , the discretization of Y (T ) in Equation (2.4), ²T , the truncation
in Equations (2.14) and (2.15), ²C , due to the cosine expansion in Equations (2.13) and
(2.16) and ²M , the numerical computation of I M in Equation (2.20).
Error ²D . This error occurs because the SABR’s time-integrated variance isR approxi-
t
mated by its discrete equivalent, i.e. Y (T ) ≈ Ỹ (T ). Note that the process Y (t ) = 0 σ2 (s)ds
is the solution of the SDE
dY (s) = σ2 (s)ds, Y (0) = 0,
which, by employing the same time discretization as for Ỹ (T ) in Equation (2.4), can be
written as
Ȳ (t j ) − Ȳ (t j −1 ) = σ2 (t j −1 )∆t .
This is essentially the Euler-Maruyama discretization scheme. Indeed, it is easy to see

that, if we sum the right-hand side of the equality over the M discrete time points, we
recover our discrete approximation Ỹ (T ) as follows
M M ¡
σ2 (t j −1 )∆t =
X X ¢
Ȳ (t j ) − Ȳ (t j −1 ) = Ȳ (t M ) − Ȳ (0) = Ỹ (T ).
j =1 j =1
10 0
Error ǫD
-i
Error bound, γ=1/2

E -log Ŷ (T ) − log Y (T )-
-
Error bound, γ=1
10 -1
2
10 -2
h-
-
10 -3
10 1 10 2 10 3
M
(a) Error ²D (b) F log Ỹ (T ) (x)
Figure 2.2: Convergence analysis in terms of the number of discrete points, M .
The error (or convergence) of the Euler-Maruyama discretization scheme has been
intensively studied in the literature (see [83], for example). Two errors can be considered
when a stochastic process is discretized: strong and weak. It was proved by Kloeden and
Platen in [83] that, under some conditions of regularity, the Euler-Maruyama scheme
has strong convergence with order γ = 12 and weak convergence with order γ = 1. In our
particular case, we focus on the strong error between Y (T ) and Ỹ (T ). Then, the error ²D
can be computed and bounded by employing the usual strong convergence expression
as µ ¶γ
T
²D := E Ỹ (T ) − Y (T ) ≤ C
£¯ ¯¤
¯ ¯ ,
M
for some constant C > 0.
In our actual experiments, we consider log Ỹ (T ) and log Y (T ). This will reduce the
error further. In order to numerically show the order of convergence of our approxima-
tion, in Figure 2.2(a) the strong error2 of log Ỹ (T ) with respect to the number of discrete
time points, M , is depicted. Additionally, “upper” (C = 1 and γ = 12 ) and “lower” (C = 1
and γ = 1) bounds are presented as references. In order to further test the convergence,
the resulting CDFs, F log Ỹ (T ) , are presented in Figure 2.2(b), again varying the number of
discrete time points, M . Both experiments perform as expected.
Error ²T . The use of cosine expansions and the COS method implies a truncation of the
integration range to interval [a, b]. This gives rise to a truncation error, which is defined
by Z
²T (H ) := f H (y)dy,
R\[a,b]
where H corresponds to log Ỹ (T ) and Y j , according to Equations (2.14) and (2.15), re-
spectively. By a sufficiently large integration range [a, b], this error can be reduced and
does not dominate the overall error.
2 As log Y (T ) is unknown, the reference value is computed by using a very fine discretization, i.e. M = 5000.
Error ²C . The convergence due to the number of terms N in the cosine series expan-
sion was derived in [49]. For any f (y|x) ∈ C∞ [a, b], ²C can be bounded by
|²C (N , [a, b])| ≤ P ∗ (N ) exp(−(N − 1)ν),
2 being ν > 0 a constant and P ∗ (N ) a term which varies less than exponentially with re-
spect to N . As the PDF of the time-integrated variance is a very smooth function, ²C will
decay exponentially with respect to N .
Error ²M . We have described two different numerical methods to compute integral I M :

Clenshaw-Curtis quadrature and the piecewise linear approximation.
Regarding the Clenshaw-Curtis quadrature in Equation (2.22) as stated in [137], the
error can be bounded by O((2n q )−m /m) for an m-times differentiable integrand, with
n q the number of quadrature terms. When m is bounded, we achieve algebraic conver-
gence while, otherwise, the error converges exponentially with respect to n q . In Equa-
tion (2.20), the integrand of I M belongs to C∞ [a, b] thus, by Clenshaw-Curtis quadrature,
error ²M can be reduced with exponential convergence in n q .
When the approximation in Equation (2.23) is employed, we observe a reduction of
the error by a factor four, when going from L = 25 to L = 50 integration points in Table
2.1. However, with L = 100 the error reduces more drastically due to the choice of optimal
integration points.
P ERFORMANCE ANALYSIS
We perform an efficiency analysis considering the terms that control the evaluated er-
rors. Error ²T is not taken into account here, as we always employ a sufficiently large
integration interval.
Error ²D can be reduced by increasing M in Equation (2.4). We find that the value
of M hardly affects the computational cost, so it can be increased without a significant
impact on the performance.
Error ²C is governed by the number of series expansion terms, N . As ²C exhibits an
exponential decay w.r.t. N , a relatively small value (N ≈ 100) already results in highly
accurate approximations. Larger N values have a great impact on the performance be-
cause I M is calculated N × N times. As this calculation dominates the computational
cost, the execution time increases by O(N 2 ).
Error ²M can be controlled by the number of quadrature terms, n q , in the case of the
Clenshaw-Curtis method or by the number of sub-intervals, L, when using the piece-
wise linear approximation. Again, the impact can be significant because I M is com-
puted many times. To achieve a similar order of accuracy (∼ 10−7 ) in a generic interval
[a, b], the piecewise linear approximation is approximately three times faster than the
Clenshaw-Curtis quadrature. Therefore, we will use the piecewise linear approximation
in the following.
2.4. S IMULATION OF Y (T )|σ(T ): COPULA APPROACH

The approximated CDF of the marginal distribution of Y (T ) (in the log-space, see Sec-
tion 2.3) is employed to simulate Y (T )|σ(T ) by means of a copula. A d -dimensional
2.4. S IMULATION OF Y (T )|σ(T ): COPULA APPROACH 29
copula is a joint distribution function on a unit hyper-cube [0, 1]d with d marginal dis-
tributions. A distribution with a copula contains essentially the same information as the
joint distribution, like the marginal properties and the dependence structure, but this
information can be decoupled. The importance of copulas is represented by two parts
of the Sklar theorem, introduced by Abe Sklar in [122]. The first part states that a copula
can be derived from any joint distribution function, and the second part states that any 2
copula can be combined with any set of marginal distributions to result in a multivariate
distribution function. More formally, Sklar’s theorem reads
• Let F be a joint distribution function and F j , j = 1, . . . , d , be the marginal distribu-

tions. Then there exists a copula C : [0, 1]d → [0, 1] such that
F (x 1 , . . . , x d ) = C (F 1 (x 1 ), . . . , F d (x d )),
for all x j ∈ [−∞, ∞]. Moreover, if the margins are continuous, then C is unique;
otherwise C is uniquely determined on Ran(F 1 ) × · · · × Ran(F d ), where Ran(F j ) =
F j ([−∞, ∞]) is the range of F j .
• The converse is also true. That is, if C is a copula and F 1 , . . . , F d are univariate dis-
tribution functions, then the multivariate function defined by the preceding equa-
tion is a joint distribution function with marginal distributions F j , j = 1, . . . , d .
2.4.1. FAMILIES OF COPULAS

Copulas can be classified into so-called fundamental, implicit and explicit copulas. Fun-
damental copulas are copulas that represent either perfect positive dependence, inde-
pendence or perfect negative dependence. Implicit copulas are copulas extracted from
well-known multivariate distributions, like the normal, Gaussian or Student t distribu-
tions, that do not have closed-form expressions. Explicit or Archimedean copulas have
simple closed-form expressions. The Clayton, Frank and Gumbel copulas are the most
commonly employed ones.
Here, we briefly describe these copulas, considering F 1 . . . , F d ∈ [0, 1]d as the marginal
distributions:
• Gaussian: given a correlation matrix, R ∈ Rd ×d , the Gaussian copula with param-

eter R is defined by
C R (F 1 . . . , F d ) = ΦR (Φ−1 (F 1 ), . . . , Φ−1 (F d )),
where Φ is the distribution function of a standard normal random variable and

ΦR is the d -variate standard normal distribution with zero mean vector and co-
variance matrix R.
• Student t: given correlation matrix R ∈ Rd ×d and the degrees of freedom parame-

ter ν ∈ (0, ∞), the t copula is defined by
C R,ν (F 1 . . . , F d ) = tR,ν (t−1 −1

ν (F 1 ), . . . , t (F d )),
where tν and tR,ν are univariate and the d -variate Student t distributions, respec-
tively.
• Archimedean: is governed by the representation
C θ (F 1 . . . , F d ) = ψ−1
θ (ψθ (F 1 ) + · · · + ψθ (F d )),
where ψθ : [0, 1] → [0, ∞] is the so-called generator function which is supposed

2 to be a continuous, strictly decreasing and convex function such that ψθ (1) = 0,
ψθ (0) = ∞ and its inverse ψ−1
θ
is monotonic on [0, ∞). A variety of Archimedean
copulas can be defined, like the following well-known copulas.
The Clayton copula, for which the generator function is given by ψθ (u) = θ −1 (u −θ −
1), θ > 0, reads
Ã !−1/θ
d
−θ
C θ (F 1 . . . , F d ) =
X
Fi − d + 1 . (2.24)
i =1
³ ´
exp(−θu)−1
The Frank copula, with generator function given by ψθ (u) = − log exp(−θ)−1 , reads
Ã Qd !
1 i =1 (exp(−θF i ) − 1)
C θ (F 1 . . . , F d ) = − log 1 + , (2.25)
θ (exp(−θ) − 1)d −1
being θ ∈ R\{0} for d = 2 and θ > 0 for d ≥ 3.

¢θ
The Gumbel copula, with generator function ψθ (u) = − log(u) , θ > 0, reads
¡
Ã Ã !1/θ !
d ¡ ¢θ
C θ (F 1 . . . , F d ) = exp −
X
− log(F i ) . (2.26)
i =1
In order to calibrate parameter θ in (2.24), (2.25) or (2.26) a relation between θ and the
rank correlation coefficient Kendall’s τ can be employed. This relation is available in
closed-form for the Clayton copula, i.e. θ = 2τ/(1 − τ), and the Gumbel copula, i.e. θ =
1/(1 − τ). There is no analytic expression of θ for the Frank copula.
To apply any of these copulas, an approximation for the correlation between the in-
volved distributions needs to be determined. Here, Pearson’s correlation coefficient, P ,
is chosen since it is employed directly in the Gaussian and Student t copulas and a rela-
tion with Kendall’s τ exists ³π ´
P = sin τ .
2
In Subsection 2.4.2, an approximation for Pearson’s coefficient between log Y (T ) and
log σ(T ) of the SABR dynamics is derived.
2.4.2. C ORRELATION : log Y (T ) AND log σ(T )

For the problem at hand, we need to derive an expression for Pearson’s correlation coef-
ficient P log Y (T ),log σ(T ) . By definition, we have
h R i
T
· Z T ¸ cov log 0 σ2 (s)ds, log σ(T )
P log Y (T ),log σ(T ) = corr log σ2 (s)ds, log σ(T ) = r .
0
h R i
T
Var log 0 σ2 (s)ds Var log σ(T )
£ ¤
We employ the following approximation

Z T Z T Z T
log σ2 (s)ds ≈ log σ2 (s)ds = 2 log σ(s)ds,
0 0 0
where the logarithm and the integral are interchanged. Since the log function is concave, 2
this approximation forms a lower bound for the true value. This can be seen by applying
Jensen’s inequality, i.e.
Z T Z T
log σ2 (s)ds ≥ log σ2 (s)ds.
0 0
We will numerically show that, under our model settings with an option contract time to
maturity of less than two years, this is a highly satisfactory approximation. The correla-
tion coefficient can now be computed as
hR i
T
cov 0 log σ(s)ds, log σ(T )
P log Y (T ),log σ(T ) ≈ r hR i .
T
Var 0 log σ(s)ds Var log σ(T )
£ ¤
The quantity Var log σ(T ) is known by Equation (2.3), and we derive expressions
£ ¤
the other two quantities. Starting with the covariance, we have

·Z T ¸ ·µZ T ¶ ¸ ·Z T ¸
log σ(s)ds, log σ(T ) = E log σ(s)ds log σ(T ) −E log σ(s)ds E log σ(T ) .
£ ¤
cov
0 0 0
From Equation (2.3), we find
1
log σ(T ) = log σ0 − α2 T + αW (T ) =: η(T ) + αW (T ),
2
and
Z T 1
Z T Z T
log σ(s)ds = T log σ0 − α2 T 2 + α W (s)ds =: γ(T ) + α (T − s)dW (s).
0 4 0 0
Based on these equations, with

·Z T ¸
E log σ(T ) = η(T ), E log σ(s)ds = γ(T ),
£ ¤
0
we obtain the following expressions for the required expectation,

·µZ T ¶ ·µ ¸ Z T ¶ ¸
E log σ(s)ds log σ(T ) = E γ(T ) + α (T − s)dW (s) η(T ) + αW (T )
¢ ¡
0 0
·Z T Z T ¸
2
= γ(T )η(T ) + α E dW (s) (T − s) dW (s)
0 0
·Z T ¸
1
= γ(T )η(T ) + α2 E (T − s) ds = γ(T )η(T ) + α2 T 2 .
0 2
RT
In order to determine the variance of 0 log σ(s)ds, we compute
"µZ
T ¶2 # "µ Z T ¶2 #
E log σ(s)ds = E γ(T ) + α (T − s) dW (s)
0 0
2 T
·Z ¸
= γ2 (T ) + α2 E (T − s)2 ds
0
1
= γ (T ) + α2 T 3 ,
2
3
and the variance reads
·Z T ¸ "µZ
T ¶2 # µ ·Z T ¸¶2
1
Var log σ(s)ds = E log σ(s)ds − E log σ(s)ds = α2 T 3 .
0 0 0 3
Pearson’s correlation coefficient is then obtained as

h³R ´ i hR i £
T T
E 0 log σ(s)ds log σ(T ) − E 0 log σ(s)ds E log σ(T )
¤
P log Y (T ),log σ(T ) ≈ r hR i
T
Var 0 log σ(s)ds Var log σ(T )
£ ¤
p
γ(T )η(T ) + 21 α2 T 2 − γ(T )η(T ) 1 2 2
2α T 3
= q¡ =q = .
1 2 3 1 2
3α T α2 T 3α T
¢¡ ¢
4 4
The approximation obtained is a constant value. We will show numerically that, for
the problems at hand, this appears to be a very reasonable approximation. The cor-
relation between log Y (T ) and log σ(T ) is affected by the maturity time T , and by the
volatility-of-volatility parameter α. In Figure 2.3, the empirical and approximated (red
grid) correlations are depicted for typical values of T and α. Because we focus on short
maturity options, we restrict T ∈ [0, 2] and α ∈ (0, 1]. The experiment shows that, only in
rather extreme cases (α > 0.7), the differences between the empirical and approximated
correlation increase, but remain within five basis points. The approximation is therefore
sufficiently accurate for our purposes, since a small error in the correlation (of less than
ten basis points) does not affect the performed simulation significantly.
2.4.3. S AMPLING Y (T )|σ(T )

The following steps describe the complete procedure for the simulation of Y (T ) given
σ(T ) by using a bivariate copula approach:
1. Determine F log σ(T ) (analytically) and F log Ỹ (T ) via Equation (2.14).
2. Define the bivariate copula distribution by employing the marginal distributions

and the correlation approximation (Subsection 2.4.2).
3. Generate correlated uniform samples, Ulog σ(T ) and Ulog Ỹ (T ) by means of the bi-
variate copula distribution.
4. From Ulog σ(T ) and Ulog Ỹ (T ) invert the original marginals, F log σ(T ) and F log Ỹ (T ) . The
resulting quantiles should be correlated under the chosen correlation coefficient.
2
Plog Y (T ),log σ(T )
1
0.9
0.8
0.7
0.6
0.5
0.2 1
0.4 0.9
0.6 0.8
0.8 0.7
1 0.6
T 1.2
1.4 0.4
0.5
α
1.6 0.3
1.8 0.2
2 0.1
Figure 2.3: Pearson’s correlation coefficient: Empirical (surface) vs. approximation (red grid).
5. Finally, the samples of Y (T )|σ(T ) are obtained by taking exponentials.
The inversion procedure in the fourth step is straightforward in the case of σ(T ). The
case of Ỹ (T ) is more involved, and we propose a direct inversion procedure based on lin-
ear interpolation. The insight is that, by rotating CDF F log Ỹ (T ) , we can interpolate over
probabilities instead of x−values. This is possible since F log Ỹ (T ) is a smoothly increasing
function. The interpolation polynomial provides the quantiles of the original distribu-
tion, F log Ỹ (T ) , from the probabilities Ulog Ỹ (T ) . This inversion procedure is very fast (at
almost no cost) and sufficiently accurate.
RT
2.4.4. S IMULATION OF S(T ) GIVEN S(0), σ(T ) AND 0 σ2 (s)ds
We complete the one-step SABR simulation by the conditional sampling of S(T ). The
most commonly used techniques can be classified into two categories: direct inversion
of SABR’s CDF given in Equation (2.2) and moment-matching approaches. The direct
inversion procedure has a higher computational cost because of the evaluation of the
non-central χ2 distribution. However, some recent developments make this computa-
tion affordable. In [21], Chen et al. proposed a forward asset simulation based on a
combination of moment-matching (quadratic Gaussian) and enhanced direct inversion
procedures. We employ this technique also here. Note, however, that, for R Tsome specific
values of β, the simulation of the conditional S(T ) given S(0), σ(T ) and 0 σ2 (s)ds gives
rise to analytic expressions.
C ASE β = 0
For β = 0, it is easily seen from Equation (2.1) that the asset process (in integral form)
becomes Z T q Z T
S(T ) = S(0) + ρ σ(s)dW̃σ (s) + 1 − ρ 2 σ(s)dW̃S (s),
0 0
2 where dW̃S (t ) and dW̃σ (t ) are independent Brownian motions and with
Z T 1
σ(s)dW̃σ (s) = (σ(T ) − σ(0)) , (2.27)
0 α
and  s 
Z T Z T
σ(s)dW̃S (s)|σ(T ) ∼ N 0, σ2 (s)ds  . (2.28)
0 0
C ASE β = 1
In the case of β = 1, the asset dynamics in Equation (2.1) are log-normal and the solution
is given by
1 T 2
µ Z Z T q Z T ¶
S(T ) = S(0) exp − σ (s)ds + ρ σ(s)dW̃σ (s) + (1 − ρ )
2 σ(s)dW̃S (s) .
2 0 0 0
If we take the log-transform,
1 T 2
Z T Z T
S(T )
µ ¶ Z q
log =− σ (s)ds + ρ σ(s)dW̃σ (s) + (1 − ρ 2 ) σ(s)dW̃S (s),
S(0) 2 0 0 0
and by³ employing

´ again Equation (2.27) and Equation (2.28), we obtain the distribution
) RT 2
of log S(T
S(0) | 0 σ (s)ds, σ(T ), σ(0), which reads
 s 
1 T ρ T
Z Z
2
N − σ (s)ds + (σ(T ) − σ(0)) , (1 − ρ 2 ) σ2 (s)ds  . (2.29)
2 0 α 0
By employing Equation (2.29), the asset dynamics S(t ) can be sampled from
 s 
ρ
Z T Z T
1 2
S(T ) ∼ S(0) exp − σ (s)ds + (σ(T ) − σ(0)) + X (1 − ρ 2 ) σ2 (s)ds  , (2.30)
2 0 α 0
where X is the standard normal.
C ASE β 6= 0, β 6= 1
As mentioned before, for the generic case of β ∈ (0, 1), we employ the enhanced inver-
sion of the SABR asset price distribution in Equation (2.2) as introduced in [21]. We
briefly summarize this approach. The asset simulation is performed either by a moment-
matched quadratic Gaussian approximation or by an enhanced direct inversion. The au-
thors in [21] proposed a threshold value to choose the suitable technique among these
two. The moment-matching approach relies on the fact that, for S(0) À 0, the distribu-
tion function in Equation (2.2) can be approximated by
µ Z t ¶
Pr S(t ) ≤ K |S(0) > 0, σ(t ), σ (s)ds ≈ χ2 (c; 2 − b, a).
2
0
By definition, the mean, µ and the variance, κ2 , of a generic non-central chi-square

2
distribution χ2 (x; δ, λ) are µ := δ + λ and κ2 := 2(δ + 2λ), respectively. A variable V ∼
χ2 (x; δ, λ) is accurately approximated by a quadratic Gaussian distribution, as follows
d
V ≈ d (e + X )2 , X ∼ N (0, 1),
with
κ2 µ
q q
ψ := , e 2 = 2ψ−1 − 1 + 2ψ−1 2ψ−1 − 1 ≥ 0, d = .
µ2 1 + e2
Applying this to our case in Equation (2.2), we can sample the conditional asset dynam-
ics, T > 0, by
µ Z T 1
¶ 2(1−β)
2 2 2 2
S(T ) = (1 − β) (1 − ρ )d (e + X ) σ (s)ds .
0
The moment-matching approximation is accurately applicable when 0 < ψ ≤ 2 and µ ≥ 0
[21]. Otherwise, a direct inversion procedure must be employed. A root-finding Newton
method is then used. In order to reduce the number of Newton iterations (and the ex-
pensive evaluation of the non-central chi-square CDF), the first iterations are based on
an approximated formula for the non-central chi-square distribution, which is based on
the normal CDF and derived by Sankaran in [114]. Then, the obtained value is employed
as the initial guess for the Newton algorithm. The result is a significant reduction in
the number of iterations and, hence, in the computational cost. Furthermore, this root-
finding procedure consists of only basic operations, so that the whole procedure can be
easily vectorized, leading to a further efficiency improvement. The resulting sampling
procedure is robust and efficient.
M ARTINGALE CORRECTION
As pointed out in [2] and [21], the exact simulation of the asset price can, in certain
SV models, give rise to loss of the martingale property (because of the approximation
of a continuous process by its discrete equivalent). This is especially seen, when the
size of the time-step is large and for certain configurations of the SABR parameters, like
small β values, close-to-zero initial asset values S 0 , high vol-vol parameter α or large
initial volatility σ0 . As we deal with one large time-step, a martingale correction needs
to be applied. We incorporate a simple but effective numerical martingale correction, as
follows
1 Xn
S(T ) = S(T ) − S p (T ) + E[S(T )],
n p=1
1 Xn
= S(T ) − S p (T ) + S 0 ,
n p=1
where S p (·) represents the forward asset value for the p−th Monte Carlo path.
Note, however, that, for practical SABR parameters, the martingale error is very small.
S0 σ0 α β ρ T
Set I 1.0 0.5 0.4 0.7 0.0 2
Set II 0.05 0.1 0.4 0.0 −0.8 0.5
Set III 0.04 0.4 0.8 1.0 −0.5 2
2 Table 2.2: Data sets.
2.5. N UMERICAL EXPERIMENTS

In this section, different numerical experiments will be carried out. In Subsection 2.5.1,
we compare the different copulas from Subsection 2.4.1 for the simulation of Y (T )|σ(T ).
After that, in Subsection 2.5.2, we employ the best performing copulas in a SABR pricing
experiment. We consider several representative SABR data sets with special characteris-
tics, like a zero correlation (Set I), a normal SABR model (Set II) and a high volatility-of-
volatility SABR set (Set III). The parameter values are shown in Table 2.2. Other parame-
ters in our one-step SABR method include:
• Number of discrete time points: M = 1, 000.
• Number of COS elements: N = 150.
• Number of intervals for piecewise linear approximation of I M : L = 100.
The experiments were performed in a computer with CPU Intel Core i7-4720HQ 2.6
GHz and RAM memory of 16 Gigabytes. The employed software package was Matlab
r2015b.
2.5.1. C OPULA APPROACH ANALYSIS

RT
As Y (T ) = 0 σ2 (s)ds is a summation of squared log-normal processes, Gaussian-based
copulas may fit well. However, we also consider Archimedean copulas as these may
be superior. In order to choose an optimal copula, we assess a so-called goodness-of-
fit (GOF) criterion for copulas. There are different GOF tests in the literature, see an
overview in [9], for example. These measures are based on a comparison between some
empirical accurate approximations (that are usually computationally expensive) and the
analyzed copula. According to the work of Durrleman et al. in [45], we split the analysis,
first evaluating the Archimedean copulas and subsequently, after choosing the optimal
copula from this class, performing an analysis of the remaining copulas. The GOF testing
for Archimedean copulas is a graphic procedure based on Kendall’s processes, proposed
by Genest and Rivest [55] and Barbe et al. [6]. They defined a function K by
K (u) = P r [C (U1 . . . ,Ud ) ≤ u]
where C is the Archimedean copula and u ∈ [0, 1]. A non-parametric estimate3 of K is

available as
1X n 1 X n
K˜ (u) = 1[ϑi ≤u] , ϑi = 1 j i j i ,
n i =1 n − 1 j =1 [x1 <x1 ,...,xd <xd ]
3 An analytic solution is known, but impracticable in terms of computational effort.
2.5. N UMERICAL EXPERIMENTS 37
0 0
Empirical λ(u) Empirical λ(u)
Clayton Clayton
Frank Frank
-0.05 Gumbel -0.05 Gumbel
λ̂(u)
λ̂(u)
2
-0.1 -0.1
-0.15 -0.15
0 0.5 1 0 0.5 1
u u
(a) Set I. (b) Set II.
0
Empirical λ(u)
Clayton
Frank
-0.05 Gumbel
λ̂(u)
-0.1
-0.15
0 0.5 1
u
(c) Set III.
Figure 2.4: Archimedean GOF test: λ̂(u) vs. empirical λ(u).
with n the number of samples and x di the i -th sample of the d -th marginal distribution.
Since we deal with a bivariate case, we have d = 2. The graphical procedure of the GOF
test consists in comparing the distance λ̂(u) := u − K˜ (u) and the empirical distance,
λ(u), obtained by using an empirical estimation of K (u).
In Figure 2.4, the distances λ̂(u) between three Archimedean copulas (Clayton, Frank
and Gumbel) for the graphical GOF test are depicted. The empirical λ(u) (black line) is
employed as the reference. The experiment is performed for each data set in Table 2.2.
As the measurable quantity, the MSE of λ̂(u) − λ(u) is presented in Table 2.3. We use
n = 100, 000 simulations.
From the GOF results for the Archimedean copulas, we find that the Gumbel copula
Clayton Frank Gumbel

Set I 1.3469 × 10−3 2.9909 × 10−4 5.1723 × 10−5
Set II 1.0885 × 10−3 2.1249 × 10−4 8.4834 × 10−5
Set III 2.1151 × 10−3 7.5271 × 10−4 2.6664 × 10−4
Table 2.3: MSE of λ̂(u) − λ(u).

Gaussian Student t (ν = 5) Gumbel

−3 −3
Set I 5.0323 × 10 5.0242 × 10 3.8063 × 10−3
Set II 3.1049 × 10−3 3.0659 × 10−3 4.5703 × 10−3
Set III 5.9439 × 10−3 6.0041 × 10−3 4.3210 × 10−3
2 Table 2.4: Generic GOF: D 2 .
fits best in our framework. Once we have determined the most suitable Archimedean
copula, we perform a GOF test including the Gaussian, Student t and Gumbel copu-
las. For an objective comparison, also a measurable GOF technique is applied. Among
the several families of available generic GOF tests in the literature, in [9], Berg suggests
that methods that rely on the so-called Deheuvels or empirical copula [42] give reliable
results. Hence, we will perform a test which is based on the distances between the De-
heuvels copula, C d , and the analyzed copula C (see [45]). By using the discrete L 2 norm,
this GOF measure reads
D d (C d , C ) = kC d − C kL 2 ,
where d is the number of random variables to build the copula, d = 2. For this GOF
test, we reduce the number of samples to n = 1, 000 due to the computational (time and
memory) cost of the Deheuvels copula calculations. The other parameters remain the
same. In Table 2.4, the obtained distances, D 2 , between the tested copulas (Gaussian,
Student t and Gumbel) and the Deheuvels copula are shown.
According to the GOF results, the three copulas perform very similarly. When longer
maturities are considered, the Gumbel copula exhibits smallest errors in the GOF tests.
In terms of speed, the Gaussian copula is around three times faster than the Gumbel
copula, although the impact in the overall method is very small. The Student t copula
is discarded since its accuracy and performance are very similar to the Gaussian copula
and the calibration of the ν parameter adds extra complexity. Therefore, as a general
strategy, we conclude that the Gumbel copula is the most robust choice. When short
maturities are considered, the Gaussian copula may be a satisfactory alternative.
2.5.2. P RICING E UROPEAN OPTIONS BY THE ONE - STEP SABR METHOD

In Subsection 2.2.1, the Monte Carlo simulation of the SABR model was described. Now
that all method components have been defined, we wish to test the one-step SABR sim-
ulation scheme by pricing European call options.
For the experiment, we consider the data sets in Table 2.2, the strike values K i are
chosen following the expression by Piterbarg, in [106],
p
K i (T ) = S(0) exp(0.1 × T × δi ),
δi = −1.5, −1.0, −0.5, 0.0, 0.5, 1.0, 1.5.
As usual, the option prices are obtained by averaging the maximum of the forward
asset values at maturity minus the strike and zero, i.e. max(S(T ) − K i (T ), 0) (call option).
Subsequently, the Black-Scholes formula is inverted to determine the implied volatili-
ties. Note that, once the forward asset is simulated by our method, we can price options
at multiple strikes simultaneously.
1.2 4
n=100 n=100
n=10000 n=10000
1 3.5
n=1000000 n=1000000
Reference Reference
Implied volatility
Implied volatility
0.8 3
0.6 2.5
0.4 2
2
0.2 1.5
0 1
0 100 200 300 400 500 0 20 40 60 80 100 120
Time-steps Time-steps
(a) Set I (b) Set II
Figure 2.5: Monte Carlo Milstein convergence test. Strike K 1 .
We will compare the implied volatilities obtained by means of different approaches:

Hagan formula, Monte Carlo simulation with a Milstein discretization and the one-step
SABR method. Milstein-based Monte Carlo simulation is employed here as a reference
for our one time-step simulation. In [21], the authors have shown that the Milstein
scheme is relatively stable when applied for the SABR model compared to the Euler-
Maruyama or log-Euler schemes. In Figure 2.5, the convergence of the Milstein Monte
Carlo simulation is numerically shown. Figure 2.5(a) gives convergence results for Set I,
where an analytic solution is given by Antonov [4] (ρ = 0). In Figure 2.5(b), the Monte
Carlo convergence for Set II is depicted (in [86], Korn and Tang provide a reference price
for β = 0). In order to have a reliable reference, we set n = 1, 000, 000 paths and 250 × T
time-steps for the pricing experiment.
Now, we analyse the accuracy and the performance of the one-step SABR method
with the Gaussian and Gumbel copulas. In Table 2.5, the convergence of our method
when the number of samples, n, is increased is empirically shown. We present the mean
and the standard deviations of the error in basis points between the implied volatili-
ties given by our one-step method and the reference price (Antonov) when Set I and
K 1 are employed. We observe a reduction in the error (both mean and standard devia-
p
tion) according to the expected Monte Carlo ratio (1/ n). Regarding the computational
time, also in Table 2.5, the execution times of the one-step SABR method are shown. We
can see that, up to n = 100, 000, the number of simulations does not affect the perfor-
mance. Also when larger numbers of samples are required, the computational time of
the one time-step method does not increase drastically. As stated before, sampling with
the Gumbel copula is somewhat slower than with the Gaussian copula, but the overall
difference is negligible. The parameter M , the number of discrete time points, is not
varied here because, as stated in Subsection 2.3.3, it has no significant impact on the
execution time and can be set to a very high value.
To further test the one-step SABR method, in Table 2.6, the differences (in basis
points) between the obtained implied volatilities with different considered methods and
several strikes are presented. The number of paths is set to n = 1, 000, 000.
As the pricing experiments show, our copula-based one-step method achieves a very
high accuracy. Furthermore, since it is a one time-step technique, it is a fast (≈ 0.5 sec.)
2 n = 1000 n = 10000 n = 100000 n = 1000000

Gaussian (Set I, K 1 )
Error 519.58(204.02) 132.39(68.03) 37.42(16.55) 16.23(7.66)
Time 0.3386 0.3440 0.3857 0.5733
Gumbel (Set I, K 1 )
Error 151.44(199.36) −123.76(86.33) 34.14(17.03) 11.59(6.58)
Time 0.3492 0.3561 0.3874 0.6663
Table 2.5: Convergence in n: mean and standard deviation of the error (basis points) and time (sec.).
Strikes K1 K2 K3 K4 K5 K6 K7
Set I (Reference: Antonov [4])
Hagan 55.07 52.34 50.08 N/A 47.04 46.26 45.97
MC 23.50 21.41 19.38 N/A 16.59 15.58 14.63
Gaussian 16.23 20.79 24.95 N/A 33.40 37.03 40.72
Gumbel 11.59 15.57 19.12 N/A 25.41 28.66 31.79
Set II (Reference: Korn [86])
Hagan −558.82 −492.37 −432.11 −377.47 −327.92 −282.98 −242.22
MC 5.30 6.50 7.85 9.32 10.82 12.25 13.66
Gaussian 9.93 9.98 10.02 10.20 10.57 10.73 11.04
Gumbel −9.93 −9.38 −8.94 −8.35 −7.69 −6.83 −5.79
Set III (Reference: MC Milstein)
Hagan 287.05 252.91 220.39 190.36 163.87 141.88 126.39
Gaussian 16.10 16.76 16.62 15.22 13.85 12.29 10.67
Gumbel 6.99 3.79 0.67 −2.27 −5.57 −9.79 −14.06
Table 2.6: One-step SABR method with Gaussian and Gumbel copulas - Implied volatility: differences in basis
points.
2.6. C ONCLUSIONS 41
alternative for pricing of European options up to maturities of two years under the SABR
model. Our one-step approach works particularly well in the case of low strike values and
higher volatilities, that pose some problems for the Hagan implied volatility formula.
2.6. C ONCLUSIONS 2
In this chapter, an efficient method to obtain samples of the SABR dynamics based on the
time-integrated variance has been developed. The technique employs a Fourier method
to derive the CDF of the time-integrated variance. By means of copulas, the conditional
sampling technique is obtained. Its application gives us a fast and accurate one time-
step Monte Carlo method for the SABR model simulation. By numerical experiments, we
have shown that our method does not suffer from well-known problems of the Hagan
formula in the case of small strike values and higher volatilities.
A generalization of our one-step SABR method to a multiple time-step version will be
presented in the next chapter. This will allow us to deal with problems with longer ma-
turities (more than two years) and also with more involved exotic options (early-exercise
and path-dependent options). We need to introduce new method components to deal
with the increasing computational complexity in that case.
The hybrid aspect of the method proposed in this chapter is represented by the combined
use of Monte Carlo and Fourier techniques. This fact has allowed us to develop an accu-
rate and highly efficient one time-step simulation scheme that can be used for calibration
purposes, where Monte Carlo-based methods are typically discarded.
CHAPTER 3
Multiple time-step Monte Carlo simulation of the SABR
model
In this chapter, we will present the multiple time-step Monte Carlo simulation technique
for pricing options under the SABR model. The proposed method is an extension of the one
time-step Monte Carlo method that we proposed in the previous chapter, for pricing Eu-
ropean options in the context of the model calibration. A highly efficient method results,
with many highly interesting and non-trivial components, like Fourier inversion for the
sum of log-normals, stochastic collocation, Gumbel copula, correlation approximation,
that are not yet seen in combination within a Monte Carlo simulation. The present multi-
ple time-step Monte Carlo method is especially useful for long-term options and for exotic
options.
3.1. I NTRODUCTION
The SABR model [67] is an established SDE model which, in practice, is often used for
interest rates and FX modelling. It is based on a parametric local volatility component in
terms of a model parameter, β, and reads

(3.1)
dσ(t ) = ασ(t )dWσ (t ), σ(0) = σ0 .
Here S(t ) = S̄(t ) exp (r (T − t )) denotes the forward price of the underlying asset S̄(t ), with
r an interest rate, S 0 the spot price and T the maturity. Further, σ(t ) represents a stochas-
tic volatility process, with σ(0) = σ0 , WS (t ) and Wσ (t ) are correlated Brownian motions
with constant correlation coefficient ρ (i.e. WS Wσ = ρt ). The parameters of the SABR
model are α > 0 (the volatility of volatility, vol-vol), 0 ≤ β ≤ 1 (the variance elasticity) and
ρ (the correlation coefficient).
By the one time-step SABR method in Chapter 2, we can efficiently calculate accurate
implied volatilities for any strike, however, only for short time to maturity (less than two
years). This fits well to the context of FX markets. Here, we extend the one time-step
Monte Carlo method and propose a multiple time-step Monte Carlo simulation tech-
nique for pricing options under the SABR dynamics. We call it the mSABR method. With
this new simulation approach, we can price options with longer maturities, payoffs re-
lying on the transitional distributions and exotic options (like path-dependent options)
under the SABR dynamics.
This chapter is based on the article “On an efficient multiple time-step Monte Carlo simulation of the SABR
model”. Published in Quantitative Finance, 2017 [89].
43
44 3. M ULTIPLE TIME - STEP M ONTE C ARLO SIMULATION OF THE SABR MODEL
Because a robust and efficient SABR simulation scheme may be involved and quite
expensive, we aim for using as few time-steps as possible. This fact, together with the
long time horizon assumption, implies that we focus on larger time-steps. In this con-
text, the important research line is called exact simulation, where, rather than Taylor-
based simulation techniques, the PDF of the SDE under consideration is highly accu-
rately approximated. Point-of-departure is Broadie and Kaya’s exact simulation for the
Heston SV model [15]. That method is however time-consuming because of multiple
evaluations of Bessel functions and the use of Fourier inversion techniques on each
Monte Carlo path. Inspired by this approach, in [124] and [130], computational im-
3 provements were proposed. Furthermore, Andersen in [2] presented two efficient al-
ternatives to the Broadie and Kaya scheme, the so-called Truncated Gaussian (TG) and
the Quadratic Exponential (QE) schemes.
For the SABR model, exact Monte Carlo simulation is nontrivial because the forward
process in (3.1) is governed by CEV dynamics [31, 117]. In [75], the connection between
the CEV and a squared Bessel processes was explained, and an analytic approximation
for the SABR conditional distribution based on the non-central χ2 distribution was pre-
sented. The expression relies on the stochastic volatility dynamics, but also on the time-
integrated variance process (which itself also depends on the volatility).
An (approximately) exact SABR simulation can then be subdivided in the simulation
of the volatility process, the simulation of the time-integrated variance (conditional on
the volatility process) and the simulation of the underlying CEV process (conditional on
the volatility and the time-integrated variance processes). Based on this, Chen et al. pro-
posed in [21] a low-biased Monte Carlo scheme with moment-matching (following the
techniques of the QE scheme by Andersen in [2]) and a direct inversion approach. For
the simulation of the time-integrated variance, they also employed moment-matching
based on conditional moments that were approximated by a small disturbance expan-
sion. In [78], Kennedy and Pham provided the moments of the time-integrated vari-
ance for specific SABR models (like the normal, log-normal and displaced diffusion SABR
models) and derived an approximation of the SABR distribution. In [95], a summary of
different approaches was presented.
In this chapter, we approximate1 the distribution of the time-integrated variance
conditional on the volatility dynamics by joining the two involved marginal distribu-
tions, i.e. the stochastic volatility and time-integrated variance distributions, by means
of a copula technique. With both marginal distributions available, we use the Gumbel
copula to define a multivariate distribution which approximates the conditional distri-
bution of the time-integrated variance given the stochastic volatility.
An approximation of the CDF of the time-integrated variance can be obtained by a
recursive procedure, as described in [137], originally employed to price arithmetic Asian
options. This iterative technique is based on the derivation of the ChF of the time-
integrated variance process and Fourier inversion to recover the PDF.
The fact that we need to apply recursion and compute the corresponding ChF, PDF
and CDF of the time-integrated variance for each Monte Carlo sample at each time-step,
makes this approach however computationally very expensive (even with the improve-
ments already made in Chapter 2), like the Heston exact simulation. In order to reduce
1 An analytical expression exists but is not practically tractable.
3.2. ‘A LMOST EXACT ’ SABR SIMULATION 45
the use of this procedure as much as possible, highly efficient sampling from the CDF of
the time-integrated variance is proposed, which is based on the so-called Stochastic Col-
location Monte Carlo sampler (SCMC), see [65]. The technique employs a sophisticated
interpolation (based on Lagrange polynomials and collocation points) of the CDF under
consideration.
As will be seen, the resulting ‘almost exact’ Monte Carlo SABR simulation method
that we present here (the mSABR method), contains several interesting (and not com-
monly used) components, like the Gumbel copula, a recursion plus Fourier inversion
to approximate the CDF of the time-integrated variance, and efficient interpolation by
means of the SCMC sampler. The proposed mSABR scheme allows for fast and accu-
3
rate option pricing under the SABR dynamics, providing a better ratio of accuracy and
computational cost than Taylor-based simulation schemes. Compared to the other ap-
proaches mentioned in this introduction, we provide a highly accurate SABR simulation
scheme, based on only a few time-steps.
The chapter is organized as follows. In Section 3.2, SABR model simulation is briefly
introduced. In Section 3.3, the different parts of the multiple time-step copula-based
technique are described, including the derivation of the marginal distributions and the
application of the copula. Some numerical experiments are presented in Section 3.4. We
conclude in Section 3.5.
3.2. ‘A LMOST EXACT ’ SABR SIMULATION

The SABR model with dependent Brownian motions was already given in Equation (3.1).
The multiple time-step Monte Carlo SABR simulation is based on the corresponding SDE
system with independent Brownian motions dW̃S (t ) and dW̃σ (t ), i.e.
µ q ¶
dS(t ) = σ(t )S β (t ) ρdW̃σ (t ) + 1 − ρ 2 dW̃S (t ) , S(0) = S 0 exp (r T ) ,
(3.2)
dσ(t ) = ασ(t )dW̃σ (t ), σ(0) = σ0 .
The forward dynamics are governed by a CEV process. Based on this fact and the
works by Schroder [117] and Islah [75], an analytic approximation for the CDF of the
SABR conditional process has been obtained. For some generic time interval [s, t ], 0 ≤
s < t ≤ T , assuming S(s) > 0, the conditional cumulative distribution
Rt for forward S(t )
with an absorbing boundary at S(t ) = 0, given σ(s), σ(t ) and s σ2 (z)dz, is given by
µ Z t ¶
Pr S(t ) ≤ K |S(s) > 0, σ(s), σ(t ), σ2 (z)dz = 1 − χ2 (a; b, c), (3.3)
s
where Ã !2
1 S(s)1−β ρ K 2(1−β)
a= + (σ(t ) − σ(s)) , c= ,
ν(t ) (1 − β) α (1 − β)2 ν(t )
1 − 2β − ρ 2 (1 − β)
Z t
b = 2− , ν(t ) = (1 − ρ 2 ) σ2 (z)dz,
(1 − β)(1 − ρ 2 ) s
2
and χ (x; δ, λ) is the non-central chi-square CDF.
This formula is exact in the case of ρ = 0 and constitutes an approximation otherwise

(because the CEV process is approximated by a shifted process with an approximated
initial condition for ρ 6= 0 in the derivation). Based on Equation (3.3), an approximately
exact simulation of SABR model is feasible by inverting the conditional SABR cumulative
distribution when the conditional time-integrated variance is known, see Section 3.2.1.
3.2.1. SABR M ONTE C ARLO SIMULATION

In order to apply an ‘almost exact’ multiple time-step Monte Carlo method for the SABR
3 model, several steps need to be performed, that are described in the following:
• Simulation of the SABR volatility process, σ(t ) given σ(s). By Equation (3.2), the
stochastic volatility process of the SABR model exhibits a log-normal distribution.
The solution is a GBM, i.e. the exact simulation of σ(t )|σ(s) reads
µ ¶
1
σ(t ) ∼ σ(s) exp αW̃σ (t ) − α2 (t − s) . (3.4)
2
Rt
• Simulation of the SABR time-integrated variance process, s σ2 (z)dz|σ(s), σ(t ). This
conditional distribution is not efficiently tractable. We will therefore derive an ap-
proximation of the conditional distribution of the SABR time-integrated variance
given σ(t ) and σ(s). The time-integrated variance sampling can be done by simply
inverting it.
• Simulation of the SABR forward price process. The forward price S(t ) can be sim-
ulated by inverting the CDF in Equation (3.3). By this, we avoid negative forward
prices in the simulation, as an absorbing boundary at zero is considered. There is
no analytic expression for the inverse distribution and therefore this inversion has
to be computed by means of some numerical approximation.
The multiple time-step Monte Carlo simulation for the SABR model is thus summa-
rized in Algorithm 2. Input parameters are the initial conditions (S 0 and σ0 ), the maturity
time, T , the SABR parameters α, β and ρ, the number of Monte Carlo samples (n) and
the number of time-steps (m).
Algorithm 2: Multiple time-step SABR Monte Carlo simulation.

Data: S 0 , σ0 , T, α, β, ρ, n, m.
S(0) = S 0 exp (r T ), σ(0) = σ0
for Path p = 1 . . . n do
for Step k = 1 . . . m do
T
s = (k − 1) m
T
t =km
Sampling σ(t ) given σ(s).
Rt
Sampling s σ2 (z)dz given σ(s) and σ(t ).
Rt
Sampling S(t ) given S(s), σ(s), σ(t ) and s σ2 (z)dz.
3.2. ‘A LMOST EXACT ’ SABR SIMULATION 47
Rt
In Section 3.3.4, we describe the procedure to sample s σ2 (z)dz|σ(s), σ(t ). This rep-
resents the main difference with respect to the technique presented in Chapter 2, where
the conditionality is solely on the final volatility (the initial volatility is constant). The
double condition implies that the marginal distributions that will be employed within
the copula are conditional distributions themselves. Although this is, conceptually, a
minor change, it has a drastic impact on the efficiency of the procedure, increasing the
computational cost.
As a copula approach is again employed, the CDF of the time-integrated variance
given the initial volatility, σ(s), (as a marginal distribution) must be derived. In Section
3.3.1, a recursive technique to obtain an approximation of this CDF is presented. Be- 3
cause
R t 2 we need to apply this recursion to approximate the ChF, the PDF and the CDF of
s σ (z)dz|σ(s) for each sample of σ(s) at each time-step, this multiple time-step ap-
proach is very expensive in terms of execution time. To overcome this drawback, an
efficient alternative will be employed here, based on Lagrange interpolation, as in the
SCMC sampler [65]. In Section 3.2.2, this methodology is briefly described.
3.2.2. S TOCHASTIC C OLLOCATION M ONTE C ARLO SAMPLER

In a multiple time-step exact simulation Monte Carlo method we need to perform a sig-
nificant number of computations each time-step on each Monte Carlo path. This often
hampers the applicability of the exact simulation for systems of SDEs. One of the impor-
tant components of the multiple time-step Monte Carlo SABR simulation is therefore an
accurate and highly efficient interpolation, as proposed in the SCMC sampler in [65].
In this section, we will introduce the SCMC technique. The details in the SABR con-
ditional time-integrated variance context will be presented in Section 3.3.2.
The SCMC technique relies on the property that a CDF of a distribution (if it exists)
is uniformly distributed. A well-known standard approach to sample from a given distri-
bution, Y , with CDF F Y , reads
d
F Y (Y ) = U thus y n = F Y−1 (u n ),
d
where = means equality in distributional sense, U ∼ U ([0, 1]) and u n are samples from
U ([0, 1]). The computational cost of this approach highly depends on the cost of the
inversion F Y−1 , which is assumed to be expensive.
We therefore consider another, ‘cheap’, random variable X , whose inversion, F X−1 , is
computationally much less expensive. In this framework, the following relation holds
d d
F Y (Y ) = U = F X (X ).
The samples y n of Y , and x n of X , are thus related via the following inversion,
y n = F Y−1 (F X (x n )).
However, this does not yet imply any improvement since the number of expensive
inversions F Y−1 remains the same. The goal is to compute y n by using a function g (·) =
F Y−1 (F X (·)), such that
d
F X (x) = F Y (g (x)) and Y = g (X ),
where evaluations of function g (·) do not require many inversions F Y−1 (·).
In [65], function g (·) is approximated by means of Lagrange interpolation, which is a
well-known interpolation also used in the Uncertainty Quantification (UQ) context. The
result is a polynomial, g NY (·), which approximates function g (·) = F Y−1 (F X (·)), and the
samples y n can be obtained by employing g NY (·) as
N Y N Y x n − x̄ j
y i ì (x n ), ì (x n ) =
X Y
y n ≈ g NY (x n ) = ,
i =1 j =1, j 6=i x̄ i − x̄ j
3 where x n is a vector of samples from X and x̄ i and x̄ j are so-called collocation points. NY
represents the number of collocation points and y i the exact inversion value of F Y at the
collocation point x̄ i , i.e. y i = F Y−1 (F X (x̄ i )). By applying this interpolation, the number
of inversions is reduced and, with only NY expensive inversions F Y−1 (F X (x̄ i )), we can
generate any number of samples by evaluating the polynomial g NY (x n ). This constitutes
the 1D version of SCMC sampler in [65].
According to the definition of the SCMC technique, a crucial aspect for the computa-
tional cost is parameter NY : the fewer collocation points are required, the more efficient
will be the sampling procedure. The collocation points must be optimally chosen in
a way to minimize their number. For any random variable X , the optimal collocation
points will be based on the moments of X . The optimal collocation points are here cho-
sen to be Gauss quadrature points that are defined as the zeros of the related orthogonal
polynomial. This approach leads to a stable interpolation under the probability distri-
bution of X . The complete procedure to compute the collocation points is described in
[65].
In Section
Rt 3.3.2, the application of the SCMC technique to generate samples from the
CDF of s σ2 (z)dz|σ(s) is presented. Since we deal with a conditional distribution, the
2D version of SCMC needs to be used.
3.3. C OMPONENTS OF THE M SABR METHOD

In this section, we will discuss the different components of the mSABR method. R t Again,
hereafter, we denote the SABR’s time-integrated variance process by Y (s, t ) := s σ2 (z)dz.
We will explain how to efficiently sample the time-integrated variance given the initial
and the final volatility, i.e. Y (s, t )|σ(t ), σ(s), as well as its use in a complete SABR simula-
tion.
In Section 3.3.4, we employ an accurate sampling method based on copula theory
[122], which is employed to approximate the required conditional distributions. The
copula relies on the availability of the marginal distributions to simulate the joint dis-
tribution. As the marginal distributions, Y (s, t )|σ(s) and σ(t )|σ(s) appear as the natural
choices. In Section 3.3.1, a procedure to recover the CDF of the time-integrated variance
process given the initial volatility is presented.
Rt
3.3.1. CDF OF s σ2 (z)dz|σ(s) USING THE COS METHOD
We present a technique to approximate the CDF of Y (s, t )|σ(s), i.e. F Y |σ(s) . We will work
in the log-space, so an approximated CDF of log Y (s, t )| log σ(s), F log Y | log σ(s) , will be de-
rived. We have employed this technique in the one time-step SABR method (Section 2.3),
3.3. C OMPONENTS OF THE M SABR METHOD 49
but the multiple time-step version is somewhat more involved. We approximate Y (s, t )
by its discrete equivalent, i.e.
Z t M
σ2 (z)dz ≈ ∆t σ2 (t j ) =: Ỹ (s, t ),
X
Y (s, t ) := (3.5)
s j =1
where M is the number of intermediate or discrete time-points, ∆t = tM −s

and t j = s+ j ∆t ,
j = 1, . . . , M . So, we now have m large time-steps, for the multiple time-step Monte Carlo
simulation, and M intermediate dates for the time-integrated variance, M >> m. We
suggest to prescribe ∆t , and use M = t∆t −s
, in order to avoid over-discretization issues.
3
This makes the value of M dependent on the interval size, t − s. Ỹ (T ) is subsequently
transformed to the logarithmic domain, with
Z x
F log Ỹ | log σ(s) (x) = f log Ỹ | log σ(s) (y)dy, (3.6)
−∞
and f log Ỹ | log σ(s) the PDF of log Ỹ (s, t )| log σ(s).
Density function f log Ỹ | log σ(s) is, in turn, recovered by approximating the associated
ChF, fˆ
log Ỹ | log σ(s) , and applying a Fourier inversion procedure.
R ECURSIVE PROCEDURE TO RECOVER fˆlog Ỹ | log σ(s)

We follow a similar procedure to approximate fˆ log Ỹ | log σ(s) as in Chapter 2. However, as
stated before, an extra computational issues will appear. We start again by defining the
sequence,
σ2 (t j )
Ã !
R j = log 2 , j = 1, . . . , M , (3.7)
σ (t j −1 )
where R j is the logarithmic increment of σ2 (t ) between t j and t j −1 , j = 1, . . . , M . As

the volatility process follows log-normal dynamics, and increments of Brownian motion
are independent and identically distributed, the R j are also independent and identically
d
distributed, i.e. R j = R. In addition, the ChF of R j is well-known and reads, ∀u, j ,
fˆR j (u) = fˆR (u) = exp(−i uα2 ∆t − 2u 2 α2 ∆t ), (3.8)

p
with α as in (3.1) and i = −1 the imaginary unit. By the definition of R j in Equation
(3.7), we write σ2 (t j ) as
σ2 (t j ) = σ2 (t 0 ) exp(R 1 + R 2 + · · · + R j ). (3.9)
At this point, a backward recursion procedure in terms of R j will be defined by which

we can recover fˆlog Ỹ | log σ(s) . We define
Y1 = R M , Y j = R M +1− j + Z j −1 , j = 2, . . . , M , (3.10)
with Z j = log(1 + exp(Y j )).

By Equations (3.9) and (3.10), the discrete time-integrated variance can be expressed
as
M
σ2 (t i )∆t = ∆t σ2 (s) exp(Y M ).
X
Ỹ (s, t ) = (3.11)
i =1
From Equation (3.11) and by applying the definition of ChF, we determine fˆlog Ỹ | log σ(s) ,
as follows
fˆlog Ỹ | log σ(s) (u) = E[exp(i u log Ỹ (s, t ))| log σ(s)]
3 = exp i u log(∆t σ2 (s)) E[exp (i uY M ) | log σ(s)]

¡ ¢
(3.12)
= exp i u log(∆t σ2 (s)) fˆYM (u).
¡ ¢
As Y M is defined recursively, its ChF can be obtained by a recursion as well. In Sec-

tions 2.3.1 and 2.3.2 of Chapter 2, we have already proposed a procedure to efficiently ap-
proximate fˆYM . The methodology is based on a Fourier cosine series expansion and the
use of a numerical piecewise linear approach in the approximation of a computational
expensive integral (Equation (2.20)). The experiments presented in Chapter 2 have em-
pirically shown the good balance between accuracy and computational effort provided
by this approach.
R ECOVERING f log Ỹ | log σ(s) BY COS METHOD

Once the approximation of fˆYM , denoted by fˆY∗M , has been efficiently derived, we can
recover f from fˆ
log Ỹ | log σ(s) by employing the COS method [49], as follows
log Ỹ | log σ(s)
2 −10
NX µ
kπ
¶
f log Ỹ | log σ(s) (x) ≈ C k cos (x − ã) , (3.13)
b̃ − ã k=0 b̃ − ã
with
kπ ãkπ
µ µ ¶ µ ¶¶
C k = ℜ fˆlog Ỹ | log σ(s) exp −i ,
b̃ − ã b̃ − ã
and
kπ kπ kπ
µ ¶ µ ¶ µ ¶
fˆlog Ỹ | log σ(s) log ∆t σ2 (s) fˆYM
¡ ¢
= exp i
b̃ − ã b̃ − ã b̃ − ã
kπ kπ
µ ¶ µ ¶
log ∆t σ2 (s) fˆY∗M
¡ ¢
≈ exp i ,
b̃ − ã b̃ − ã
where N is the number of COS terms, [ã, b̃] is the support2 of log Ỹ (s, t )| log σ(s) and the
prime 0 and ℜ symbols in Equation (3.13) mean division of the first term in the summa-
tion by two and taking the real part of the complex-valued expressions in the brackets, re-
spectively. CDF F log Ỹ | log σ(s) can be obtained by integration, plugging the approximated
f log Ỹ | log σ(s) from Equation (3.13) into Equation (3.6).
The ChF fˆ and PDF f
log Ỹ | log σ(s) need to be derived for each sample of
log Ỹ | log σ(s)
log σ(s) and this makes the computation of F log Ỹ | log σ(s) (and also its subsequent inver-
sion) very expensive. In order to overcome this issue, the SCMC technique (see Section
3.2.2) will be employed approximating these rather expensive computations by means
of an accurate interpolation.
2 It can be calculated given the support of Y
M , [a, b].
3.3.2. E FFICIENT SAMPLING OF log Ỹ | log σ(s)

By employing the SCMC technique, instead of directly computing F log Ỹ | log σ(s) for each
sample of log σ(s), we only have to compute F log Ỹ | log σ(s) at the collocation points. In
general, only a few collocation points are sufficient to obtain accurate approximations,
which is a well-known fact from the UQ research field. This fact allows us to drastically
reduce the computational cost of sampling the required distribution.
For the problem at hand, we require samples from the time-integrated variance con-
ditional on the initial volatility, log Ỹ (s, t )| log σ(s). Therefore, we need to make use of the
2D version of the SCMC technique. Two levels of collocation points need to be chosen,
one for each dimension. If we denote them by NỸ and Nσ , respectively, the resulting 3
number of inversions equals NỸ · Nσ . The formal definition of the 2D SCMC technique
applied in our context reads
N Nσ
Ỹ X
−1
X
y n |v n ≈ g NỸ ,Nσ (x n ) = F log Ỹ | log σ(s)=v̄
(F X (x̄ i ))ì (x n )` j (v n ), (3.14)
j
i =1 j =1
where x n are the samples from X ∼ N (0, 1), which is used as the cheap variable, and v n
the samples of log σ(s); x̄ i and v̄ j are the collocation points for approximating variables
log Y and log σ(s), respectively. The Lagrange polynomials ì and ` j are defined by
N Nσ
Ỹ x n − x̄ k v n − v̄ k
ì (x n ) = , ` j (v n ) =
Y Y
.
k=1,k6=i x̄ i − x̄ k k=1,k6= j v̄ i − v̄ k
The collocation points x̄ i and v̄ j are calculated based on the moments of the ran-
dom variables. The two involved variables in the application of the 2D SCMC technique
p
are the collocation variables, X ∼ N (0, 1), and log σ(s) ∼ N (log σ0 + 12 α2 s, α s). The
moments of a normal variable, X ∼ N (µ X , s X ) are analytically available, and are given
by (
£ p¤ 0 if p is odd,
E X = p
s X (p − 1)!! if p is even,
where p is the moment order and the expression !! represents the double factorial. In
order to compute the optimal collocation points, v̄ j , the precalculated collocation points
by means of the log σ(s) moments need to be shifted according to the mean, i.e. log σ0 +
1 2
2 α s.
We first test numerically the accuracy of the SCMC technique for our problem. In
Figure 3.1, two experiments are presented. In Figure 3.1(a), we compare the samples ob-
tained by direct inversion and by the SCMC technique. The fit is highly accurate, already
with only seven collocation points. In Figure 3.1(b), the empirical CDFs are depicted,
where the excellent match is confirmed.
In order to show the performance of the SCMC technique for the simulation of the
distribution of log Ỹ | log σ(s), the execution times of generating different numbers of
samples are presented in Table 3.1.
3.3.3. E RROR ANALYSIS

We have proposed a method to compute the approximated CDF for the conditional time-
integrated variance, i.e. Y (s, t )|σ(s). In this section, the different sources of error due to
0 1
DI Without SCMC
SCMC With SCMC
-0.5
0.8
-1
Flog Ŷ | log σ(s) (x)

0.6
-1.5
log Ŷ
-2
0.4
-2.5
0.2
-3
3 -3.5 0
-2 -1.5 -1 -0.5 -5 -4 -3 -2 -1 0 1 2
log σ(s) x
(a) log Ỹ | log σ(s) - Direct inversion vs. SCMC. (b) F log Ỹ | log σ(s) (x).
Figure 3.1: (a) Points cloud log Ỹ | log σ(s) by direct inversion (DI) sampling (red dots) and SCMC sampling
(black circles). (b) Empirical CDFs of log Y | log σ(s) with and without using SCMC.
Samples Without SCMC With SCMC

NỸ = Nσ = 3 NỸ = Nσ = 7 NỸ = Nσ = 11
100 1.0695 0.0449 0.0466 0.0660
10000 16.3483 0.0518 0.0588 0.0798
1000000 1624.3019 0.2648 0.5882 1.0940
Table 3.1: SCMC time in seconds.
the approximations made in the Sections 3.3.1 and 3.3.2 are analysed, indicating how
to reduce or bound the errors. The sources of error in the approximation include ²D ,
the discretization of Y (s, t ) in Equation (3.5), ²T , the truncation in Equations (2.16) and
(3.13), ²C , due to the cosine expansion in Equations (2.16) and (3.13), ²M , the numerical
computation of I M in Equation (2.23) and ²S due to the application of SCMC in Equation
(3.14). The errors ²T , ²C and ²M were already studied in Section 2.3.3 of Chapter 2. The
error ²D is updated according to the multiple time-step version.
Error ²D . This error occurs because SABR’s time-integrated variance is approximated Rt

by its discrete equivalent, i.e. Y (s, t ) ≈ Ỹ (s, t ). Note that the process Y (s, t ) = s σ2 (z)dz
is the solution of the SDE
dY (s, z) = σ2 (z)dz, Y (s, s) = 0,
which, by employing the same time discretization as for Ỹ (s, t ) in Equation (3.5), can be
written as
Ȳ (s, t j ) − Ȳ (s, t j −1 ) = σ2 (t j −1 )∆t .
This is again the Euler-Maruyama discretization scheme. Indeed, it is easy to see

that, if we sum the right-hand side of the equality over the M discrete time points, we
recover the discrete approximation Ỹ (s, t ), as follows

M M ¡
σ2 (t j −1 )∆t =
X X ¢
Ȳ (s, t j ) − Ȳ (s, t j −1 ) = Ȳ (s, t M ) − Ȳ (s, t 0 ) = Ỹ (s, t ).
j =1 j =1
Then, error ²D can be computed and bounded by employing the usual strong con-
vergence expression as
t −s γ
µ ¶
²D := E Ỹ (s, t ) − Y (s, t ) ≤ C
£¯ ¯¤
¯ ¯ ,
M 3
for some constant C > 0.
In the actual experiments, we consider log Ỹ (s, t ) and log Y (s, t ). This will reduce the
error further.
Error ²S . This error appears in the application of the interpolation-based SCMC tech-
nique to get samples of log Ỹ | log σ(s). In [65], where the SCMC sampler was proposed,
an error analysis of the technique was carried out. Here we summarize the most impor-
tant results related to our context and refer the reader to the original paper for further
details.
To measure the error which results from the collocation method in a more general
case (see Section 3.2.2), we can consider either the difference between functions g (X )
and g N (X ) (X the cheap variable) or the error associated with the approximated CDF.
The first type of error is due to Lagrange interpolation, so the corresponding error esti-
mate is well-known. The error ²(ξn ) with NY collocation points given by the standard
Lagrange interpolation reads
¯ ¯
¯ ¯¯ 1 ∂NY g (x) ¯¯ N Y ¯
² X (ξn ) = g (ξn ) − g NY (ξn ) = ¯
¯ Y
(ξ − x̄ ) ¯, (3.15)
¯
n i
¯ ¯
¯ NY ! ∂x NY x=ξ i =1
¯
¯
where x̄ i are the collocation points, and ξ̃ ∈ [min(x), max(x)], x = (x̄ 1 , . . . , x̄ NỸ )T . The error
can be bounded by defining ξ as the point where ¯∂NY g (x)/∂x NY ¯ has its maximum. A
¯ ¯
small probability of large errors in the tails can be observed by deriving the error ²U (ξn ),
by substituting the uniformly distributed random variable u n in the previous equation,
using ξn = F X−1 (u n ),
¯ ¯
¯ ¯¯ 1 ∂NY g (x) ¯¯ N Y ¯
−1 −1 −1
²U (u n ) = ¯g (F X (u n )) − g NY (F X (u n ))¯ = ¯
¯ Y
(F (u ) − x̄ ) ¯.
¯
X n i
¯ NY ! ∂x NY x=ξ i =1
¯
¯
A “close-to-linear” relation between the involved stochastic variables, meaning that

∂NY g (x)/∂x NY is small, gives a small approximation error. On the other hand, when ap-
proximating the CDF of the expensive variable Y , we have F Y (y) = F Y (g (x)) ≈ F Y (g NY (x)),
which is exact at the collocation points x̄ i ,
F Y (y i ) = F Y (g (x̄ i )) = F Y (g NY (x̄ i )).
A deeper analysis can be found in the original paper of SCMC [65], including a theo-
retical convergence in L 2 .
We have chosen X to be standard normal, and Y = log Ỹ | log σ(s) can be seen as the
logarithm of a sum of log-normal variables. Therefore, it is a “close-to-normal” distri-
bution which can be efficiently approximated by a linear combination of standard nor-
malR tdistributions. In order to measure the error when SCMC is applied to the sampling
of s σ2 (z)dz|σ(s) (see Section 3.3.2), we can consider the difference between the func-
tions g (x) = F −1 (F X (x)) and g NỸ ,Nσ (x) in Equation (3.14). By following the same
log Ỹ | log σ(s)
derivation as in Equation (3.15), we have
N
¯ ¯
¯ ¯¯ 1 ∂NỸ g (x) ¯¯ Nσ
3 Ỹ ¯
²S := ¯g (ξn ) − g N
¯ Y Y
(ξ ) = (ξ − x̄ ) (v − v̄ ) ¯,
¯
,N n n i n j
¯
σ
Ỹ ¯ NỸ ! ∂x NỸ x=ξ i =1
¯ ¯
j =1
¯
where NỸ and Nσ are the number of collocation points in each dimension and v̄ j are the
collocation points corresponding to the conditional random variable log σ(s).
Rt
3.3.4. C OPULA - BASED SIMULATION OF s σ2 (z)dz|σ(t ), σ(s)
In this section we present the algorithm for the simulation of the time-integrated vari-
ance given σ(t ) and σ(s) by means of a copula. In order to obtain the joint distribution,
we require the marginal distributions. In our case, the required CDFs are F log Y | log σ(s)
and F log σ(t )| log σ(s) . In Section 3.3.1, an approximated CDF of log Y | log σ(s), F log Ỹ | log σ(s) ,
has been derived. Since σ(t ) follows a log-normal distribution, by definition, log σ(t ) is
normally distributed, and the conditional process log σ(t ) given log σ(s), follows a nor-
mal distribution,
p
µ ¶
1
log σ(t )| log σ(s) ∼ N log σ(s) − α2 (t − s), α t − s . (3.16)
2
P EARSON ’ S CORRELATION COEFFICIENT

For any copula some measure of the correlation between the marginal distributions is
needed. We will employ the Pearson correlation coefficient for log Y (s, t ) and log σ(t ).
For this quantity, an approximated analytic formula can be derived. By definition, we
have
£ Rt
cov log s σ2 (z)dz, log σ(t )
· Z t ¸ ¤
2
P log Y ,log σ(t ) = corr log σ (z)dz, log σ(t ) = q £ Rt ¤.
s
Var log s σ2 (z)dz Var log σ(t )
¤ £
We employ the following approximation

Z t Z t Z t
log σ2 (z)dz ≈ log σ2 (z)dz = 2 log σ(z)dz,
s s s
where the logarithm and the integral are interchanged. Since the log function is concave,
this approximation forms a lower bound for the true value. This can be seen by applying
Jensen’s inequality, i.e.
Z t Z t
log σ2 (z)dz ≥ log σ2 (z)dz.
s s
It has been numerically shown that, under the model settings with an interval size t − s
less than two years, the results based on this approximation are highly satisfactory (fur-
ther details in Chapter 2, Section 2.4.2). The correlation coefficient can then be approxi-
mated by
£R t
cov s log σ(z)dz, log σ(t )
¤
P log Y (T ),log σ(t ) ≈ q £R t ¤.
Var s log σ(z)dz Var log σ(t )
¤ £
£¡R t ¤ £R t
the covariance, we need E s log σ(z)dz log σ(t ) , E £s log σ(z)dz 3
¢ ¤
In order
£ to compute
and E log σ(t ) . From Equation (3.4), the last expectation as well as Var log σ(t ) are
¤ ¤
known. We find that
1
log σ(t ) = log σ0 − α2 t + αW (t ) =: η(t ) + αW (t ),
2
and
Z t 1
Z t
log σ(z)dz = log σ0 (t − s) − α2 (t 2 − s 2 ) + α W (z)dz
s 4 s
µ Z t ¶
=: γ(s, t ) + α tW (t ) − sW (s) − zdW (z) .
s
Based on these equations, using

·Z t ¸
E log σ(T ) = η(t ), E log σ(z)dz = γ(s, t ),
£ ¤
s
we obtain the following expressions for the remaining expectation,

·µZ t ¶ ¸ ·µ µ Z t ¶¶ ¸
E log σ(z)dz log σ(t ) = E γ(s, t ) + α tW (t ) − sW (s) − zdW (z) η(t ) + αW (t )
¡ ¢
s s
µ ·Z t ¸¶
= η(t )γ(s, t ) + α2 t 2 − s 2 − E zdW 2 (z)
s
1
= η(t )γ(s, t ) + α2 (t 2 − s 2 ).
2
Rt
For the variance of s log σ(z)dz, we first compute
"µZ ¶2 # "µ ¶¶2 #
t µ Z t
E log σ(z)dz = E γ(s, t ) + α tW (t ) − sW (s) − zdW (z)
s s
"µZ ¶2 #
t
= γ (s, t ) + α E (tW (t ) − sW (s))2 + α2 E
2 2
£ ¤
zdW (z)
s
· Z t ¸
− 2α2 E (tW (t ) − sW (s)) zdW (z) ,
s
where
E (tW (t ) − sW (s))2 = t 2 E (W (t ))2 + s 2 E (W (s))2 − 2st E [W (s)W (t )]
£ ¤ £ ¤ £ ¤
= t 3 + s 3 − 2s 2 t ,
"µZ ¶2 #
t Z t
1
E zdW (z) = z 2 dz = (t 3 − s 3 ),
s s 3
· Z t ¸ · Z t ¸ · Z t ¸
E (tW (t ) − sW (s)) zdW (z) = t E W (t ) zdW (z) − sE W (s) zdW (z)
s s s
3 ·Z t Z t ¸ ·Z s Z t ¸
= tE dW (z) zdW (z) − sE dW (z) zdW (z)
s s s s
1
= t (t 2 − s 2 ),
2
and the variance reads
·Z t "µZ ¶2 # µ ·Z t ¸¶2
¸ t
Var log σ(z)dz = E log σ(z)dz − E log σ(z)dz
s s s
µ ¶
1
2 3 3 2
= α t + s − 2s t + (t 3 − s 3 ) − t (t 2 − s 2 )
3
µ ¶
1 2
= α2 t 3 + s 3 − s 2 t .
3 3
An approximation of the Pearson correlation coefficient is then obtained as
η(t )γ(s, t ) + 21 α2 (t 2 − s 2 ) − η(t )γ(s, t )
P log Y ,log σ(t ) ≈ q ¡
α2 31 t 3 + 23 s 3 − s 2 t α2 t
¢¡ ¢
(3.17)
t 2 − s2
= q¡
1 4 2 3
¢.
2 t + t s − t 2s2
3 3
Some copulas do not employ, in their definition, Pearson’s coefficient as the correla-
tion measure. For example, Archimedean copulas employ the Kendall’s τ rank correla-
tion coefficient. However, a relation between both Pearson’s and Kendall’s τ coefficients
is available, ³π ´
P = sin τ .
2
G UMBEL COPULA
In this chapter, we will use Archimedean Gumbel copula. The Gaussian copula Rt may
seem a natural choice, as σ(t ) follows a log-normal distribution and Y (T ) = s σ2 (z)dz
can be seen as summation of squared log-normal processes. However, in Section 2.5.1,
we have empirically shown that the Gumbel copula performs most accurately in this
context for a variety of SABR parameter values. The formal definition of the Archimedean
Gumbel copula, considering F 1 . . . , F d ∈ [0, 1]d as the marginal distributions, reads
Ã Ã !1/θ !
d ¡
X ¢θ
C θ (F 1 . . . , F d ) = exp − − log(F i ) , (3.18)
i =1
¢θ
where − log(·) , θ > 0 is the so-called generator function of the Gumbel copula. In or-
¡
der to calibrate parameter θ, a relation between θ and the rank correlation coefficient
Kendall’s τ can be employed, i.e. θ = 1/(1 − τ).
So, we have derived approximations for all the components required to apply the
copula-based
Rt technique for the time-integrated variance simulation. The algorithm to
sample s σ2 (z)dz given σ(t ) and σ(s) then consists of the following steps:
1. Determine F log σ(t )| log σ(s) by Equation (3.16).
2. Determine F log Ỹ | log σ(s) by Equation (3.6). 3

3. Determine the correlation between log Y (s, t ) and log σ(t ) by Equation (3.17).
4. Generate correlated uniforms, Ulog σ(t )| log σ(s) and Ulog Ỹ | log σ(s) from the Gumbel
copula by Equation (3.18).
5. From Ulog σ(t )| log σ(s) , invert F log σ(t )| log σ(s) to get the samples σ̄n of log σ(t )| log σ(s).
This procedure is straightforward since the normal distribution inversion is ana-
lytically available.
6. From Ulog Ỹ | log σ(s) , invert F log Ỹ | log σ(s) to get the samples ȳ n of log Ỹ | log σ(s). We
propose an inversion procedure based on linear interpolation. First we evalu-
ate the CDF function at some discrete points. Then, the insight is that, by ro-
tating the CDF under consideration, we can interpolate over probabilities. This
is possible when the CDF function is a smoothly increasing function. The inter-
polation polynomial provides the quantiles of the original distribution from some
given probabilities. Since F log Ỹ | log σ(s) is indeed a smooth and increasing function,
the interpolation-based inversion is definitely applicable. This procedure together
with the use of 2D SCMC sampler (see Section 3.3.2) results in a fast and accurate
inversion.
Rt
7. The samples σn of σ(t )|σ(s) and y n of Y (s, t ) = s σ2 (z)dz|σ(t ), σ(s) are obtained
by simply taking exponentials as
σn = exp(σ̄n ), y n = exp( ȳ n ).
Rt
3.3.5. S IMULATION OF S(t ) GIVEN S(s), σ(s), σ(t ) AND s σ2 (z)dz
Once the samples from the time-integrated variance are obtained, the simulation of the
conditional asset dynamics represents the final step of the mSABR method. For that, we
follow the same approach as in Section 2.4.4 of the previous chapter, distinguishing the
cases β = 0, β = 1 and β 6= 0, β 6= 1.
However, as the mSABR method is a multiple time-step scheme, a generic interval
[s, t ] (instead of the fixed interval [0, T ]) is employed and some extra considerations must
be taken into account.
When β 6= 1, i.e. the S(t ) process does not follow log-normal dynamics, and negative
asset prices can be obtained. This contradicts one of the main assumptions for the va-
lidity of the CDF expression for the ‘exact’ SABR simulation in Equation (3.3). Therefore,
in the case of β 6= 1, the simulation of S(t ) must be corrected to be consistent within the
S0 σ0 α β ρ T
Set I [65] 0.5 0.5 0.4 0.5 0.0 4
Set II [21] 0.04 0.2 0.3 1.0 −0.5 5
Set III [4] 1.0 0.25 0.3 0.6 −0.5 20
Set IV [5] 0.0056 0.011 1.080 0.167 0.999 1
Table 3.2: Data sets.
3 whole simulation procedure. If the price at the initial time, s, is less or equal to zero, i.e.
S(s) ≤ 0, the price S(t ) must remain zero for all time. Otherwise, the absorbing boundary
at zero needs to be prescribed to avoid negative prices as
S(t ) = max(S(t ), 0).
If negative values are permitted, this must be handled in a different way, as it has
been recently studied in [68] or [5], for example.
As was also mentioned in Section 2.4.4, a loss of the martingale property can appear
when a continuous process is approximated by its discrete equivalent, especially if the
discretization steps are large. In the case of the SABR model, this issue appears in the
case of small β and/or S 0 values and high α and/or σ0 . Note however that, in most
situations appearing in practice, this martingale error is very small and can be easily
controlled by an increasing number of time-steps.
Since our goal is to use as few time-steps as possible, the application of a martingale
correction becomes again essential and needs to be considered in each time-step of the
simulation. We use the same martingale correction as proposed in Chapter 2.
3.4. N UMERICAL EXPERIMENTS

In this section, we benchmark the mSABR method by pricing options under the SABR dy-
namics. We will consider several parameter configurations, extracted from the literature
(see Table 3.2), a zero-correlation set as in [65] (set I), a set under log-normal dynamics
as in [21] (set II), a medium-long time to maturity set as in [4] (set III) and a more extreme
set as in [5] (set IV). The strikes, K i , are chosen following the expression
p
K i (T ) = S(0) exp(0.1 × T × δi ), δ = {−1.5, −1.0, −0.5, 0.0, 0.5, 1.0, 1.5},
introduced by Piterbarg in [105].

Other parameters include:
• Step size in Equation (3.5): ∆t = 5 × 10−5 .
• Number of terms in Equations (2.16) and (3.13): N = 150.
• Number of intervals for piecewise linear approximation in Equation (2.23): L =

100.
• Number of collocation points in Equation (3.14): NỸ = Nσ = 3.

Antonov 73.34% 71.73% 70.17% N/A 67.23% 65.87% 64.59%
m = T /4 73.13% 71.75% 70.41% 69.11% 67.85% 66.64% 65.48%
Error(bp) −21.51 2.54 24.38 N/A 61.71 76.66 89.26
m = T /2 73.30% 71.78% 70.29% 68.86% 67.49% 66.17% 64.93%
Error(bp) −4.12 4.94 12.71 N/A 25.48 30.40 34.73
m=T 73.25% 71.67% 70.14% 68.66% 67.24% 65.89% 64.62%
Error(bp) −9.56 −5.93 −2.79 N/A 0.92 2.21 3.17
m = 2T
Error(bp)
73.32%
−2.08
71.71%
−1.56
70.16%
−1.20
68.65%
N/A
67.22%
−1.65
65.85%
−2.35
64.55%
−3.36
3
m = 4T 73.34% 71.73% 70.18% 68.67% 67.24% 65.87% 64.58%
Error(bp) 0.15 0.58 0.78 N/A 0.43 0.04 −0.48
Table 3.3: Implied volatility, increasing m: Antonov vs. mSABR. Set I.
• Number of samples: n = 1, 000, 000.
The experiments were performed on a computer with CPU Intel Core i7-4720HQ 2.6 GHz
and RAM memory of 16 Gigabytes. The employed software package was Matlab r2016b.
As usual, the European option prices are obtained by averaging the maximum of the
forward asset values at maturity minus the strike and zero, i.e. max(S(T ) − K i (T ), 0) (call
option). Subsequently, the Black-Scholes formula is inverted to determine the implied
volatilities. Note that, once the forward asset is simulated by the mSABR method, we can
price options at multiple strikes simultaneously.
3.4.1. C ONVERGENCE TEST

We perform a convergence study in terms of the number of time-steps, m and, subse-
quently, in terms of the number of samples, n. Set I is considered suitable for this exper-
iment, since an analytic solution is available by Antonov et al. [4] when ρ = 0, and it can
be used as a reference. Also, as Equation (3.3) is employed to sample the forward asset
dynamics, the case ρ = 0 results in an exact simulation.
In Table 3.3, the implied volatilities obtained by the mSABR method for several choices
of the number of time-steps m are presented. We observe by the errors in basis points
that, by increasing the number of time-steps, the technique converges very rapidly. Fur-
thermore, only a few time-steps are required to obtain accurate results: with only m = 4
time-steps (one per year) the error is already less than ten basis points and with m = 4T
(the time-step size is a quarter of year), the error remains below one basis point. Com-
pared to the low-bias SABR simulation scheme proposed in [21], the mSABR method is
more accurate, since it relies on the exact simulation of the time-integrated variance,
whereas the low-bias scheme approximates the time-integrated variance by a small dis-
turbance expansion and moment-matching techniques. That scheme performs worse
when bigger time-steps are considered.
We also perform a convergence test in the number of samples, n. According to the
previous experiment and in order to guarantee that the errors do not increase by the time
discretization, we set m = 4T . In Table 3.4, the implied volatilities for increasing n are
Antonov 73.34% 71.73% 70.17% N/A 67.23% 65.87% 64.59%
n = 102 67.29% 65.55% 63.84% 62.20% 60.63% 59.01% 57.65%
RE 8.24 × 10−2 8.61 × 10−2 9.01 × 10−2 N/A 9.82 × 10−2 1.04 × 10−1 1.07 × 10−1
n = 104 73.41% 71.87% 70.36% 68.91% 67.51% 66.19% 64.94%
RE 9.65 × 10−4 1.94 × 10−3 2.75 × 10−3 N/A 4.08 × 10−3 4.93 × 10−3 5.48 × 10−3
n = 106 73.34% 71.73% 70.18% 68.67% 67.24% 65.87% 64.58%
RE 2.04 × 10−5 8.08 × 10−5 1.11 × 10−4 N/A 6.39 × 10−5 6.07 × 10−6 7.43 × 10−5
Table 3.4: Implied volatility, increasing n: Antonov vs. mSABR. Set I.

3
0.9 0.8
Antonov Antonov
mSABR mSABR
Implied Volatility
0.8
Implied Volatility
0.7
0.7
0.6
0.5
0.4 0.6
10 2 10 3 10 4 10 5 10 6 10 2 10 3 10 4 10 5 10 6
n n
(a) Strike K 2 . (b) Strike K 6 .
Figure 3.2: Convergence of the mSABR method.
presented. By the relative error (RE), we can see that, as expected, the error is approx-
p
imately reduced by a factor of 1/ n. Furthermore, we wish to show how the number
of samples affects the variance of the mSABR method. In Figure 3.2, the standard devia-
tions3 for several choices of n are presented. Two strikes are evaluated, one in-the-money
strike, K 2 , and one out-of-the-money strike4 , K 6 . An identical behaviour can be observed
for all other strikes as well. Again, we see a fast convergence and variance reduction in
terms of n.
3.4.2. P ERFORMANCE TEST

Next to the convergence of the mSABR method in terms of m and n, another impor-
tant aspect is the computational cost. Specifically, we will measure the execution times
of an option pricing experiment using different alternative Monte Carlo-based meth-
ods and compare them with the mSABR method. Again, parameter Set I is employed
since a reference value is available for this case. We consider a plain Monte Carlo Euler-
Maruyama (MC Euler) discretization scheme for the forward asset, S(t ). Note that the
absorbing boundary at zero must be handled with care. Since our main contribution
here is the exact simulation of the time-integrated variance, it is natural to also com-
3 Computed by performing 100 realizations.

4 We consider European call options.
Error < 100 bp < 50 bp < 25 bp < 10 bp

MC Euler 6.85(200) 10.71(300) 27.42(800) 42.90(1200)
Y -Euler 2.18(4) 6.55(16) 11.85(32) 45.12(128)
Y -trpz 2.17(3) 4.24(8) 7.25(16) 14.47(32)
mSABR 3.46(1) 2.98(2) 3.72(3) 4.89(4)
Table 3.5: Execution times and time-steps, m (parentheses).
Error < 100 bp < 50 bp < 25 bp < 10 bp

MC Euler 1.98 3.59 7.37 8.77
3
Y -Euler 0.63 2.19 3.18 9.22
Y -trpz 0.62 1.42 1.94 2.95
Table 3.6: Speedups provided by the mSABR method.
pare the performance with other, less involved, techniques. We therefore also present a
comparison between the mSABR method R t and two techniques where the time-integrated
variance is either approximated by s σ2 (z)dz ≈ σ2 (s)(t − s) (left rectangular rule) or
Rt 2 1 2 2
s σ (z)dz ≈ 2 (σ (t ) + σ (s))(t − s) (trapezoidal rule). We denote these alternatives as
the Y -Euler and Y -trpz schemes, respectively. The simulation of S(t ) is, in both cases,
carried out as explained in Section 3.3.5. In Table 3.5, we present execution times (in sec-
onds) of the MC Euler, Y -Euler, Y -trpz and mSABR schemes, to achieve a certain level of
accuracy (measured as the error of the implied volatilities in basis points). In addition,
the required number of time-steps is presented in parentheses to provide an insight in
the efficiency. Furthermore, the speedup obtained by our method is included in Table
3.6. We observe that the mSABR method is superior in terms of the ratio between com-
putational cost and accuracy. The accuracy of Y -Euler or Y -trpz can be improved by
adding intermediate time-steps, but then also the computational cost increases.
3.4.3. ‘A LMOST EXACT ’ SABR SIMULATION BY VARYING ρ

As a next experiment we test the stability of the mSABR method when correlation pa-
rameter ρ varies from negative to positive values. As stated before, Equation (3.3) is only
exact for ρ = 0. For that, we use Set II since the conditional forward asset simulation
enables a closed-form solution (see Section 2.4.4) for β = 1. The considered values are
ρ = {−0.5, 0.0, 0.5}. Following the outcome of the previous experiment, we set m = 4T .
As a reference, a Monte Carlo (MC) simulation with a very fine Euler-Maruyama time
discretization (1000T time-steps) in the forward asset process is employed. In Table 3.7,
the resulting implied volatilities are provided. The differences between the Monte Carlo
volatilities and the ones given by the mSABR method are within ten basis points for all
choices of ρ.
3.4.4. P RICING BARRIER OPTIONS

We will price barrier options with the mSABR method. The up-and-out call option is
considered here, with the barrier level, B , B > S 0 , B > K i . The price of this type of barrier
ρ = −0.5
MC 22.17% 21.25% 20.38% 19.57% 18.88% 18.33% 17.95%
mSABR 22.21% 21.28% 20.39% 19.58% 18.88% 18.32% 17.94%
Error(bp) 3.59 2.86 1.78 0.95 −0.19 −0.96 −1.10
ρ = 0.0
MC 21.35% 20.96% 20.71% 20.63% 20.71% 20.96% 21.34%
mSABR 21.35% 20.95% 20.69% 20.60% 20.68% 20.93% 21.32%
Error(bp) 0.04 −1.04 −2.51 −3.02 −3.33 −3.19 −2.56
3 ρ = 0.5
MC 19.66% 20.04% 20.61% 21.34% 22.20% 23.14% 24.16%
mSABR 19.59% 19.96% 20.54% 21.28% 22.15% 23.11% 24.11%
Error(bp) −6.93 −7.36 −6.77 −5.53 −4.35 −3.76 −4.05
Table 3.7: Implied volatility, varying ρ: Euler Monte Carlo vs. mSABR. Set II.
option reads
· ¸
Vi (T, S, B ) = exp (−r T ) E (S(T ) − K i )1( max S(t k ) > B ) ,
0<t k ≤T
where 1(·) represents the indicator function and t k are the predefined times where the
barrier condition (whether or not it is hit) is checked.
As a reference, a Monte Carlo method with very fine Euler-Maruyama time discretiza-
tion is employed, as in the previous experiment. The number of time-steps for the mSABR
scheme is again set to m = 4T . In Tables 3.8, 3.9 and 3.10, the obtained prices are pre-
sented for the parameter sets I, II and III, respectively. The prices are multiplied by a
factor of 100. Also, the MSE is shown. We define the MSE as
1X 7 ¡ ¢2
MSE = V MC (T, S, B ) − VimS AB R (T, S, B ) ,
7 i =1 i
where ViMC (T, S, B ) and VimS AB R (T, S, B ) are the barrier option prices provided by Monte
Carlo method and by the mSABR method, respectively.
The resulting accuracy is very satisfactory. As expected, higher option prices are ob-
tained for bigger barrier levels, B , and/or lower strikes, K i .
3.4.5. N EGATIVE INTEREST RATES

The SABR model is very popular in the interest rate context. One of the most impor-
tant current features in this market is the occurence of negative interest rate values and
strikes. Approaches dealing with this issue have appeared in the literature, like [5] or [68].
To apply the mSABR scheme in the case of negative interest rates, we will use the shifted
SABR model by Schlenkrich in [116]. The shifted SABR model is defined as follows
dS(t ) = σ(t )(S(t ) + θ)β dWS (t ),

S(0) = (S 0 + θ) exp (r T ) ,
B = 1.2
MC 6.2868 5.4577 4.6330 3.8263 3.0551 2.3358 1.6869
mSABR 6.3264 5.4890 4.6566 3.8436 3.0677 2.3452 1.6953
MSE 5.3196 × 10−8
B = 1.5 3
MC 10.2271 9.1568 8.0718 6.9856 5.9142 4.8753 3.8899
mSABR 10.2249 9.1507 8.0609 6.9702 5.8960 4.8572 3.8737
MSE 1.8884 × 10−8
B = 1.8
MC 13.5740 12.3571 11.1118 9.8513 8.5910 7.3477 6.1403
mSABR 13.5915 12.3686 11.1174 9.8507 8.5851 7.3380 6.1276
MSE 1.0851 × 10−8
Table 3.8: Pricing barrier options with mSABR: Vi (T, S, B ) × 100. Set I.
B = 0.08
MC 1.1702 0.9465 0.7268 0.5215 0.3423 0.1996 0.0987
mSABR 1.1724 0.9486 0.7285 0.5226 0.3428 0.1997 0.0986
MSE 1.8910 × 10−10
B = 0.1
MC 1.3099 1.0766 0.8462 0.6290 0.4367 0.2794 0.1626
mSABR 1.3092 1.0761 0.8456 0.6282 0.4355 0.2782 0.1618
MSE 7.5542 × 10−11
B = 0.12
MC 1.3521 1.1168 0.8841 0.6644 0.4695 0.3093 0.1891
mSABR 1.3518 1.1166 0.8838 0.6639 0.4686 0.3080 0.1880
MSE 6.3648 × 10−11
Table 3.9: Pricing barrier options with mSABR: Vi (T, S, B ) × 100. Set II.
B = 2.0
MC 29.1174 23.4804 17.2273 10.7825 5.0203 1.1750 0.0036
mSABR 29.2346 23.5828 17.3086 10.8327 5.0385 1.1805 0.0036
MSE 4.8146 × 10−7
B = 2.5
MC 41.3833 34.5497 26.8311 18.6089 10.7281 4.4893 0.9434
mSABR 41.3394 34.5097 26.7948 18.5747 10.6943 4.4546 0.9320
MSE 1.2131 × 10−7
3 B = 3.0
MC 48.5254 41.1652 32.7980 23.7807 14.9344 7.5364 2.6692
mSABR 48.5008 41.1515 32.7888 23.7655 14.9097 7.5117 2.6549
MSE 3.6201 × 10−8
Table 3.10: Pricing barrier options with mSABR: Vi (T, S, B ) × 100. Set III.
where θ > 0 is a displacement, or shift, in the underlying. The volatility process, σ(t ),
remains invariant (see Equation (3.1)). Parameter θ can be seen as the maximum level of
negativity that is expected in the rates. This generalization of the SABR model is widely
used by market practitioners due to its simplicity, and, precisely, the advantage of keep-
ing the existing simulation schemes. In mSABR, the assumption S(t ) > 0 is required to
employ the method.
In order to test the mSABR scheme in the context of negative interest rates, we em-
ploy the set IV. The SABR model parameters were obtained by Antonov et al. [5] after a
calibration process to real data5 under the shifted SABR model. Note that this config-
uration corresponds to an extreme situation with high values of the correlation, ρ, and
vol-vol, α. As a shift parameter, θ = 0.02 is chosen. We compare the normal implied
volatilities6 obtained by employing Monte Carlo with very fine (1000T ) time discretiza-
tion (as a reference) and the mSABR method with m = 4T . In Figure 3.3(a), these curves
are depicted and a perfect fit is observed. The strikes, K are chosen to permit negative
values. We perform an even more extreme experiment, by setting S 0 = 0. The resulting
normal implied volatilities are presented in Figure 3.3(b), again with a excellent fit.
Due to the complexity of the negative interest rates experiment and this particular set
of parameters (ρ ≈ 1), we wish to test the performance of our method under these con-
ditions. In Tables 3.11 and 3.12 the execution times obtained are presented. Again, the
small number of time-steps due to the “almost” exact simulation of the mSABR method
provides an important gain in terms of performance.
3.5. C ONCLUSIONS
In this chapter, we have proposed an accurate and robust multiple time-step Monte
Carlo method to simulate the SABR model with only a few time-steps. The mSABR
method employs many nontrivial components that are not yet seen before in combina-
5 1Y15Y Swiss franc swaption from February 10, 2015.
6 Determined by using the normal Bachelier model instead of the log-normal Black-Scholes model.
Normal Implied Volatility (bp)
Normal Implied Volatility (bp)

150 150
MC MC
mSABR mSABR
100 100
50 50
3
0 0
-0.5 0 0.5 1 1.5 2.0 2.5 -1 -0.5 0 0.5 1 1.5 2
K (bp) K (bp)
(a) (b)
Figure 3.3: The mSABR method in the context of negative interest rates.
Error < 25 bp < 10 bp < 5 bp < 1 bp

Y-Euler 2.47(4) 9.64(16) 14.32(24) 19.25(32)
Y-trpz 1.32(2) 4.99(8) 9.75(16) 14.52(24)
mSABR 1.61(1) 2.57(2) 3.52(3) 4.57(4)
Table 3.11: Execution times and time-steps, m (parentheses).
Error < 25 bp < 10 bp < 5 bp < 1 bp

Y-Euler 1.57 3.75 4.07 4.21
Y-trpz 0.82 1.94 2.77 3.17
Table 3.12: Speedups provided by the mSABR method.

tion within a Monte Carlo simulation. A copula methodology to generate samples from
the conditional SABR’s time-integrated variance process has been used. The marginal
distribution of the time-integrated variance has been derived by employing Fourier tech-
niques. We have dealt with an increasing computational complexity, due to the multi-
ple time-steps and the almost exact simulation, by applying an interpolation based on
stochastic collocation. This has resulted in an efficient SABR simulation scheme. We
have numerically shown the convergence and the stability of the method. In terms of
convergence, the mSABR method requires only very few time-steps to achieve high ac-
curacy, due to the focus on almost exact simulation (in contrast to the low-bias SABR
3 scheme [21]). This fact impacts the performance, obtaining an excellent ratio between
accuracy and computational cost. As a multiple time-step method, it can be employed
to price path-dependent options. As an example, a pricing experiment for barrier op-
tions has been carried out. Furthermore, we have shown that the mSABR scheme is also
suitable in the context of negative interest rates, in combination with a shifted model.
As a generalization of the technique presented in Chapter 2, the mSABR method combines

the Monte Carlo simulation and Fourier inversion, resulting again in a hybrid solution.
However, due to the extra complexity introduced by the use of multiple time-steps, an ad-
ditional component is required to make the technique tractable in terms of computational
cost. We have employed an advanced interpolation method based on stochastic colloca-
tion, SCMC, which provides an enormous gain regarding the execution time.
CHAPTER 4
The data-driven COS method
In this chapter, we consider a completely different hybrid solution method, which relies on
a combination of Monte Carlo methods and Fourier inversion techniques, but employing
a data-based approach. We therefore present the data-driven COS method, ddCOS. It is a
financial option valuation method which assumes the availability of asset data samples:
a ChF of the underlying asset is not required. As such, the method represents a general-
ization of the well-known COS method[49]. Convergence with respect to the number of
asset samples is according the convergence of Monte Carlo methods for pricing financial
derivatives. The ddCOS method is particularly interesting for PDF recovery and also for
the efficient computation of the option’s sensitivities Delta and Gamma. These are often
used in risk management, and can be obtained at a higher accuracy with ddCOS than
with plain Monte Carlo methods.
4.1. I NTRODUCTION
In quantitative finance, statistical distributions are commonly used for the valuation of
financial derivatives and within risk management. The underlying assets are often mod-
elled by means of SDEs. Except for the classical and most simple asset models, the cor-
responding PDF and CDF are typically not known and need to be approximated.
In order to compute derivative prices, and to approximate statistical distributions,
Fourier-based methods are often used numerical techniques. They are based on the
connection between the PDF and the ChF, which is the Fourier transform of the PDF. The
ChF is often available, and sometimes even in closed form, for the broad class of regu-
lar diffusions and also for Lévy processes. Some representative efficient Fourier pric-
ing methods include those by Carr and Madan [19], Boyarchenko and Levendorskii [13],
Lewis [92] and Fang and Oosterlee [49]. Here, we focus on the COS method from [49],
which is based on an approximation of the PDF by means of a cosine series expansion.
Still, however, the asset dynamics for which the ChF are known is not exhaustive,
and for many relevant asset price processes we do not have such information to recover
the PDF. In recent years several successful attempts have been made to employ Fourier
pricing methods without the explicit knowledge of the ChF. In Grzelak and Oosterlee [64],
for example, a hybrid model with stochastic volatility and stochastic interest rate was
linearized by means of expectation operators to cast the approximate system of SDEs in
the framework of affine diffusions. Ruijter and Oosterlee [111] discretized the governing
asset SDEs first and then worked with the ChF of the discrete asset process, within the
framework of the COS method. Borovykh et al. [12] used the Taylor expansion to derive
This chapter is based on the article “On the data-driven COS method”. Submitted for publication, 2017 [91].
67
68 4. T HE DATA - DRIVEN COS METHOD
a ChF for which they could even price Bermudan options highly efficiently. In this work,
we extend the applicability of the COS method to the situation where only data (like
samples from an unknown underlying risk neutral asset distribution) are available.
The density estimation problem, using a data-driven PDF, has been intensively stud-
ied in the last decades, particularly since it is a component in the machine learning
framework [10]. Basically, density estimators can be classified into parametric and non-
parametric estimators. The first type relies on the fact that prior knowledge is available
(like moments) to determine the relevant parameters, while for non-parametric estima-
tors the parameters need to be determined solely from the samples themselves. Within
this second type of estimators we can find histograms, kernel density and orthogonal
series estimators. A thorough description of these estimators is provided in [120]. More
recently, some applications in finance have also appeared, see [48, 97, 118], for example.
4 For the valuation of financial derivatives, we will combine density estimators with
Fourier-based methods, so orthogonal series form a natural basis. We will focus on the
framework of statistical learning, see [131]. In statistical learning, a regularization is em-
ployed to derive an expression for the data-driven empirical PDF. By representing the
unknown PDF as a cosine series expansion, a closed-form solution of the regularization
problem is known [131], which forms the basis of the data-driven COS method (ddCOS).
The use of the COS method gives us expressions for option prices and, in particular,
for the option sensitivities or Greeks. These option Greeks are the derivatives of option
price with respect to a variable or parameter. The efficient computation of the Greeks is
a challenging problem when only asset samples are available. Existing approaches are
based on Monte Carlo-based techniques, like on finite-differences (bump and revalue),
pathwise or likelihood ratio techniques, for which details can be found in [59], chapter 7.
Several extensions and improvements of these approaches have appeared, for example,
based on adjoint formulations [56], the ChF [60, 61], Malliavin calculus [41, 53], algorith-
mic differentiation [17, 43] or combinations of these [18, 22, 58]. All in all, the computa-
tion of the Greeks can be quite involved. The ddCOS method is not directly superior to
Monte Carlo methods for option valuation, but it is competitive for the computation of
the corresponding sensitivities. We derive simple expressions for the Greeks Delta and
Gamma.
The importance of Delta and Gamma in dynamic hedging and risk management is
well-known. A useful application is found in the Delta-Gamma approach [14] to quantify
market risk. The approximation of risk measures like Value-at-Risk (VaR) and Expected
Shortfall (ES) under the Delta-Gamma approach is still nontrivial. Next to Monte Carlo
methods, Fourier techniques have been employed in this context, when the ChF of the
change in the value of the option portfolio is known (see [23, 121]). For example, the
COS method has been applied in [101] to efficiently compute the VaR and ES under the
Delta-Gamma approach. The ddCOS method may generalize the applicability to the
case where only data is available.
This paper is organized as follows. The ddCOS method, and the origins in statistical
learning and Fourier-based option pricing, are presented in Section 4.2. Variance reduc-
tion techniques can also be used within the ddCOS method, providing an additional con-
vergence improvement. We provide insight and determine values for the method’s open
parameters in Section 4.3. Numerical experiments, with a focus on the option Greeks,
4.2. T HE DATA - DRIVEN COS METHOD 69
are presented in Section 4.4. We conclude in Section 4.5.
4.2. T HE DATA - DRIVEN COS METHOD

In this section we will discuss the ddCOS method, in which aspects of the Monte Carlo
method, density estimators and the COS method are combined to approximate, in par-
ticular, the option Greeks Delta and Gamma. We will focus on European options here.
The COS method in [49] is a Fourier-based method by which option prices and sensi-
tivities can be computed for various options under different models. The method relies
heavily on the availability of the ChF, i.e., the Fourier transform of the PDF. In the present
work, we assume that only asset samples are available, not the ChF, resulting in the data-
driven COS method. It is based on regularization in the context of the statistical learning
theory, presented briefly in Section 4.2.2. The connection with the COS method is found
in the fact that the data-driven PDF appears as a cosine series expansion. 4
4.2.1. T HE COS METHOD
The starting point for the well-known COS method is the risk-neutral option valuation
formula, where the value of a European option at time t , V (t , x), is an expectation under
the risk neutral pricing measure, i.e.,
Z
V (t , x) = e−r (T −t ) E h(T, y)|x = e−r (T −t ) h(T, y) f (y|x)dy,
£ ¤
(4.1)
R
with r the risk-free rate, T the maturity time, and f (y|x) the PDF of the underlying pro-
cess, and h(T, y) represents the option value at maturity time, being the payoff function.
Typically, x and y are chosen to be scaled variables,
S(0) S(T )
µ ¶ µ ¶
x := log and y := log ,
K K
where S(t ) is the underlying asset process at time t , and K is the strike price.
f (y|x) is unknown in most cases and in the COS method it is approximated, on a
finite interval [a, b], by a cosine series expansions, i.e.,
Ã !
1 X∞ ³ y −a´
f (y|x) = A0 + 2 A k (x) · cos kπ ,
b−a k=1 b−a
Z b ³ y −a´
A 0 = 1, A k (x) = f (y|x) cos kπ dy, k = 1, 2, . . . .
a b−a
By substituting this expression in Equation (4.1), interchanging the summation and
integration operators using Fubini’s Theorem, and introducing the following definition,
Z b ³ y −a´
2
Hk := h(T, y) cos kπ dy,
b−a a b−a
we find that the option value is given by
∞
0
V (t , x) ≈ e−r (T −t )
X
A k (x)Hk , (4.2)
k=0
where 0 indicates that the first term is divided by two. So, the product of two real-valued
functions in Equation (4.1) is transformed into the product of their cosine expansion
coefficients, A k and Hk . Coefficients A k can be computed by the ChF and Hk is known
analytically (for many types of options).
Closed-form expressions for the option Greeks can also be derived. From the COS
option value formula, ∆ and Γ are obtained by
∞
∂V (t , x) 1 ∂V (t , x) 0 ∂A k (x) Hk
∆=
X
= ≈ exp(−r (T − t )) ,
∂S S(0) ∂x k=0 ∂x S(0)
∂2 V (t , x) ∂V (t , x) ∂2 V (t , x)
µ ¶
1
Γ= = − + (4.3)
∂S 2 S 2 (0) ∂x ∂x 2
∞ 2
X0 ∂A k (x) ∂ A k (x) Hk
µ ¶
≈ exp(−r (T − t )) − + .
4 k=0 ∂x ∂x 2 S 2 (0)
Due to the rapid decay of the coefficients, V (t , x), ∆ and Γ can be approximated with
high accuracy by truncating the infinite summation in Equations (4.2) and (4.3) to N
terms. Under suitable assumptions, exponential convergence is proved and numerically
observed.
4.2.2. S TATISTICAL LEARNING THEORY FOR DENSITY ESTIMATION

In the setting of this paper, we assume a vector of n independent and identically dis-
tributed (i.i.d.) samples, X 1 , X 2 , . . . , X n . Based on these samples, we wish to find an accu-
rate approximation of the PDF estimator, f n (x), which should approximate PDF f (x).
By definition, the PDF is related to its CDF F (x),
Z x
f (y)dy = F (x). (4.4)
−∞
Function F (x) is approximated by the empirical approximation,
1 X n
F n (x) = η(x − X j ), (4.5)
n j =1
where η(·) is a step function. This approximation converges to the “true CDF” with rate
p
O (1/ n). Rewriting Equation (4.4) as a linear operator equation, gives us,
C f = F ≈ Fn ,
Rx
where the operator C h := −∞ h(z)dz.
As explained in [131], this matrix equation represents an ill-posed problem, and a
risk functional should be constructed, with a regularization term, as follows
R γn ( f , F n ) = L 2H (C f , F n ) + γn W ( f ), (4.6)
where L H is a metric of the space H , γn > 0 is a parameter which gives a weight to the
regularization term W ( f ), with γn → 0 as n → ∞. The solution of C f = F n belongs to D,
the domain of definition of W ( f ). Functional W ( f ) takes real non-negative values in D.
Furthermore, Mc = { f : W ( f ) ≤ c} is a compact set in H (the space where the solution

exists and is unique).
The solution f n , minimizing the functional in Equation (4.6), converges almost surely
to the desired PDF. For the ill-posed density estimation problem, other conditions im-
posed for consistency (see details in [131], chapter 7), include
nγn → ∞ as n → ∞, and
n (4.7)
γn → ∞ as n → ∞.
log n
4.2.3. R EGULARIZATION AND F OURIER- BASED DENSITY ESTIMATORS

A relation exists between the regularization approach in Equation (4.6) and Fourier-
based PDF approximation, more specifically, cosine series expansion estimators. By spe-
cific choices for the metric and the regularization term in Equation (4.6), i.e., L H = L 2 , 4
and Z µZ ¶ 2
W (f ) = K (z − x) f (x)dx dz,
R R
with kernel K (z − x), the functional reads
Z µZ x ¶2 Z µZ ¶2
R γn ( f , F n ) = f (y)dy − F n (x) dx + γn K (z − x) f (x)dx dz. (4.8)
R 0 R R
Denoting by fˆ(u), F̂ n (u) and Kˆ (u) the Fourier transforms of f (x), F n (x) and K (x),
respectively, an expression for F̂ n (u) can be derived by applying Fourier transformation
to Equation (4.5),
1
Z
F̂ n (u) = F n (x)e−i ux dx
2π R
1
Z X n 1 X n exp(−i uX )
j
= η(x − X j )e−i ux dx = ,
2nπ R j =1 n j =1 iu
p
where i = −1 is the imaginary unit.
By employing the convolution theorem and Parseval’s identity, Equation (4.8) can be
rewritten as
°ˆ °2
° f (u) − 1 n exp(−i uX j ) °
P
n j =1 °2
° + γn °Kˆ (u) fˆ(u)°L 2 .
°
R γn ( f , F n ) = °
° °
° iu °
L2
The condition to minimize R γn ( f , F n ) is given by,
fˆ(u) 1 X n
− exp(−i uX j ) + γn Kˆ (u)Kˆ (−u) fˆ(u) = 0,
u2 nu 2 j =1
which gives us,

µ
1
¶
1 Xn
fˆn (u) = exp(−i uX j ). (4.9)
1 + γn u 2 Kˆ (u)Kˆ (−u) n j =1
When kernel K is the p-th derivative of the Dirac delta function, i.e., K (x) = δ(p) (x),
and the desired PDF, f (x), belongs to the class of functions whose p-th derivative (p ≥ 0)
belongs to L 2 (0, π), the risk functional becomes
Z π µZ x ¶2 Z π¡ ¢2
R γn ( f , F n ) = f (y)dy − F n (x) dx + γn f (p) (x) dx. (4.10)
0 0 0
Given a series expansion in orthonormal functions, ψ1 (θ), . . . , ψk (θ), . . . , the approxi-

mation to the unknown PDF will be of the form
1 2 X∞
f n (θ) = + Ã k ψk (θ), (4.11)
π π k=1
4 with Ã 0 , Ã 1 , . . . , Ã k , . . . expansion coefficients, defined as Ã k =< f n , ψk >.

We need to compute the expansion coefficients so that the functional in Equation
(4.10) is minimized. The coefficients Ã k cannot be directly computed from the definition
since the unknown PDF, f n , is implicitly involved in the expression, i.e.,
Ã k =< f n , ψk >=< fˆn , ψ̂k >

Z π Ãµ ¶ n
!
1 1 X
= exp(−i uθ j ) · ψ̂k (u)du.
0 1 + γn u 2 Kˆ (u)Kˆ (−u) n j =1
Using cosine series expansions, i.e., ψk (θ) = cos(kθ), it is well-known that ψ̂k (u) =
1
2 (δ(u−k)+δ(u+k)). This facilitates the computation of the series coefficients, Ã k , avoid-
ing the calculation of the integral. Thus, the minimum of the functional using cosine
series expansions is obtained when
Ãµ ¶ n
1 1 X
Ã k = exp(i kθ j )
2n 1 + γn (−k)2 Kˆ (−k)Kˆ (k) j =1
µ ¶ n !
1 X
+ exp(−i kθ j )
1 + γn k 2 Kˆ (k)Kˆ (−k) j =1
(4.12)
1 1 X n
= cos(kθ j )
1 + γn k 2 Kˆ (k)Kˆ (−k) n j =1
1 1 X n
= cos(kθ j ),
1 + γn k 2(p+1) n j =1
where θ j ∈ (0, π) are given samples of the unknown distribution. In the last step, Kˆ (u) =
(i u)p is used.
Assuming that the samples are given, the solution contains two free parameters: reg-
ularization parameter γn , and smoothing parameter p.
In Section 4.3, we will discuss the impact of the regularization parameter on the con-
vergence to the PDF in terms of the number of samples. We will use p = 0 here.
True
0.4
p=0
p=1
0.3 p=2
p=3
0.2
0.1
-4 -2 0 2 4 4
Figure 4.1: Smoothing of the PDF approximation in relation to parameter p.
S MOOTHING PARAMETER EXAMPLE

In order to give an insight in the influence of parameter p on the approximation, in Fig-
ure 4.1 standard normal densities obtained for several values of p are shown. With p
increasing, the densities get increasingly smoothed and flat. The choice p = 0 (regular-
izing the PDF itself and not imposing regularization upon its derivatives) appears most
appropriate in our context of smooth densities.
4.2.4. T HE DD COS METHOD

We are now ready to present the ddCOS method, where we employ the series expan-
sion coefficients from the regularization approach. We replace the A k -coefficients from
Equation (4.2) by those coefficients based on data, Ã k in Equation (4.12).
So, suppose we have risk neutral samples (or values) from an underlying asset at a
future time t , i.e., S 1 (t ), S 2 (t ), . . . , S n (t ). We compute the value of a European option with
maturity time T and strike price K , and require therefore the samples S j (T ). With a
logarithmic transformation, we have
S j (T )
µ ¶
Y j := log .
K
Before employing these samples in the regularization approach and because the so-
lution is defined in (0, π), we need to transform the samples by the following change of
variables,
Yj − a
θj = π ,
b−a
where the boundaries a and b are defined as
a := min (Y j ), b := max (Y j ).
1≤ j ≤n 1≤ j ≤n
The A k coefficients in Equation (4.2) are replaced by the data-driven Ã k in Equation

(4.12), ³ Y −a ´
1 Pn j
n j =1 cos kπ b−a
A k ≈ Ã k = .
1 + γn k 2(p+1)
The ddCOS pricing formula for European options based on risk neutral data is now
obtained as ³ Y −a ´
∞ 1 n cos kπ
P j
−r (T −t )
X 0 n j =1 b−a
Ṽ (t , x) = e · Hk
k=0 1 + γn k 2(p+1)
(4.13)
∞
−r (T −t )
X 0
=e Ã k Hk .
k=0
4 As in the original COS method, we must truncate the infinite sum in Equation (4.13)
to a finite number of terms N , i.e.,
N
0
Ṽ (t , x) = e−r (T −t )
X
Ã k Hk , (4.14)
k=0
which completes the ddCOS pricing formula.

The samples Y j should originate from one initial state, i.e. the dependency on the
state x is implicitly assumed. In the case of European options this is typically fulfilled.
In the Monte Carlo method, ³ for´ example, all simulated asset paths depart from the same
point S(0), so that x := log S(0)
K .
Regarding the Greeks, we can also derive data-driven expressions for the ∆ and Γ
sensitivities. We first define the corresponding sine coefficients as
³ Y −a ´
1 Pn j
n j =1 sin kπ b−a
B̃ k := .
1 + γn k 2(p+1)
Taking derivatives in Equation (4.14) w.r.t the samples, Y j , and following the COS
expression for the sensitivities in Equation (4.3), the data-driven Greeks, ∆ ˜ and Γ̃, can be
obtained by
N
kπ Hk
µ ¶
˜ = e−r (T −t ) 0 B̃ k · −
∆
X
· ,
k=0 b − a S(0)
N µ
kπ kπ 2 Hk
µ ¶ ¶
0
Γ̃ = e−r (T −t )
X
B̃ k · − Ã k · · 2 .
k=0 b−a b−a S (0)
A PPLICATION OF VARIANCE REDUCTION

Because of the focus on asset path data, the ddCOS method is related to the Monte Carlo
method. Variance reduction in Monte Carlo methods is typically achieved by the use of
variance reduction techniques. The ddCOS method also admits an additional variance
reduction, in this case, for the computation of the expansion coefficients, Ã k . We show
how to introduce antithetic variates (AV) to our method. Since one of the assumptions
for the regularization approach is that the samples are i.i.d., an immediate application
4.3. C HOICE OF PARAMETERS IN DD COS M ETHOD 75
of AV is not possible. Therefore, if we assume that antithetic samples, Yi0 , to the original
samples Yi , can be computed without any serious computational effort, a new estimator
for the coefficients can be defined as
1¡
Ã k + Ã 0k ,
¢
Ā k :=
2
where we denote by Ã 0k the corresponding “antithetic coefficients”, obtained by Yi0 . By a
similar derivation as for the standard AV technique, it can be proved that the use of coef-
ficients Ā k will give us a variance reduction compared to using the Ã k coefficients. Other
variance reduction techniques may also be considered for the ddCOS method under the
assumption of i.i.d. samples.
In order to reduce the variance of any estimator, additional information may be in-
troduced. A well-known property to fulfil is the martingale property. To preserve this
property, a simple transformation of the samples can be made by 4
1 X n
S(T ) = S(T ) − S j (T ) + E[S(T )],
n j =1
1 X n
= S(T ) − S j (T ) + S(0) exp(r T ).
n j =1
As this modification is performed over the samples, it can also be used in the context
of the ddCOS method.
4.3. C HOICE OF PARAMETERS IN DD COS M ETHOD

In this section, the selection and the influence of the regularization parameter γn in the
ddCOS method is studied.
4.3.1. R EGULARIZATION PARAMETER γn

The regularization parameter γn plays an important role in the empirical PDF f n . With-
out the inclusion of the regularization term, the PDF approximation provided by Equa-
tion (4.10), would give us a standard orthogonal series estimator. The choice of param-
eter γn impacts the efficiency of the data-driven COS method, since it is related to the
required number of data samples, and by reducing the number of samples, the overall
computational cost can be reduced.
The first option for choosing the regularization parameter, γn , which was proposed
in [131], is given by the following rule,
log log n
γn = . (4.15)
n
As proved in [131], this rule provides a robust asymptotic rate of convergence un-
der the assumption of a compactly supported PDF. It implies, with probability one, uni-
formly converging approximations f n to the unknown PDF. Note, however, that the reg-
ularization parameter does not satisfy the second condition in Equation (4.7).
Although Equation (4.15) ensures an optimal asymptotic convergence in terms of
n, it may not be the optimal γn -value for density estimation with a given fixed amount
of samples. For that purpose we can exploit the relation between the empirical CDF,
F n (x), and the unknown CDF, F (x). This relation can be modelled by means of different
statistical laws. Some examples include the Kolmogorov-Smirnov, Anderson-Darling,
Kuiper and the Smirnov-Cramér-von Mises laws, by which a measure of the distance, or,
goodness-of-fit, between F n (x) and F (x) can be defined.
We are interested in a statistic which has a distribution, independent of the actual
CDF and the number of samples n, and consider the Smirnov-Cramér-von Mises (SCvM)
statistic [32, 123], defined as
Z
ω := n (F (x) − F n (x))2 dF (x).
2
R
Based on an approximation of the desired PDF, f γn (depending on γn ) and thus the

CDF, F γn , we choose the regularization parameter such that F γn satisfies the SCvM statis-
4 tic optimally, by solving (in terms of γn ) the following equation:
m 2
Z
F γn (x) − F n (x) dF (x) = ω ,
¡ ¢2
R n
where m ω2 is the mean of the SCvM statistic, ω2 . In the one-dimensional case, a simpli-
fied expression can be derived [3, 123], i.e.,
n i − 0.5 2
µ ¶
X 1
F γn ( X̄ j ) − = m ω2 − , (4.16)
j =1 n 12n
with X̄ 1 , X̄ 2 , . . . , X̄ n , the ordered array of samples X 1 , X 2 , . . . , X n . It can be proved (details

in [131]) that, by solving Equation (4.16) under the assumption that the solution is in the
form of a cosine series expansion, a regularization parameter can be determined, which
provides an almost optimal rate of convergence in n towards the desired PDF.
Next to the improvement of the method’s convergence, the quality of the PDF ap-
proximation, in terms of the considered expansion coefficients, is also influenced by the
regularization parameter.
In order to assess the impact of γn on the quality of approximation, we employ the
well-known Mean Integrated Squared Error (MISE), which is commonly used in density
estimation (also known as the risk function). The formal definition reads
h° ·Z ¸
°2 i ¢2
E ° f n − f °2 = E
¡
f n (x) − f (x) dx .
R
In our case, the MISE can be decomposed into two terms (see [87]), as follows,
·Z π¡ ¢2
¸ N ∞
E Var Ã k + A 2k ,
X £ ¤ X
f n (x) − f (x) dx =
0 k=1 k=N +1
where the A k are the “true” coefficients from Equation (4.2), and the Ã k are from Equa-
tion (4.12). The MISE, as it is defined, is the summation of the bias and the variance of
the estimator. In the Fourier cosine expansions context, an increasing N implies smaller
bias (but bigger variance). The opposite also holds i.e. small N produces more bias and
smaller variance. We need to compute the variance of the data-driven coefficients, Ã k .

By Equation (4.12), we have
" #
1 1 X n
Var Ã k = Var
£ ¤
cos(kθ j )
1 + γn k 2(p+1) n j =1
1 1 Xn
Var cos(kθ j ) .
£ ¤
=¡ ¢2 2
1 + γn k 2(p+1) n j =1
By basic trigonometric properties, this variance can be computed as
Var [cos(kx)] = E cos2 (kx) − (E [cos(kx)])2

£ ¤
Z π µZ π ¶2
= cos2 (kx) f (x)dx − cos(kx) f (x)dx
0
Z πµ
1 + cos(2kx)
¶
0
1 A 2k
4
= f (x)dx − A 2k = + − A 2k ,
0 2 2 2
where the definition of the expansion coefficients is used in steps 2 and 3.
The expression for the variance of Ã k then reads
µ ¶
1 1 1 1 2
Var Ã k = ¡
£ ¤
¢2 + A 2k − A k ,
1 + γn k 2(p+1) n 2 2
and the MISE is thus given by
1 XN 1
µ
1 1
¶ ∞
2
A 2k .
X
MISE = ¢2 + A 2k − A k + (4.17)
n k=1 1 + γn k 2(p+1) 2 2
¡
k=N +1
E XAMPLE
The error measure defined in Equation (4.17) is employed to analyse the influence of the
regularization. We use the standard normal distribution as a reference test case. The
coefficients that are based on the available analytic solution are replaced by the corre-
sponding data-driven coefficients that depend on γn .
In Figure 4.2(a), we present the convergence results for different regularization pa-
rameters. Next to the rules suggested by Equations (4.15) and (4.16), we also include
the case γn = 0 to highlight the benefits of employing the regularization approach. The
obtained results confirm the improvements provided by both γn -rules, with the almost
optimal γn given by the SCvM statistic.
A second aspect which is influenced by γn is the accuracy with respect to the number
of expansion terms N in Equation (4.14). For this, in Figure 4.2(b) we present the MISE
for the standard normal distribution when different coefficients A k are employed: A k by
the γn -rule (4.15) (red lines), A k by the SCvM (4.16) (blue lines) and, as a reference, the
A k -coefficients obtained by the ChF (black dashed line). We notice that, when γn = 0
is used in the MISE formula (all dashed lines), for increasing values of N , the approxi-
mation deteriorates, resulting in increasing approximation errors. In contrast, when the
corresponding γn is used (regular lines), the error stabilizes. Since the number of expan-
sion coefficients is typically chosen high, this property of the regularization approach is
useful.
10 2 10 0
n
"rule" A k real
10 1 SCvM -1 A k rule
n 10
0 n
=0 A k rule, n
=0
10
10 -2 A k SCvM
10 -1 A k SCvM, n =0
10 -3
10 -2
10 -3 10 -4
10 -4 10 -5
10 1 10 2 10 3 10 4 10 5 0 50 100 150 200
(a) Convergence in terms of n. (b) Accuracy in terms of N .
4 Figure 4.2: Influence of γn on the convergence w.r.t. the number of samples n and the number of terms N .
O PTIMAL N - VALUES
As mentioned, with an increasing number of series expansion coefficients N , the ap-
proximation based on the regularization approach does not improve any further. This
fact indicates that we need to determine an optimal value of N , i.e. the smallest value
of N for which the MISE stabilizes, see Figure 4.2(b). Using small N -values is important
within the data-driven methodology, since parameter N considerably affects the perfor-
mance of the method.
We propose an empirical procedure to compute the optimal value of N . The MISE
in Equation (4.17) depends on the number of samples n, the number of coefficients N ,
and the coefficients themselves A k . Since we wish to compute no more coefficients than
necessary, we focus on the parameters n and N . Regarding the influence of the number
of samples, see also the curves in Figure 4.3(a), higher n-values also require higher N -
values to ensure stabilizing errors in the MISE curve. To determine a relation between n
and N , we need to simplify the MISE formula as we desire a closed-form expression. We
discard the second part in Equation (4.17), ¡as it goes to zero¢ when N increases. Within
the first part, we approximate the quantity 21 + 12 A 2k − A 2k ≈ 21 . Then, the approximate
formula for the MISE is found to be
N 1
1 X 2
MISE ≈ ¢ =: MISEN ,
n k=1 1 + γn k 2(p+1) 2
¡
where n and N are directly connected, but also γn appears. It is possible to prove that
the above MISE proxy, MISEN , is an upper bound for the first part in Equation (4.17).
From Figure 4.3(b), we observe two important facts: the MISE proxy provides a highly
satisfactory approximation for the first addend of the MISE, which converges to the MISE
when N increases. By combining these two observations, we will employ the MISE proxy
to determine the optimal number of terms N . Since the computation of γn by Equation
(4.16) involves N , we use the case where γn is determined by Equation (4.15) (which only
depends on n). Figure 4.3(b) shows that the MISEN (to a different level of accuracy) is
very similar in both cases, where the γn rule appears conservative, i.e. bigger N -values
10 0 10 0
n = 10 1 rule
n
2
n = 10 rule - addend 1
-1 n
n = 10 3
10
10 -1 n = 10 4 n
rule - proxy
n = 10 5 SCvM
10 -2
n
n
SCvM - addend 1
-2
10 SCvM - proxy
n
10 -3
10 -3
10 -4
10 -4 10 -5
0 5 10 15 20 25 0 50 100 150 200
(a) MISE for several values of n. (b) Comparison between MISE and MISEN .
Figure 4.3: Influence of the number of samples n, and the number of coefficients N in the MISE and in the
4
MISE proxy, MISEN .
are required to reach the non-decreasing error region. The proposed procedure itera-
tively determines whether or not we reached error stabilization by checking the differ-
ences in MISEN between two consecutive N -values. When this difference is less than
a predefined tolerance, ², we have approximated the optimal N -value. Since N should
grow with n, we propose to use ² := p1n , i.e. the expected order of accuracy for the PDF
approximation should be given in terms of the number of samples.
By collecting all described components, the approximately optimal N -value becomes
a function only of n. The iterative methodology is described in Algorithm 3. In Figure 4.4,
we observe that the resulting optimal N function is an increasing staircase function (with
a predefined floor of N = 5).
Algorithm 3: Optimal N -values.
Data: n, γn
Nmi n = 5
Nmax = ∞
² = p1n
MISE∗ = ∞
for N = Nmi n : Nmax do
1
MISEN = n1 k=1
PN 2
(1+γn k 2(p+1) )2
²N = |MISE N −MISE∗ |
|MISE | N
if ²N > ² then
Nop = N
else
Break
MISE∗ = MISEN
Now, we have described the techniques to determine values for the regularization
18
16
14
12
N
10
4
1 2 3
10 10 10
n
Figure 4.4: Optimal N -values in terms of n, computed by Algorithm 3.

4
parameter γn , and for the number of coefficients, N . By these, the ddCOS method is
defined with Monte Carlo samples, or other data values, as the input.
4.4. A PPLICATIONS OF THE DD COS METHOD

In this section, we present some applications of the ddCOS method. The first applica-
tion is an option pricing experiment, where we show the method’s convergence. Sub-
sequently, we present the performance regarding the computation of the Greeks, where
ddCOS exhibits a stable convergence and can be employed with involved models, as we
only need asset samples. We also compute the Greeks under the SABR model. Once the
Greeks have been efficiently approximated, they can be used for the computation of the
VaR and ES risk measures within the Delta-Gamma approach. All steps in this method-
ology can be performed by the ddCOS method.
The experiments have been carried out on a computer system with the following
characteristics: CPU Intel Core i7-4720HQ 2.6GHz and RAM memory of 16GB RAM. The
employed software package is Matlab r2016b.
4.4.1. O PTION VALUATION AND G REEKS

First of all, we numerically test the convergence of the ddCOS method in an option val-
uation experiment. The GBM asset dynamics are employed, since a reference value for
the option value is available by the Black-Scholes formula. The regularization parameter
γn is set as in Equation (4.15), as for option valuation experiments the difference be-
tween this rule and γn based on the SCvM statistic in Equation (4.16) is not significant.
Moreover, the use of the γn rule provides faster ddCOS estimators.
As is common in Monte Carlo experiments, the MSE is considered as the error mea-
sure. In the convergence tests, the reported values are computed as the average of 50
experiments.
p
The expected order of convergence for the option values is O (1/ n), according to
the convergence of the empirical CDF towards the true CDF in Equation (4.5). In Section
4.2.4, the application of antithetic variates in the ddCOS framework has been presented.
In Figure 4.5, we confirm that this variance reduction technique provides a similar im-
4.4. A PPLICATIONS OF THE DD COS METHOD 81
10 1 10 1
ddCOS ddCOS
ddCOS, AV ddCOS, AV
MC MC
MC, AV MC, AV
10 0 10 0
10 -1 10 -1
10 -2 10 -2
10 1 10 2 10 3 10 4 10 5 10 1 10 2 10 3 10 4 10 5
(a) Call: Strike K = 100. (b) Put: Strike K = 100.

4
Figure 4.5: Convergence in prices of the ddCOS method: Antithetic Variates (AV); GBM, S(0) = 100, r = 0.1,
σ = 0.3 and T = 2.
provement in terms of precision as when it is applied to the plain Monte Carlo method.
Another observation is that, under this particular setting, the estimators (both ddCOS
and Monte Carlo) of the put option value result in smaller variances than the call option
estimators. In terms of accuracy, it is thus worth computing the put value and then use
the put-call parity formula for call options. In addition, the use of the put together with
the put-call parity is recommended since call payoff functions grow exponentially and
may give rise to cancellation errors.
We have empirically shown in Figure 4.5 that the ddCOS method converges to the
p
true price with the expected convergence rate O (1/ n), which resembles the plain Monte
Carlo convergence. However, by the ddCOS method, not only the option value but also
the sensitivities can readily be obtained. This is an advantage w.r.t Monte Carlo-based
methods for estimating sensitivities, where often, additional simulations, intermediate
time-steps or prior knowledge are required.
Thus, a similar convergence test is performed for the ∆ and Γ sensitivities, see Figure
4.6. As Monte Carlo-based method for the Greeks calculation we consider the Finite
Difference method (bump and revalue, denoted as MCFD). We have chosen MCFD for
the comparison because it is flexible and it does not require prior knowledge. MCFD may
require one or two extra simulations, starting at S(0) + ν and S(0) − ν, and the choice of
optimal shift parameter, ν, may not be trivial. The reference Delta and Gamma values are
given by the Black-Scholes formula. In Figure 4.6 we observe the expected convergence
and the reduction in the variance due to the use of AV. In both experiments, while the ∆
is very well approximated by the ddCOS and MCFD methods, the second derivative, Γ,
appears more complicated for the MCFD method. This fact was already pointed out by
Glasserman in [59]. The ddCOS estimator, however, is accurate and stable as it is based
on the data-driven PDF and the ddCOS machinery.
Using n = 100, 000, in Table 4.1 we now compare the ∆ and Γ estimations obtained
under the GBM dynamics for several strikes. The performance of the ddCOS method is
very satisfactory as it is accurate, with small RE (averaged over K ) and reproduces the
10 -1
ddCOS ddCOS
ddCOS , AV ddCOS , AV
MCFD 10 -2 MCFD
MCFD , AV MCFD , AV
10 -2
-3
10
10 -3
10 -4
-4
10
10 1 10 2 10 3 10 4 10 5 10 1 10 2 10 3 10 4 10 5
(a) ∆ (Call): Strike K = 100. (b) Γ: Strike K = 100.

4
Figure 4.6: Convergence in Greeks of the ddCOS method: Antithetic Variates (AV); GBM, S(0) = 100, r = 0.1,
σ = 0.3 and T = 2.
reference values very well. The difficulties of the MCFD estimating Γ are more clearly
visible.
We wish to test the ddCOS method in a more complex situation, by adding jumps
in the form of a Merton jump-diffusion asset price process. To accurately compute the
option sensitivities in this case gives rise to difficulties for Monte Carlo-based methods.
We perform a similar experiment as before, where now the underlying asset follows the
Merton jump-diffusion model, and the obtained ∆ and Γ are presented in Table 4.2. In
this case, the reference value is provided by the COS method at a high accuracy.
In terms of computational cost, the ddCOS method is a competitive alternative, as
additional simulations are not needed. Notice that in these latter experiments AV tech-
niques are not employed. Under the Merton dynamics, the ddCOS method takes 0.1813
seconds and MCFD 0.3149 seconds. In this case, the use of the ddCOS method reduces
the computational costs, as the cost of an individual simulation by the Merton model is
significantly higher than for GBM asset dynamics.
4.4.2. T HE SABR MODEL

The SABR model [67] is interesting within the ddCOS framework since the ChF is not
known and, furthermore, the asset path Monte Carlo simulation is not trivial. The model
is a stochastic-local volatility model which is widely used in FX modelling, and is given
by
(4.18)
dσ(t ) = ασ(t )dWσ (t ), σ(0) = σ0 ,
where S(t ) = S̄(t ) exp (r (T − t )) is the forward value of the underlying S̄(t ), with r the
interest rate, S 0 the spot price and T maturity time. The stochastic volatility process is
denoted by σ(t ), with σ(0) = σ0 , WS (t ) and Wσ (t ) are two correlated Brownian motions
with correlation ρ (i.e. WS Wσ = ρt ). The parameters of the SABR model are α > 0 (the
volatility of the volatility), 0 ≤ β ≤ 1 (the elasticity) and ρ (the correlation coefficient).
K (% of S(0)) 80% 90% 100% 110% 120%

0.1 ∆
Ref. 0.8868 0.8243 0.7529 0.6768 0.6002
ddCOS 0.8867 0.8240 0.7528 0.6769 0.6002
RE 1.1012 × 10−4
MCFD 0.8876 0.8247 0.7534 0.6773 0.6006
RE 7.5168 × 10−4
Γ
Ref. 0.0045 0.0061 0.0074 0.0085 0.0091
ddCOS 0.0045 0.0062 0.0075 0.0084 0.0090 4
RE 8.5423 × 10−3
MCFD 0.0045 0.0059 0.0071 0.0079 0.0083
RE 4.9554 × 10−2
Table 4.1: GBM option Greeks: Call, S(0) = 100, r = 0.1, σ = 0.3 and T = 2.
K (% of S(0)) 80% 90% 100% 110% 120%

∆
Ref. 0.8385 0.8114 0.7847 0.7584 0.7328
ddCOS 0.8383 0.8113 0.7846 0.7585 0.7333
RE 2.7155 × 10−4
MCFD 0.8387 0.8118 0.7850 0.7586 0.7330
RE 3.1265 × 10−4
Γ
Ref. 0.0022 0.0024 0.0027 0.0029 0.0030
ddCOS 0.0022 0.0024 0.0027 0.0029 0.0030
RE 8.2711 × 10−3
MCFD 0.0023 0.0026 0.0028 0.0031 0.0033
RE 6.118 × 10−2
Table 4.2: Merton jump-diffusion option Greeks: Call, S(0) = 100, r = 0.1, σ = 0.3, µ j = −0.2, σ j = 0.2 and λ = 8
and T = 2.
K (% of S(0)) 80% 90% 100% 110% 120%

∆
Ref. 0.9914 0.9284 0.5371 0.0720 0.0058
ddCOS 0.9916 0.9282 0.5363 0.0732 0.0058
RE 5.2775 × 10−3
MCFD 0.9911 0.9279 0.5368 0.0737 0.0058
RE 5.5039 × 10−3
Table 4.3: Greek ∆ under the SABR model: Call, S(0) = 100, r = 0, σ0 = 0.3, α = 0.4, β = 0.6, ρ = −0.25 and T = 2.
K (% of S(0)) 80% 90% 100% 110% 120%

∆
4 Ref. 0.8384 0.7728 0.6931 0.6027 0.5086
ddCOS 0.8364 0.7703 0.6902 0.6006 0.5084
RE 2.7855 × 10−3
Hagan 0.8577 0.7955 0.7170 0.6249 0.5265
RE 3.1751 × 10−2
Table 4.4: Greek ∆ under SABR model: Call, S(0) = 0.04, r = 0.0, σ0 = 0.4, α = 0.8, β = 1.0, ρ = −0.5 and T = 2.
The calculation of the Greeks under the SABR model becomes challenging but can
be addressed by the ddCOS method. To employ the method, we need samples of the
underlying asset at time T . Here, we make use of the one time-step SABR Monte Carlo
simulation introduced in Chapter 2. This one time-step SABR simulation is based on the
expression for the CDF of the conditional SABR process [75]. Thus, the ddCOS method
will be combined with the one time-step SABR simulation to efficiently compute ∆ and
Γ under the SABR dynamics.
For the numerical experiments, we consider two parameter settings. First of all, a
basic parameter set is taken, where the Hagan formula by [67] is valid and can be used
as a reference. The results are presented in Table 4.3. For the second test we use a more
difficult set of parameters (i.e., Set III in Table 2.2 of Chapter 2), where the Hagan for-
mula does not provide accurate results anymore. In Table 4.4, we observe that the dd-
COS provides accurate ∆-values in this case, without any problems. The reference value
has been computed by the MCFD in combination with the mSABR simulation scheme
from Chapter 3, with a large number of Monte Carlo paths (n = 10, 000, 000) and time
steps (4T ). The convergence in n of the ddCOS ∆ estimator under the SABR dynamics
is shown in Figure 4.7(a). The calculation of Γ when the underlying is governed by the
SABR model is again involved and the MCFD estimation is not reliable. In Figure 4.7(b),
the convergence of the ddCOS Γ estimator is presented, where we observe convergence,
with impressive variance reduction.
4.4.3. VA R, ES AND THE D ELTA -G AMMA APPROACH

In the evaluation of market risk, the computation of risk measures is important, and
even mandatory for regulatory purposes to estimate the risk of large losses. With the risk
0.75 150
ddCOS ddCOS
Ref. 100
50
0.7
0
-50
-100
0.65
101 10 2
10 3
10 4
10 5
101 102 103 104 105
(a) ∆: Strike K = 0.04. (b) Γ: Strike K = 0.04.

4
Figure 4.7: The ddCOS method and SABR model: Greeks convergence test.
factors denoted by S and a time horizon ∆t , we define the change in S at time ∆t by ∆S.
The variation in S directly affects the value of a portfolio Π(t , S), containing derivatives
of S. We denote the changes in the value of the portfolio by ∆Π, so that the definition of
the loss in interval [t , ∆t ] is given by
L := −∆Π = Π(t , S) − Π(t + ∆t , S + ∆S).
In order to manage possible large losses, we are interested in the distribution of L,

specifically in the CDF, F L (x) = Pr(L < x), which can be employed to compute the risk
measures VaR or ES. The formal definition of the VaR reads
Pr(∆Π < VaR(q)) = 1 − F L (VaR(q)) = q,
with q a predefined confidence level, whereas, given the VaR, the ES measure is com-
puted as
ES := E[∆Π|∆Π > VaR(q)].
So, VaR is given as a quantile of the loss distribution, while ES is the average of the
largest possible losses.
Although simple in definition, the practical computation of these risk measures is
a challenging and computationally expensive problem, especially when the changes in
Π cannot be assumed linear in S. Then, VaR and ES estimation is often performed by
means of an Monte Carlo method. In order to find a balance between accuracy and
tractability, one of the most employed methodologies is the Delta-Gamma approxima-
tion which combines Monte Carlo path generation, a second-order Taylor expansion and
the sensitivities to reduce the computational cost and capture the non-linearity in port-
folio changes. The delta-gamma approximation of ∆Π (in the case of only one risk factor)
is given by
M ∂Πi 1X M ∂2 Πi
∆Π ≈ ∆S + (∆S)2 ,
X
wi wi
i =1 ∂S 2 i =1 ∂S 2
1 2
ddCOS ddCOS
COS COS
0.8 1.5
0.6
1
0.4
0.5
0.2
0
0
-2 -1 0 1 2 -2 0 2 4
(a) PDF Portfolio 1. (b) PDF Portfolio 2.

4
Figure 4.8: Recovered densities of L: ddCOS vs. COS.
with M the number of assets depending on risk factor S, w i and Πi the amount and the
value of asset i , respectively. The partial derivatives are evaluated at initial time t . In
the case of options contracts, these partial derivatives correspond to the ∆ and Γ sensi-
tivities. It is usually assumed that the distribution of ∆S is known (normal, Student’s t,
etc). Then, by applying the Delta-Gamma approach, the distribution of the losses, F L ,
and therefore the VaR and the ES are easily calculated.
The use of the ddCOS method in the context of the Delta-Gamma approach general-
izes its applicability. Since the use of the ChF is not longer required, we can assume non-
trivial dynamics for ∆S, where the use of Fourier inversion methods (as in [101]) would
be a limitation (it may be impossible to obtain a ChF). As we have seen, by employing the
ddCOS method, ∆ and Γ can be computed at once and, therefore, be directly employed
within the Delta-Gamma approximation. The ddCOS method can thus be used to re-
cover the distribution of ∆Π, whenever samples are available. This can be useful when
historical data is employed, and no particular distribution is assumed.
In order to show the performance of the ddCOS method within the Delta-Gamma ap-
proach, we first repeat the experiments from [101]. Two portfolios are considered, both
with the same composition (one European call and half a European put under the same
underlying asset, maturity 60 days and strike K = 101) but different time horizons, i.e. 1
day and 10 days. We denote them by Portfolio 1 and Portfolio 2, respectively. The under-
lying asset follows a GBM with S(0) = 100, r = 0.1 and σ = 0.3. Change ∆S is assumed to
be normally distributed.
In Figure 4.8 the recovered densities by the COS and ddCOS methods are depicted.
An almost perfect fit is observed, with the expected small-sized oscillations in the data-
driven approach. Since the computational domain is also driven by data, the ddCOS
recovered PDF remains within the defined domain, avoiding incorrect estimations out-
side the domain (see the COS curve in Figure 4.8(b)).
We also employ the ddCOS method to compute the risk measures VaR and ES. The
convergence of VaR and ES in terms of n is presented in Figure 4.9, with reference values
p
provided by [101]. As expected, the convergence rate for both estimators is O (1/ n).
10 0 10 1
VaR VaR
ES ES
10 -1 10 0
10 -2 10 -1
10 -3 10 -2
10 1 10 2 10 3 10 4 10 5 10 1 10 2 10 3 10 4 10 5
(a) Portfolio 1: q = 99%. (b) Portfolio 2: q = 90%.

4
Figure 4.9: VaR and ES convergence in n.
S MOOTHING THE PDF OF L

As seen in Figure 4.8, the densities estimated by the ddCOS method exhibit some ar-
tificial oscillations due to the lack of data in particular regions and the so-called Gibbs
phenomenon. Two possibilities to avoid the appearance of these oscillations are increas-
ing smoothing parameter p, and the application of so-called spectral filters within the
ddCOS formula (4.13). By parameter p we can include derivatives of the PDF into the
regularization (see Equation (4.10)). We analyze the use of p = 1. Filtering was already
successfully applied in the context of the Delta-Gamma approximation in [101], based
on the work by Ruijter et al. [112], and we refer to the references for the filter details.
Adding the filter is almost trivial as it merely implies a multiplication with a specific filter
term. Based on the references, we here choose the so-called 6-th order exponential filter
within the ddCOS formula.
In Figure 4.10, the resulting densities from the application of both alternatives are
presented. Whereas both smoothing techniques give highly satisfactory results for Port-
folio 1, the spectral filters are superior in the case of Portfolio 2. Based on these tests,
we suggest the use of a spectral filter to obtain smooth densities. Note, however, that
the application of these smoothing procedures does not give us an improvement in the
convergence, which is still dominated by the order of convergence in Equation (4.5).
D ELTA -G AMMA APPROACH UNDER THE SABR MODEL

In order to further test the ddCOS method in the context of the Delta-Gamma approach,
we now assume the dynamics of the underlying asset and ∆S to be governed by the SABR
dynamics. In Figure 4.11(a), the obtained VaR and ES when varying n are presented. No
reference is available here, since the MCFD is unstable for the Γ computation under the
SABR model. We observe that, already for n = 1, 000, a stable Γ-value is found and, even
more important, the variance is negligible. By the ddCOS method, a closed-form expres-
sion for the loss distribution is also obtained. The recovered F L and the corresponding
PDF f L are depicted in Figure 4.11(b) (for n = 100, 000), where we also include the result-
ing densities when employing the exponential spectral filter.
1 2
COS COS
ddCOS, p=1 ddCOS, p=1
0.8 ddCOS, filter 1.5 ddCOS, filter
0.6
1
0.4
0.5
0.2
0
4 0
-2 -1 0 1 2 -2 0 2 4
(a) PDF Portfolio 1. (b) PDF Portfolio 2.
Figure 4.10: Smoothed densities of L.
3 1
ddCOS f L
ddCOS F L
0.8 ddCOS f L, filter
2.5
ddCOS F L, filter
0.6
2
0.4
1.5
ddCOS VaR 0.2
ddCOS ES
1
101 102 103 104 105 0
-4 -2 0 2 4
(a) VaR and ES: q = 99%. (b) F L and f L .
Figure 4.11: Delta-Gamma approach under the SABR model. Setting: S(0) = 100, K = 100, r = 0.0, σ0 = 0.4,
α = 0.8, β = 1.0, ρ = −0.5, T = 2, q = 99% and ∆t = 1/365.
q 10% 30% 50% 70% 90%

VaR −1.4742 −0.5917 −0.0022 0.5789 1.3862
ES 0.1972 0.5345 0.8644 1.2517 1.8744
Table 4.5: VaR and ES under SABR model. Setting: S(0) = 100, K = 100, r = 0.0, σ0 = 0.4, α = 0.8, β = 1.0,
ρ = −0.5, T = 2, and ∆t = 1/365.
In Table 4.5, the VaR and ES under SABR are presented for several choices of q, rang-
ing from 10% to 90%. Again, the results seem to be coherent.
4.5. C ONCLUSIONS
In this chapter, the ddCOS method has been introduced. The method extends the COS
method applicability to cases when only data samples of the underlying asset are avail- 4
able. The method exploits a closed-form solution, in terms of Fourier cosine expansions,
of a PDF. The use of the COS machinery in combination with density estimation allowed
us to develop a data-driven method which can be employed for option pricing and risk
management. The ddCOS method particularly results in an efficient method for the ∆
and Γ sensitivities computation, based solely on the samples. Therefore, it can be em-
ployed within the Delta-Gamma approximation for calculating risk measures. Through
several numerical examples, we have empirically shown the convergence of our method.
In some cases, in order to get monotonic densities, it may be beneficial to add a filter
term to the ddCOS method.
A possible future extension may be the use of other basis functions. Haar wavelets
are for example interesting since they provide positive densities and allow an efficient
treatment of dynamic data.
The hybrid aspect presented in this chapter is expressed as a combination of data-driven

density estimation, Fourier series and Monte Carlo sampling. We have thus generalized
the COS method to a data-driven approach, the ddCOS method, where the knowledge
of the ChF is not required. Therefore, the technique presented here can be employed as
a Monte Carlo-based alternative for challenging problems in quantitative finance, like
Greek estimation or risk measures computation.
CHAPTER 5
GPU Acceleration of the Stochastic Grid Bundling Method
In this chapter, we move in yet another type of algorithm. Here we focus on a parallel GPU
version of the Monte Carlo-based SGBM for pricing multi-dimensional early-exercise op-
tions. To extend the method’s applicability, the problem dimensions will be increased dras-
tically. This makes SGBM very expensive in terms of computational costs on conventional
hardware systems based on CPUs. A parallelization strategy of the method is developed
and the GPGPU paradigm is used to reduce the execution time. An improved technique for
bundling asset paths, which is more efficient on parallel hardware is introduced. Thanks
to the performance of the GPU version of SGBM, a general approach for computing the
early-exercise policy is proposed. Comparisons between sequential and GPU parallel ver-
sions are presented.
5.1. I NTRODUCTION
In recent years, different techniques for pricing early-exercise option contracts, as they
appear in computational finance, have been developed. In the wake of the recent fi-
nancial crisis, accurately modelling and pricing these kinds of options gained additional
importance, as they also form the basis for incorporation of the so-called counterparty
risk premium to an option value, which can be seen as an option covering the fact that a
counterparty may go into default before the end of a financial contract, and thus cannot
pay possible contractual obligations.
The early-exercise pricing methods can be classified according to different kinds of
techniques depending on whether they are based on simulation and regression, transi-
tion probabilities or duality approaches. Focusing on the first class of methods, Monte
Carlo path generation and dynamic programming to determine an optimal early-exercise
policy and the option price are typically combined. Some representative methods were
developed by Longstaff and Schwartz [94] and Tsitsiklis and Van Roy [128].
The pricing problem becomes significantly more complex when the early-exercise
options are multi- or even high-dimensional. One of the recent simulation-regression
pricing techniques for early-exercise (Bermudan) options with several underlying assets,
on which we will focus, is the SGBM, proposed by Jain and Oosterlee in [76]. The method
is a hybrid of regression and bundling approaches, as it employs regressed value func-
tions, together with bundling of the state space to approximate continuation values at
different time-steps. A high biased direct estimator and an early-exercise policy are first
computed in SGBM. The early-exercise policy is then used to determine a lower bound to
This chapter is based on the article “GPU Acceleration of the Stochastic Grid Bundling Method for Early-
Exercise options”. Published in International Journal of Computer Mathematics, 92(12):2433–2454, 2015 [90].
91
92 5. GPU A CCELERATION OF THE S TOCHASTIC G RID B UNDLING M ETHOD
the true option price which is called the path estimator. The early-exercise policy com-
putation involves the computation of expectations of certain basis functions. Usually,
these basis functions are chosen by experience in such a way that an analytic solution
for the expectations appearing in the pricing method is available.
In this chapter, we extend SGBM’s applicability by drastically increasing the problem
dimensionality, the number of bundles and, hence, the number of Monte Carlo asset
paths. As the method becomes much more time-consuming then, we parallelize SGBM
taking advantage of the GPGPU paradigm. It is known that GPGPU is very well suited for
financial purposes and Monte Carlo techniques. In the case of pricing early-exercise op-
tions, several contributions regarding GPU implementations of different techniques ap-
peared recently, Abbas-Turki and Lapeyre [1], Benguigui and Baude [7], Cvetanoska and
Stojanovski [38], Dang et al. [39], Fatica and Phillips [52] and Pagès and Wilbertz [103].
All these papers are based on a combination of a Monte Carlo method and dynamic pro-
gramming, except the Dang et al. paper, which uses PDE-based pricing methods and the
Pagès and Wilbertz paper, which employs Monte Carlo simulation together with a quan-
tization method. Our GPU version of SGBM is based on parallelization in two steps,
according to the method’s stages, i.e., forward in time (Monte Carlo path generator) fol-
5 lowed by the backward stage (early-exercise policy computation). This is a novelty with
respect to other methods since, although the parallelization of a Monte Carlo method is
immediate, the parallelization of the backward stage is not trivial for other methods, for
example not for the Least Squares Monte Carlo Method (LSM) by Longstaff and Schwartz
[94] where some load balancing issues for parallel systems can appear as only the in-the-
money paths are considered. Due to some timing and distribution issues observed in the
original bundling procedure caused by an increasing number of bundles, we present an-
other bundling technique for SGBM which is much more efficient and more suitable on
parallel hardware.
Thanks to the performance of our SGBM GPU version, we can explore different possi-
bilities to compute the expectations appearing within the SGBM formulation, generalize
towards different option contracts and, in particular, towards more general underlying
asset models. A general approach to compute the expectations needed for the early-
exercise policy is proposed and evaluated. This approach is based on the computation
of a ChF for the discretized underlying stochastic differential system. Once this ChF of
the discretized asset dynamics is determined, we can use it for the expectations calcula-
tion, for example, for multi-dimensional local volatility asset models.
The chapter is organized as follows. In Section 5.2, the Bermudan option pricing
problem is introduced. Section 5.3 describes the SGBM. The new approach to compute
the expectations for the early-exercise policy is explained in Section 5.4. Section 5.5 gives
details about the GPU implementation. In Section 5.6, numerical results and CPU/GPU
time comparisons are shown. Finally, we conclude in Section 5.7.
5.2. B ERMUDAN OPTIONS

This section defines the Bermudan option pricing problem and sets up the notations
used. A Bermudan option is an option where the buyer has the right to exercise at a
number of times, t ∈ [t 0 = 0, . . . , t m , . . . , t M = T ], before the end of the contract, T . St :=
(S 1 (t ), . . . , S d (t )) ∈ Rd defines a d -dimensional underlying process which here follows the
5.2. B ERMUDAN OPTIONS 93
dynamics given by the system of SDEs
dS 1 (t ) = µ1 (St )dt + σ1 (St )dW1 (t ),

dS 2 (t ) = µ2 (St )dt + σ2 (St )dW2 (t ),
.. (5.1)
.
dS d (t ) = µd (St )dt + σd (St )dWd (t ),
where Wδ (t ), δ = 1, 2, . . . , d , are correlated standard Brownian motions. The instanta-
neous correlation coefficient between Wi (t ) and W j (t ) is ρ i , j . The correlation matrix
is thus defined as
ρ 1,2 · · · ρ 1,d
 
1
ρ 1 · · · ρ 2,d 
 1,2 
C=  .. .. .. .. ,
 . . . . 
ρ 1,d ρ 2,d ··· 1
and the Cholesky decomposition of C reads 5
 
1 0 ··· 0
ρ L ··· 0 
 1,2 2,2 
L= . (5.2)
 .. .. .. ..
 . . . .


ρ 1,d L 2,d ··· L d ,d
Let h t := h(St ) be an adapted process representing the intrinsic value of the option,
i.e. the holder of the option receives max(h t , 0), ifR the option is exercised at time t . With
t
the risk-free savings account process B (t ) = exp( 0 r s ds), where r t denotes the instanta-
neous risk-free rate, we define
B (t m−1 )
D tm := .
B (t m )
We consider the special case where r t = r is constant. The problem is then to com-
pute
h(Sτ )
· ¸
V (t 0 , St0 ) = max E ,
τ B (τ)
where τ is a stopping time, taking values in the finite set {0, t 1 , . . . , T }. The value of the
option at the terminal time T is equal to the option’s payoff,
V (T, ST ) = max (h(ST ), 0) .

The conditional continuation value Q tm−1 , i.e. the expected payoff at time t m−1 , is
given by
Q tm−1 (Stm−1 ) = D tm E V (t m , Stm )|Stm−1 .

£ ¤
(5.3)
The Bermudan option value at time t m−1 and state Stm−1 is then given by
¡ ¡ ¢ ¡ ¢¢
V (t m−1 , Stm−1 ) = max h Stm−1 ,Q tm−1 Stm−1 .
We are interested in finding the value of the option at the initial state St0 , i.e. V (t 0 , St0 ).
5.3. S TOCHASTIC G RID B UNDLING M ETHOD

The SGBM [76] is a simulation-based Monte Carlo method for pricing early-exercise op-
tions (such as Bermudan options). SGBM first generates Monte Carlo paths forward in
time, which is followed by determining the optimal early-exercise policy, moving back-
wards in time in a dynamic programming framework, based on the Bellman principle of
optimality. The steps involved in the SGBM algorithm are briefly described in the follow-
ing paragraphs.
S TEP I: G ENERATION OF STOCHASTIC GRID POINTS

The grid points in SGBM are generated by Monte Carlo sampling, i.e., by simulating in-
5 dependent copies of sample paths, {St0 (n), . . . , St M (n)}, n = 1, . . . , N , of the underlying
process St , all starting from the same initial state St0 . The n-th grid point at exercise
time-step t m is then denoted by Stm (n), n = 1, . . . , N . Depending upon the underly-
ing stochastic process an appropriate discretization scheme (with ∆t the discretization
step), e.g. the Euler-Maruyama scheme, is used to generate sample paths. Sometimes
the diffusion process can be simulated directly, essentially because it appears in closed
form, like for the regular multi-dimensional Black–Scholes model.
S TEP II: O PTION VALUE AT TERMINAL TIME

The option value at the terminal time t M = T is given by
¡ ¢
V (t M , St M ) = max h(St M ), 0 ,
¡ ¢
with max h(St M ), 0 the payoff function.
This relation is used to compute the option value for all grid points at the final time-
step.
The following steps are subsequently performed for each time-step, t m , m ≤ M , re-
cursively, moving backwards in time, starting from t M .
S TEP III: B UNDLING

The grid points at exercise time t m−1 are clustered or bundled into Btm−1 (1), . . . , Btm−1 (ν)
non-overlapping sets or partitions. SGBM employs bundling to approximate the condi-
tional distribution in Equation (5.3) using simulation. The method samples this distri-
bution by bundling the grid points at t m−1 and then uses those paths that originate from
the corresponding bundle to obtain a conditional sample for time t m , see also in [76].
Different approaches for partitioning can be considered. Due to its importance in the
parallel case, this decision is discussed in more detail in Section 5.3.1.
5.3. S TOCHASTIC G RID B UNDLING M ETHOD 95
S TEP IV: M APPING HIGH - DIMENSIONAL STATE SPACE TO A LOW- DIMENSIONAL SPACE
Corresponding to each bundle Btm−1 (β),
³ β = 1,´ . . . , ν, a parametrized value function Z :
β β
Rd × Rb 7→ R, which assigns values Z Stm , αtm to states Stm , is computed. Here αtm ∈
Rb is a vector of free parameters. The objective is then to choose, for each t m and β, a
β
parameter vector αtm so that
β
³ ´
Z Stm , αtm ≈ V (t m , Stm ).
β
³ ´
After some approximations to be discussed in Section 5.3.2, Z Stm , αtm can be com-
puted by using ordinary least squares regression.
S TEP V: C OMPUTING THE CONTINUATION AND OPTION VALUES AT t m−1

The continuation values for S tm−1 (n) ∈ Btm−1 (β), n = 1, . . . , N , β = 1, . . . , ν, are approxi-
mated by
β
h ³ ´ i
Qbtm−1 (Stm−1 (n)) = E Z Stm , αtm |Stm−1 (n) .
5
The option value is then given by
¡ ¡ ¢ ¡ ¢¢
Vb (t m−1 , Stm−1 (n)) = max h Stm−1 (n) , Qbtm−1 Stm−1 (n) .
The Steps III, IV and V are repeated backward in time, till the initial time, t 0 is reached.
Thus, the value V (t 0 , St0 ) of the option is approximately computed.
5.3.1. B UNDLING
One of the techniques proposed (in [76]) to partition the asset path data into ν non-
overlapping sets at each time t m−1 is the k-means clustering technique. The algorithm
uses an iterative refinement algorithm, where, given an initial guess of cluster means,
first of all the algorithm assigns each data item to one specific set (i.e. bundle) by calcu-
lating the distance between the item and the cluster mean (under some measure) and
subsequently updates the sets and the cluster means. This process is repeated until
some stopping condition is satisfied. The procedure needs a pre-bundling step to set
approximated initial cluster means. The bundles obtained by using the k-means tech-
nique can be very irregular, i.e. the number of data items within each bundle may vary
significantly. This fact can be a problem when many bundles are considered, and parallel
load-balancing can be an issue too.
Since our goal is to drastically increase the number of bundles used and, in particu-
lar, the problem dimensionality, the k-means algorithm becomes too expensive in terms
of computational time and memory usage in high dimensions because the iterative pro-
cess of searching new sets (of high dimension) takes much time and d -dimensional data
points for each Monte Carlo path and for each time-step have to be stored. In addition,
it may happen that some bundles do not contain a sufficient number of data points to
compute an accurate regression function when the number of bundles increases. In or-
der to overcome these two problems of the k-means clustering technique, we propose
another bundling technique in the SGBM context which is much more efficient when
SORT SPLIT
Figure 5.1: Equal partitioning scheme.
5
taking into account our goal of efficiency in high dimensions: it does not involve an
iterative process, distributes the data equally and does not require the storage of the d -
dimensional points. The details are given in the next subsection.
E QUAL - PARTITIONING TECHNIQUE

The equal-partitioning bundling technique is particularly well-suited for parallel pro-
cessing; it involves two steps: sorting and splitting. The general idea is to sort the data
first under some convenient criterion and then split the sorted data items into sets (i.e.
bundles) of equal size. A schematic representation of this technique is shown in Figure
5.1.
With this simple approach, the drawbacks of the iterative bundling in the case of
very high dimensions and an enormous number of bundles can be avoided. The sorting
process is independent of the dimension of the problem, more efficient and less time-
consuming than an iterative search and, furthermore, it is highly parallelizable. In addi-
tion, the storage of all Monte Carlo simulation data points is avoided. Assuming that the
criterion is known in advance, we can perform the necessary computations within inside
the Monte Carlo generator. That allows us to map the d -dimensional Monte Carlo points
to a 1D vector and reduce amount of the stored data, i.e., storing a 2D matrix (N × M ) in-
stead of storing a 3D matrix (N × M × d ). The split stage assigns directly the portions
of data to bundles which will contain the same number of similar (following some cri-
terion) data items. Hence, the regression can be performed accurately even though the
number of bundles increases in a significant way. Furthermore, the equally sized bun-
dles allow for a better load balancing within the parallel implementation.
5.3.2. PARAMETERIZING THE OPTION VALUES

As we aim to increase the dimensions of the problem drastically, the option pricing prob-
lem may become intractable and requires the approximation of the value function. This
can be achieved by a parametrized value function Z : Rd × Rb 7→ R, which assigns a value
Z (Stm , α) to state Stm , where α ∈ Rb is a vector of free parameters. The objective is to

β
choose, corresponding to each bundle β at time point t m−1 , a parameter vector αtm := α
so that,
β
³ ´
V (t m , Stm ) ≈ Z Stm , αtm .
As in the original SGBM paper, we follow the approach of Tsitsiklis and Van Roy [128]
and we use basis functions to approximate the values of the options. Hence, two impor-
tant decisions have to be made: the form of the function Z and the basis functions. For
each particular problem we define several basis functions, φ1 , φ2 , . . . , φb , that are typ-
ically chosen based on experience, as in the case of the LSM method [94], aiming to
represent relevant properties of a given state, Stm . In Section 5.3.2 and in Section 5.4, we
present two possible choices for the basis
³ functions, one specific and the other one more
β
´
generally applicable. In our case, Z Stm , αtm depends on Stm only through φk (Stm ).
β β
³ ´ ³ ´
Hence, for some function f : Rb × Rb 7→ R, we can write Z Stm , αtm = f φk (Stm ), αtm ,
where
b
5
β β
³ ´ X
Z St m , αt m = αtm (k)φk (Stm ). (5.4)
k=1
β
An exact computation of the vector of free parameters, αtm , is generally not feasible.
Hence, Equation (5.4) can be approximated by
b
β β
³ ´ X
Z St m , α
b tm = α
b tm (k)φk (Stm ), (5.5)
k=1
satisfying
|Btm−1 (β)|
Ã !2
b
β
α
X X
arg min V (t m , Stm (n)) − b tm (k)φk (Stm (n)) ,
β n=1 k=1
α
b tm
for the corresponding bundle Btm−1 (β). The parametrized function in Equation (5.5) is
computed by using ordinary least squares regression, so that
β β
³ ´
V (t m , Stm (n)) = Z Stm (n), α
b t m + ²t m ,
β
where Stm−1 (n) ∈ Btm−1 (β). The regression error, ²tm , can be neglected here, as we have a
sufficiently large number of paths in each bundle.
C OMPUTING THE CONTINUATION VALUE

β
³ ´
By employing the parametrized option value function Z Stm , α b tm corresponding to bun-
dle Btm−1 (β), the continuation values for the grid points that belong to this bundle are
approximated by
β
h ³ ´ i
Qbtm−1 (Stm−1 (n)) = D tm−1 E Z Stm , α
b tm |Stm−1 = Stm−1 (n) ,
where Stm−1 (n) ∈ Btm−1 (β). Using Equation (5.5), this can be written as
"Ã ! #
b
β
Qbtm−1 (Stm−1 (n)) = D tm−1 E α
X
b tm (k)φk (Stm ) |Stm−1 = Stm−1 (n)
k=1
(5.6)
b
β
α
b tm (k)E φk (Stm )|Stm−1
X £ ¤
= D tm−1 = Stm−1 (n) .
k=1
The continuation value will give us a reference value to compute the early-exercise
policy and it will be used by the estimators in Section 5.3.3.
C HOICE OF BASIS FUNCTIONS

£ basis functions φk should
The be chosen such that the expectations in Equation (5.6),
E φk (Stm )|Stm−1 = Stm−1 (n) , are easy to calculate, i.e. they are preferably known in closed
¤
form or otherwise have analytic approximations. The intrinsic value of the option, h(·), is
usually an important and useful basis function, especially in high problem dimensions.
In this section, we describe the choices of basis functions for different Bermudan basket
5 options with the underlying processes following GBM; we thus choose in Equation (5.1),
µδ (St ) = (r t − q δ )S δ (t ), σδ (St ) = σδ S δ (t ),
where r t is the risk-free rate and q δ and σδ , δ = 1, 2, . . . , d , are the dividend yield rates and
the volatility, respectively.
In the case of a geometric Bermudan basket option, the intrinsic value of the option
is defined by
Ã !1 Ã !1
¡ ¢ d
Y d
¡ ¢ d
Y d
h St m = S δ (t m ) − K , h Stm = K − S δ (t m ) ,
δ=1 δ=1
for call or put options, respectively, where K is the strike value of the option contract.
Then, basis functions that make sense are given by
Ã ! 1 k−1
d d
φk (Stm ) = 
Y
S δ (t m )  , k = 1, . . . , b. (5.7)
δ=1
The expectation in Equation (5.6) requires the computation of the expectations of

the expression in Equation (5.7), given Stm−1 ,
´ ¶k−1
σ̄2
µ ³
µ̄+ (k−1) ∆t
E φk (Stm )|Stm−1 = P tm−1 e
£ ¤
2
, k = 1, . . . , b, (5.8)
where,
!1 !2
σ2δ
Ã Ã ! Ã
d d
1 Xd 1 Xd d
2 2
, µ̄ = , σ̄ = 2
Y X
P tm−1 = S δ (t m−1 ) r t − qδ − L ,
δ=1 d δ=1 2 d i =1 j =1 i j
with L i j − matrix element i , j of L, the Cholesky decomposition of the correlation matrix,

given by Equation (5.2).
For the arithmetic Bermudan basket options, the intrinsic values of the call and put
options are
Ã ! Ã !
¡ ¢ 1 X d ¡ ¢ 1 Xd
h St m = S δ (t m ) − K , h Stm = K − S δ (t m ) .
d δ=1 d δ=1
The basis functions are chosen as
Ã !k−1
1 Xd
φk (Stm ) = S δ (t m ) , k = 1, . . . , b.
d δ=1
Again, the continuation value, as given by Equation (5.6), requires us to compute,
"Ã !k−1 #
1 Xd
E φk (Stm )|Stm−1 = E
£ ¤
S δ (t m ) |Stm−1 , k = 1, . . . , b. (5.9)
d δ=1
The expectation in Equation (5.9) can be expressed as a linear combination of mo- 5
ments of the geometric average of the assets [76], i.e.
Ã !k Ã !
d k
(S δ (t m ))kδ ,
X X Y
S δ (t m ) =
δ=1 k 1 +k 2 +···+k d =k k1 , k2 , . . . , kd 1≤δ≤d
where,
Ã !
k k!
= ,
k1 , k2 , . . . , kd k1 ! k2 ! · · · kd !
which can be computed in a straightforward way by Equation (5.8).
5.3.3. E STIMATING THE OPTION VALUE

The estimation of the option value is the final step in SGBM. In this chapter, we follow
the original proposed idea in [76] and we consider the so-called direct estimator and
path estimator, which can give us a confidence interval for the option price. SGBM has
been developed as a so-called duality-based method, as originally introduced by Haugh
and Kogan [70] and Rogers [108]. Using a duality-based methods an upper bound on
the option value for a given exercise policy can be obtained, by adding a non-negative
quantity that penalizes potentially incorrect exercise decisions made by the sub-optimal
policy. The SGBM direct estimator is typically biased high, i.e., it is often an upper bound
estimator. The definition of the direct estimator is
¡ ¡ ¢ ¡ ¢¢
Vb (t m−1 , Stm−1 (n)) = max h Stm−1 (n) , Qbtm−1 Stm−1 (n) ,
where n = 1, . . . , N . The final option value reads
1 XN
E[Vb (t 0 , St0 )] = Vb (t 0 , St0 (n)).
N n=1
The direct estimator corresponds to Step V of the initial description in Section 5.3.
Once the optimal early-exercise policy has been obtained, the path estimator, which
is typically biased low, can be developed based on the early-exercise policy. The result-
ing confidence interval is useful, because, depending on the problem at hand, some-
times the path estimator and sometimes the direct estimator appears superior in terms
of accuracy. The obtained confidence intervals are generally small, indicating accurate
SGBM results. In order to compute the low-biased estimates, we generate a new set of
paths, as is common for duality-based Monte Carlo methods, S(n) = {St1 (n), . . . , St M (n)},
n = 1, . . . , NL , using the same scheme as followed for generating the paths in the case
of the direct estimator and the bundling stage is performed considering the new set of
paths. Along each path, the approximate optimal policy exercises at,
τb∗ (S(n)) = min{t m : h Stm (n) ≥ Qbtm Stm (n) , m = 1, . . . , M },

¡ ¢ ¡ ¢
where Qbtm (Stm (n)) is previously

¡ computed
¢ using Equation (5.6). The path estimator is
then defined by v(n) = h Sτb∗ (S(n)) . Finally, the low-biased estimate given by the path
estimator is
5 NL
1 X
V (t 0 , St0 ) = lim v(n),
NL NL n=1
where Vt0 (St0 ) is the true option value.
To summarize and clarify the general idea behind SGBM, in Algorithm 4 we show the
corresponding algorithm considering both, direct estimator and path estimator.
Algorithm 4: SGBM.
Data: St0 , K , µδ , σδ , ρ i , j , T, N , M
Pre-Bundling (only in k-means case).
Generation of the grid points (Monte Carlo). Step I.
Option value at terminal time t = M . Step II.
for Time t = (M − 1) . . . 1 do
Bundling. Step III.
for Bundle β = 1 . . . ν do
Exercise policy (Regression). Step IV.
Continuation value. Step V.
Direct estimator. Step V.
Generation of the grid points (Monte Carlo). Step I.

Option value at terminal time t = M . Step II.
for Time t = (M − 1) . . . 1 do
Bundling. Step III.
for Bundle β = 1 . . . ν do
Continuation value. Step V.
Path estimator. Step V.
5.4. C ONTINUATION VALUE COMPUTATION : NEW APPROACH 101
5.4. C ONTINUATION VALUE COMPUTATION : NEW APPROACH

In Section 5.3.2, we showed that, for certain derivative contracts (geometric and arith-
metic Bermudan basket options) and underlying processes (GBM), analytic solutions
were available for the expectations that we needed to compute within SGBM. However,
finding suitable basis functions resulting in analytic expressions for the expectations is
not always possible for all underlying models. For that, in this chapter, we propose a
different way to compute the expected values in Equation (5.6) which is more gener-
ally applicable, for example, we can also work with local volatility models with this ap-
proach. The approach consists of, first, discretizing the asset process and then obtain-
ing the multi-variate ChF of the discrete process. This procedure is based on the work
of Ruijter and Oosterlee in [111]. Once we have this ChF of the discrete process, we can
compute the required expectations for certain choices of basis functions.
5.4.1. D ISCRETIZATION , JOINT C H F AND JOINT MOMENTS

Taking into account the correlation between Brownian motions and the Cholesky de-
composition given by Equation (5.2) and using an Euler-Maruyama scheme with time-
step ∆t = t m+1 − t m , we can discretize the general system of SDEs defined by Equa- 5
tion (5.1), with independent Brownian motions, ∆W̃δ (t m+1 ) = W̃δ (t m+1 ) − W̃δ (t m ), δ =
1, 2, . . . , d , as follows
S 1 (t m+1 ) = S 1 (t m ) + µ1 (Stm )∆t + σ1 (Stm )∆W̃1 (t m+1 ),

S 2 (t m+1 ) = S 2 (t m ) + µ2 (Stm )∆t + ρ 1,2 σ2 (Stm )∆W̃1 (t m+1 ) + L 2,2 σ2 (Stm )∆W̃2 (t m+1 ),
..
.
S d (t m+1 ) = S d (t m ) + µd (Stm )∆t + ρ 1,d σd (Stm )∆W̃1 (t m+1 ) + · · · + L d ,d σd (Stm )∆W̃d (t m+1 ),
where Stm = (S 1 (t m ), S 2 (t m ), . . . , S d (t m )).

The d -variate ChF of process Stm+1 , given Stm , is given by
" Ã ! #
d
ψStm+1 u 1 , u 2 , . . . , u d |Stm = E exp
¡ ¢ X
i u j S j (t m+1 ) |Stm
j =1
" Ã Ã !! #
d j
= E exp i u j S j (t m ) + µ j (Stm )∆t + σ j (Stm ) L k, j ∆W̃k (t m+1 )
X X
|Stm
j =1 k=1
Ã ! Ã " Ã !#!
d d d
i u j S j (t m ) + µ j (Stm )∆t · E exp i u j L k, j σ j (Stm )∆W̃k (t m+1 )
X ¡ ¢ Y X
= exp
j =1 k=1 j =k
Ã ! Ã Ã !!
d d d
i u j S j (t m ) + µ j (Stm )∆t · ψN (0,∆t ) u j L k, j σ j (Stm )
X ¡ ¢ Y X
= exp ,
j =1 k=1 j =k
where i is the imaginary unit and ψN (µ,σ2 ) (u) = exp i µu − 0.5σ2 u 2 is the ChF of a nor-
¡ ¢
mal random variable with mean µ and variance σ2 .

The joint moments of the product of several random variables can be obtained by
evaluating the following expression (more details in [85] or [129]), based on the ChF of
the discrete process
M Stm+1 = E (S 1 (t m+1 ))c1 (S 2 (t m+1 ))c2 · · · (S d (t m+1 ))cd |Stm

£ ¤
" c +c +···+c
∂1 2 dψ
#
c 1 +c 2 +···+c d Stm+1 (u|St m ) (5.10)
= (−i ) c c c ,
∂u 11 ∂u 22 · · · ∂u dd u=0
being u = (u 1 , u 2 , . . . , u d ).
If we take the basis functions as used in Equation (5.5), to be a product of the under-
lying processes to some power, i.e.,
Ã !k−1
d
φk (Stm ) =
Y
S δ (t m ) , k = 1, . . . , b, (5.11)
δ=1
the expected value in Equation (5.6) can be easily computed by means of Equation (5.10).
5 The approximation obtained in this way is, in general, worse than the analytic value
because the ChF on which the expectations are based is related to the Euler-Maruyama
SDE discretization and thus less accurate. However, since we can increase the number of
bundles and, in particular, time-steps drastically (due to our GPU implementation), we
can employ a suitable combination of these to price products without analytic solution
for the ChF. More involved models for the underlying asset dynamics can be chosen, for
which we do not have analytic expressions for these expectations. In Section 5.6.4, more
details are given.
5.5. I MPLEMENTATION DETAILS

5.5.1. GPGPU: CUDA
GPGPU is the use of graphics hardware (GPUs) in order to perform computations that
are typically performed by central processing units (CPUs). In this sense, the GPU can
be seen as a co-processor of the CPU.
CUDA is a parallel computing platform and programming model developed and main-
tained by NVIDIA (see [35]) for its GPUs. CUDA eases the coding of algorithms by means
of an extension of traditional programming languages, like C. In this section, we give
some basic details about the platform that are necessary to understand the parallel ver-
sion of SGBM. More information about CUDA can be obtained from [30, 34, 51, 81, 82,
113, 134].
P ROGRAMMING MODEL
The basic operational units in CUDA are the CUDA threads. The CUDA threads are pro-
cesses that are executed in parallel. The threads are grouped into blocks of the same size
and blocks of threads are grouped into grids. Such organization eases the adaptability to
different problems. One grid executes special functions called kernels so that all threads
of the grid execute the same instructions in parallel.
5.5. I MPLEMENTATION DETAILS 103
M EMORY HIERARCHY
CUDA threads may access data from multiple memory spaces during their execution.
Each thread manages its own registers and has private local memory. Each thread block
has shared memory visible to all threads of the block and with the same lifetime as the
block. All threads have access to the same global memory. There are also two additional
read-only memory spaces accessible by all threads: the constant and texture memory
spaces. In terms of speed, the registers represent the fastest memory (but also the small-
est) while the global and local memories are the slowest. In recent GPU architectures
(Kepler GK110 and GK110B [79], for example), several levels of cache are included to
improve the accessibility.
M EMORY TRANSFERS
A key aspect in each parallel system is the memory transfer. Specifically in a GPU frame-
work, the transfers between the CPU main memory and the GPU memory space are of
great importance for the performance of the implementation. The number of transfers
and the amount of data for each transfer are important. CUDA provides different ways to
allocate and move data. In that sense, one of the CUDA 5.5 features is Unified Virtual Ad- 5
dressing (UVA) which allows asynchronous transfers and page-locked memory accesses
by using a single address space for both CPU and GPU.
5.5.2. PARALLEL SGBM

The original SGBM implementation [76] was done in Matlab. The first step of our im-
plementation was to code an efficient sequential version of the method in the C pro-
gramming language, because it eases the parallel coding in CUDA. In addition, both im-
plementations can be used to compare results and execution times. Once we obtained
the C-version, we coded the CUDA-version aiming to parallelize the suitable parts of the
method. In addition, we also carried out the implementation of SGBM with the equal-
partitioning bundling technique in C and CUDA.
Since SGBM is based on two clearly separated stages, we parallelize them separately.
First of all, the Monte Carlo path generation is parallelized (Step I). As is well-known,
Monte Carlo methods are very well suitable for parallelization, because of characteris-
tics like a very large number of simulations and data independence. In Figure 5.2(a), we
see schematically how the parallelization is done where p 0 , p 1 , . . . , p N −1 are the CUDA
threads. The second main stage of SGBM is the regression and the computation of the
continuation and option values (Steps IV and V) in each bundle, backwards in time.
Due to the data dependency between time-steps, the way to parallelize this stage of the
method is by parallelizing over the bundles, performing the calculations in each bundle
in parallel. Schematic and simplified representations with two bundles are given in Fig-
ure 5.2(b) and Figure 5.2(c) for the two considered bundling techniques, k-means and
equal-partitioning. Note that, actually, several stages of parallelization are performed,
one per time-step. Between each parallel stage, the bundling (Step III) is carried out. This
step can be also parallelized in the case of equal-partitioning. In the case of k-means,
bundling has to be done sequentially.
As mentioned, the memory transfers between CPU and GPU are key since we have
to move huge amounts of data. An increase of the number of bundles implies a drastic
(a) Monte Carlo stage.
(b) Bundling stage using k-means. (c) Bundling stage using equal-partitioning.
Figure 5.2: Parallel SGBM. Monte Carlo and bundling stages.

5.5. I MPLEMENTATION DETAILS 105
increase of the number of Monte Carlo paths. Data has to be moved to CPU memory
between the stages of parallelization, Monte Carlo path simulation and bundle compu-
tations. For example, in case of the Monte Carlo scenario generator, we need to store in
and move from GPU global memory to CPU memory N × M × d doubles1 (8 bytes).
In the following subsections, we will show more specific details of the CUDA imple-
mentation of the two SGBM versions, the original one (with k-means bundling) and the
new one (with equal-partitioning).
PARALLEL M ONTE C ARLO PATHS

In a GPU very large numbers of threads can be managed, so we launch one CUDA thread
per Monte Carlo path. The necessary random numbers are obtained “on the fly” in each
thread by means of the cuRAND library [37]. In addition, the intrinsic value of the option
and also the computation of the expectation in Equation (5.6) are performed within the
Monte Carlo generator, decreasing the number of launched loops i.e. we perform all nec-
essary computations for the bundling and pricing stages taking advantage of the parallel
Monte Carlo simulation. The intermediate results are stored in an array defined inside
the kernel which can be allocated in the registers, speeding up the memory accesses.
In the original implementation (using k-means bundling), we need to store all Monte
5
Carlo data because it will be used during the bundling stage. This is a limiting factor
since the maximum memory storage in global memory is easily reached. Furthermore,
we have to move each value obtained from GPU global memory to CPU main memory
and, with a significant number of paths, time-steps and dimensions, this transfer can
become really expensive.
When equal-partitioning bundling is considered, the main difference is that we also
perform calculations for the sorting criterion inside the Monte Carlo generator, avoiding
the storage of the complete Monte Carlo simulation and the transfer of data from GPU
global memory to CPU main memory. This approach gives us a considerable perfor-
mance improvement and allows us to increase drastically the dimensionality and also
the number of Monte Carlo paths (depending on the number of bundles).
B UNDLING SCHEMES
For the k-means clustering the computations of the distances between the cluster means
and all of the stochastic grid points have been parallelized. However, the other parts
must be performed sequentially. The very large number of bundles makes this very ex-
pensive, since the bundling must be done in each time-step.
As mentioned, equal-partitioning bundling involves two operations: sorting and split-
ting. It is well known that efficient sorting algorithms are a challenging research topic
(see [84], for example). In parallel systems, like GPUs, this is even more important. In
recent years, a great effort has taken place, see, for example, [77], [98], [115] or [126], to
adapt classical algorithms to the new parallel systems and developing new parallel tech-
niques for sorting. Several libraries for GPU programming appeared in this field, like
Thrust [127], CUB [33], Modern GPU [99], clogs [27] or VexCL [132].
In our work, we take advantage of the CUDA Data-Parallel Primitives Library (CUDPP),
described in [36]. CUDPP is a library of data-parallel algorithm primitives that are im-
1 Double-precision floating-point format.
portant building blocks for a wide variety of data-parallel algorithms, including sorting.
The sorting algorithms are based on the work of Satish, Harris and Garland [115]. The
authors showed the performance of several sorting algorithms implemented on GPUs.
Following the results of their work, we choose the parallel Radix sort which is included
in version 2.1 of CUDPP. In addition, CUDPP provides a kernel-level API2 , allowing us to
avoid the transfers between stages of parallelization.
Once the sorting stage has been performed, the splitting stage is immediate since
the size of the bundles is known, i.e. N /ν. Each CUDA thread manages a pointer which
points at the beginning of the corresponding region of global memory for each bundle.
The global memory allocation is made for all bundles which means that the bundle’s
memory areas are adjacent and the accesses are faster.
E STIMATORS
When the bundling stage is done, the exercise policy and the final option values can be
computed by means of direct and path estimators. In order to use the GPU memory
efficiently, the obtained Monte Carlo paths (once bundled) are sorted with respect to
the bundles and stored in that way. We minimize the global memory accesses and the
5 sorted version is much more efficient because, again, the bundle’s memory areas are
contiguous. For this purpose, we use again the CUDPP 2.1 library. Note that the data is
already sorted in the case of equal-partitioning bundling.
For the direct estimator, one CUDA thread per bundle is launched at each time-step.
For each bundle, the regression and option values are calculated on the GPU. All threads
collaborate in order to compute the continuation value which determines the early-
exercise policy. As we mentioned before, T /∆t stages of parallelization are performed,
i.e. one per time-step. Hence, in each stage of parallelization, we need to transfer only
the data corresponding to the current time-step from CPU memory to GPU global mem-
ory. This data cannot be transferred at once because we wish to take advantage of the
optimized parallel sorting libraries for the sorting step after the bundling stage. This
allows us to minimize and improve the global memory accesses when the sequential
bundling is employed, i.e. considering k-means clustering. This fact does not imply
a reduction of the performance since the total transferred amount is the same. Note
again that, employing equal-partitioning bundling, these transfers are avoided since this
bundling technique is fully parallelizable.
Once the early-exercise policy is determined, the path estimator can be executed.
For that, a new set of grid points has to be generated following the procedure described
in Section 5.5.2. In the case of the path estimator, the parallelization can be done over
paths because the early-exercise policy is already known (given by the previous compu-
tation of the direct estimator) and it is not needed to perform the regression again. One
CUDA thread per path is launched and it computes the optimal exercise time and the
cash flows according to the policy. Another sorting stage is needed to assign the paths
to the corresponding bundle. Again, we use the sorting functions of CUDPP 2.1 for that
purpose.
In the final stage for both estimators, a summation is necessary to determine the final
option price. We take advantage of the Thrust library [127] to perform this reduction on
2 Application Programming Interface.
5.6. R ESULTS 107
the GPU.
We present a schematic representation of parallel SGBM in Algorithm 5, highlight-
ing the parts that have been parallelized and considering the equal-partitioning version.
The variables payoffData, expData and critData correspond to three matrices in which
we store the necessary computations for the pricing, regression and sorting stages, re-
β
spectively, for each path and each exercise time. The αt values are computed in the
regression step and determine the early-exercise policy, which is later used in the path
estimator calculation.
Algorithm 5: Parallel SGBM.

Data: St0 , K , µδ , σδ , ρ i , j , T, N , M
// Generation of the grid points (Monte Carlo). Step I.
// Option value at terminal time t = M . Step II.
[payoffData, critData, expData] = MonteCarloGPU(St0 , K , µδ , σδ , ρ i , j , T, N , M );
for Time t = M . . . 1 do
// Bundling. Step III.
SortingGPU(critData[t-1]);
begin CUDAThread per bundle β = 1 . . . ν
5
// Exercise policy (Regression). Step IV.
β
αt = LeastSquaresRegression(payoffData[t]);
// Continuation value. Step V.
β
CV = ContinuationValue(αt , expData[t-1]);
// Direct estimator. Step V.
DE = DirectEstimator(CV, payoffData[t-1]);
return DE;
// Generation of the grid points (Monte Carlo). Step I.
// Option value at terminal time t = M . Step II.
[payoffData, critData, expData] = MonteCarloGPU(St0 , K , µδ , σδ , ρ i , j , T, N , M );
for Time t = M . . . 1 do
// Bundling. Step III.
SortingGPU(critData[t-1]);
begin CUDAThread per path n = 1 . . . N
// Continuation value. Step V.
β
CV[n] = ContinuationValue(αt , expData[t-1]);
// Path estimator. Step V.
PE[n] = PathEstimator(CV[n], payoffData[t-1]);
return PE;
5.6. R ESULTS
Experiments were performed on the Accelerator Island system of the Cartesius Super-
computer (more information in [20]) with the following main characteristics:
• Intel Xeon E5-2450 v2.
• NVIDIA Tesla K40m.
• C-compiler: GCC 4.4.7.
• CUDA version: 5.5.
All computations are done in double precision, because a high accuracy is required
both in the bundling as well as in the regression computations. We consider the d -
dimensional problem of pricing Bermudan basket options with the following charac-
teristics:
• Initial state: St0 = (40, 40, . . . , 40) ∈ Rd .
• Strike: K = 40.
• Risk-free interest rate: r t = 0.06.

5
• Dividend yield rate: q δ = 0.0, δ = 1, 2, . . . , d .
• Volatility: σδ = 0.2, δ = 1, 2, . . . , d .
• Correlation: ρ i , j = 0.25, j = 2, . . . , d , i = 1, . . . , j .
• Maturity: T = 1.0.
• Exercise times: M = 10.
Specifically, geometric and arithmetic Bermudan basket put options are chosen in
order to show the accuracy and performance. The number of basis functions is taken as
b = 3. As the stochastic asset model, we first choose the multi-dimensional GBM, and
for the discretization scheme we employ the Euler-Maruyama SDE discretization.
5.6.1. E QUAL - PARTITIONING : CONVERGENCE TEST

In the original SGBM paper, the authors have shown the convergence of SGBM using k-
means bundling, in dependence of the number of bundles. For the equal-partitioning
bundling, we perform a similar convergence study by pricing the previously mentioned
options. Regarding the sorting criterion needed for the equal-partitioning technique,
we choose the payoff criterion, i.e. we sort the Monte Carlo scenarios following the ge-
ometric or arithmetic average of the assets for geometric and arithmetic basket options,
respectively. Similar results can be obtained using other criteria like the product or the
sum of the assets. In Figure 5.3, we show the convergence of the calculated option prices
for both geometric and arithmetic Bermudan basket options with different dimension-
alities, i.e. d = 5, d = 10 and d = 15. In the case of the geometric basket option, we can
also specify the reference price, because a multi-dimensional geometric basket option
can be reformulated as a one-dimensional option pricing problem. In the original paper
for SGBM [76], we can also find a comparison with the LSM method [94].
5.6. R ESULTS 109
1.5 5d True price 1.5

5d Direct est. 5d Direct est.
5d Path est.
5d Path est.
10d True price
10d Direct est. 1.4 10d Direct est.
1.4 10d Path est. 10d Path est.
15d True price
15d Direct est.
15d Direct est.
1.3 15d Path est.
V (t0 , St0 )
V (t0 , St0 )
15d Path est.
1.3 1.2
1.1
1.2
1
1.1 0.9
1 4 16 1 4 16
Bundles ν Bundles ν
(a) Geometric basket put option. (b) Arithmetic basket put option.
Figure 5.3: Convergence with equal-partitioning bundling technique. Test configuration: N = 218 and ∆t =
T /M .
Geometric Bermudan basket option 5

k-means equal-partitioning
MC DE PE MC DE PE
C 82.42 234.37 203.77 101.77 41.48 59.16
CUDA 1.04 18.69 12.14 0.63 4.66 1.29
Speedup 79.25 12.88 16.78 161.54 8.90 45.86
Arithmetic Bermudan basket option
MC DE PE MC DE PE
C 78.86 226.23 203.49 79.22 39.64 58.65
CUDA 1.36 17.89 11.74 0.83 4.14 1.20
Speedup 57.98 12.64 17.33 95.44 9.57 48.87
Table 5.1: SGBM stages time (s) for the C and CUDA versions. Test configuration: N = 222 , ∆t = T /M , d = 5
and ν = 210 .
5.6.2. B UNDLING TECHNIQUES : K - MEANS VS . EQUAL - PARTITIONING

With the convergence of the equal-partitioning technique shown numerically, we now
increase drastically the number of bundles and, hence, the number of Monte Carlo paths.
For the two presented bundling techniques, we perform a time comparison between the
C and CUDA implementations for multi-dimensional geometric and arithmetic Bermu-
dan basket options. First of all, in Table 5.1, we show the execution times for the different
SGBM stages, i.e. Monte Carlo path generation (MC), direct estimator computation (DE)
and path estimator computation (PE), for the C and CUDA versions. We can see an over-
all improvement of the timing results for SGBM based on equal-partitioning bundling
due to more efficient direct and path estimators, which are around four times faster.
The performance of the CUDA versions is remarkable, reaching a speedup of 160 for the
Monte Carlo simulation.
Geometric Bermudan basket option

d =5 d = 10 d = 15 d =5 d = 10 d = 15
C 604.13 1155.63 1718.36 303.26 501.99 716.57
CUDA 35.26 112.70 259.03 8.29 9.28 10.14
Speedup 17.13 10.25 6.63 36.58 54.09 70.67
d =5 d = 10 d = 15 d =5 d = 10 d = 15
C 591.91 1332.68 2236.93 256.05 600.09 1143.06
CUDA 34.62 126.69 263.62 8.02 11.23 15.73
Speedup 17.10 10.52 8.48 31.93 53.44 72.67
Table 5.2: SGBM total time (s) for the C and CUDA versions. Test configuration: N = 222 , ∆t = T /M and ν = 210 .
5 In Table 5.2, the total execution times for the d = 5, d = 10 and d = 15 problems are
presented. We observe a significant acceleration of the CUDA versions for both bundling
techniques, with a special improvement in the case of the equal-partitioning. This is
because the iterative process of k-means bundling penalizes parallelism and memory
transfers, especially when the dimensionality increases, while equal-partitioning han-
dles these issues in a more efficient way.
5.6.3. H IGH - DIMENSIONAL PROBLEMS

The second goal is to drastically increase the problem dimensionality for Bermudan bas-
ket option pricing. In the case of the k-means bundling algorithm, this is not possible
because of memory limitations. However, we save memory using the equal-partitioning
technique which enables us to increase the problem dimensions. We can reduce the to-
tal number of Monte Carlo paths, since by equal-partitioning each bundle has the same
number of paths. This gives us accurate regression values in all bundles. The SGBM
prices for geometric and arithmetic Bermudan basket put options are shown in Table
5.3. Again, the reference price for the geometric Bermudan basket option is obtained by
the COS method [50] and we present the prices given by the direct estimator (DE) and
path estimator (PE).
In Table 5.4, the execution times for pricing geometric and arithmetic Bermudan bas-
ket put options in different dimensions and with different numbers of bundles, ν, are
shown. Note that the number of bundles hardly influences the execution times. With
the equal-partitioning technique, the performance is mainly dependent on the number
of paths and the dimensionality. For that reason, we can exploit the GPU parallelism ar-
riving at a speedup of around 75 for the 50-dimensional problem in the case of geometric
Bermudan basket option and around 100 for the 50-dimensional arithmetic Bermudan
basket option.
Taking into account the accuracy and the performance of our CUDA implementa-
tion, we can even increase the dimension of the problem to higher values.
5.6. R ESULTS 111

d = 30 d = 40 d = 50
COS 1.057655 1.041889 1.032343
SGBM DE 1.057657 1.041888 1.032339
SGBM PE 1.057341 1.041545 1.031797
d = 30 d = 40 d = 50
SGBM DE 0.937436 0.921009 0.911023
SGBM PE 0.934359 0.919695 0.909646
Table 5.3: Option price for a high-dimensional problem with equal-partitioning. Test configuration: N = 220 ,
∆t = T /M and ν = 210 .

ν = 210 ν = 214
d = 30 d = 40 d = 50 d = 30 d = 40 d = 50
C 337.61 476.16 620.11 337.06 475.12 618.98
CUDA 4.65 6.18 8.08 4.71 6.26 8.16
Speedup 72.60 77.05 76.75 71.56 75.90 75.85
ν = 210 ν = 214
d = 30 d = 40 d = 50 d = 30 d = 40 d = 50
C 993.96 1723.79 2631.95 992.29 1724.60 2631.43
CUDA 11.14 17.88 26.99 11.20 17.94 27.07
Speedup 89.22 96.41 97.51 88.60 96.13 97.21
Table 5.4: SGBM total time (s) for a high-dimensional problem with equal-partitioning. Test configuration:
N = 220 and ∆t = T /M .
1.77 1.71
Reference price Reference price
Direct estimator Direct estimator
1.765 Path estimator
1.7 Path estimator
1.76
V (t0 , St0 )
V (t0 , St0 )
1.69
1.755
1.68
1.75
1.745 1.67
1.74 1.66
10 100 1000 2000 4000 10 100 1000 2000 4000
M C Steps = T /∆t M C Steps = T /∆t
Figure 5.4: CEV model convergence, γ = 1.0. Test configuration: N = 216 , ν = 210 and d = 2.
5.6.4. E XPERIMENTS WITH MORE GENERAL APPROACH

5 Parallel SGBM can thus be used to efficiently price high-dimensional Bermudan basket
options under GBM or other processes for which the ChF is available. However, as pre-
sented in Section 5.4, we can use SGBM also when the ChF is not available. In this case,
we first discretize the SDE system and then determine the ChF, which changes with posi-
tion and/or time. In order to get accurate approximations when using this ChF of the dis-
crete process, we need to improve the approximation of the expectations resulting from
the use of the basis functions in Equation (5.11). For that, we can increase the number
of time-steps, i.e., we can reduce the differences between Monte Carlo paths within each
bundle (with this, the regression becomes more accurate). We perform a convergence
test to show this behaviour. For the experiment, we choose the GPU version of SGBM
with equal-partitioning bundling, because this is the most efficient implementation and
the performance is independent of the number of bundles. An increasing number of
time-steps makes the C version too expensive again.
We consider a parametric local volatility model called the CEV model [31]. This
model can be obtained by substituting
µδ (St ) = (r t − q δ )S δ (t ), σδ (St ) = σδ (S δ (t ))γ , (5.12)
in Equation (5.1). γ ∈ [0, 1] is a free parameter called the variance elasticity, and q δ and
σδ , δ = 1, 2, . . . , d , are constant dividend yield and volatility, respectively.
The computation of the derivatives in Equation (5.10) is performed by using Wolfram
Mathematica 8 [135].
In Figure 5.4, a convergence test regarding the number of Monte Carlo time-steps is
presented. A 2-dimensional problem is considered and γ = 1.0 is used in Equation (5.12)
to compare our approximation with the reference price given by the COS method (in
the case of a geometric Bermudan basket put option) and the original SGBM following
GBM (for an arithmetic Bermudan basket put option). The figure shows that we can get
improved approximations of the option price by increasing the number of Monte Carlo
time-steps.
1.36 1.27
Reference price Reference price
Direct estimator Direct estimator
1.355 Path estimator 1.26 Path estimator
1.25
V (t0 , St0 )
V (t0 , St0 )
1.35
1.24
1.345
1.23
1.34 1.22
1.335 1.21
10 100 1000 2000 4000 10 100 1000 2000 4000
M C Steps = T /∆t M C Steps = T /∆t
Figure 5.5: CEV model convergence, γ = 1.0. Test configuration: N = 216 , ν = 210 and d = 5.

γ = 0.25 γ = 0.5 γ = 0.75 γ = 1.0
SGBM DE 0.001420 0.055636 0.411066 1.755705
5
SGBM PE 0.001395 0.055620 0.410758 1.754869
γ = 0.25 γ = 0.5 γ = 0.75 γ = 1.0
SGBM DE 0.001417 0.055340 0.404410 1.688609
SGBM PE 0.001346 0.055249 0.400400 1.682956
Table 5.5: CEV option pricing. Test configuration: N = 216 , ∆t = T /4000, ν = 210 and d = 2.
In Figure 5.5, the same convergence test as d = 2 case is presented for a 5-dimensional
problem. Again, we can see that the approximations are better when the number of
Monte Carlo time-steps is increased. In both cases, d = 2 and d = 5, the direct estimator
gives a better approximation compared to the reference price. It is also observed that the
direct and path estimators for the new approach perform similarly as in the case of the
original SGBM and they can provide a confidence interval for the option price.
Once we find an appropriate combination of the number of Monte Carlo paths, bun-
dles and time-steps, we can use our parallel SGBM method to price products under local
volatility dynamics. In Tables 5.5 and 5.6, the results of pricing 2-dimensional and 5-
dimensional geometric and arithmetic Bermudan basket put options are presented. We
take a different values for γ in the CEV model here. Note that we did not encounter any
problems with the Euler-Maruyama discretization of the CEV dynamics in our test cases.
The execution times (s) are around 120 and 150 for 2-dimensional and 5-dimensional
problems, respectively.
5.7. C ONCLUSIONS
In this chapter, we have presented an efficient implementation of the SGBM on a GPU
architecture. Through the GPU parallelism, we could speed up the execution times when

γ = 0.25 γ = 0.5 γ = 0.75 γ = 1.0
SGBM DE 0.000291 0.029395 0.276030 1.342147
SGBM PE 0.000274 0.029322 0.275131 1.342118
γ = 0.25 γ = 0.5 γ = 0.75 γ = 1.0
SGBM DE 0.000289 0.029089 0.267943 1.241304
SGBM PE 0.000288 0.028944 0.267214 1.225359
Table 5.6: CEV option pricing. Test configuration: N = 216 , ∆t = T /4000, ν = 210 and d = 5.
the number of bundles and the dimensionality increase drastically. In addition, we have
proposed a parallel bundling technique which is more efficient in terms of memory use
and more suitable on parallel systems. These two improvements enable us to extend
the method’s applicability and explore more general ways to compute the continuation
values. A general approach for approximating expectations based on the ChF of the dis-
5 crete process was presented. This approach allowed us to use SGBM for underlying asset
models for which the continuous ChF is not available.
Several results under the CEV local volatility model were presented to show the per-
formance and the accuracy of the new proposed techniques.
Compared with other GPU parallel implementations of early-exercise option pricing
methods, our parallel SGBM is very competitive in terms of computational time and can
solve very high-dimensional problems. Furthermore, we provided a new way to paral-
lelize the backward stage, according to the bundles, which gave us a remarkable perfor-
mance improvement.
We think that the parallel SGBM technique presented forms a fine basis to deal with
Credit Valuation Adjustment of portfolios with financial derivatives in the near future.
The hybrid component in this chapter is represented by the SGBM itself (it combines Monte
Carlo, dynamic programming, bundling and local regression approaches) and the effi-
cient parallel implementation of the method on GPU-based systems. The application of
parallel computing to early-exercise option pricing methods is very challenging. Here, we
have taken advantage of the particular features of SGBM to develop an efficient parallel
GPU version, allowing the use of the method for very high-dimensional problems and un-
der more involved dynamics. In the latter, another interesting component is employed, i.e.
the ChF of the discrete process.
CHAPTER 6
Conclusions and Outlook
6.1. C ONCLUSIONS
In this thesis, several numerical methods have been presented to deal with some of the
challenging problems appearing in computational finance. We have based our compu-
tational techniques on the combination of Monte Carlo methods and other advanced
methodologies, resulting in robust and efficient hybrid methods. We have proposed an
“almost exact” simulation technique for the SABR model, which is a standard model in
the FX and interest rate markets. The data-driven COS (ddCOS) method has been intro-
duced in the thesis, which may be a useful method for applications in risk management.
We have also developed a parallel GPU version of the Stochastic Grid Bundling Method,
which opens up the applicability towards very high-dimensional problems. In the fol-
lowing paragraphs, we briefly summarize the most important findings in the thesis.
In Chapter 2, a one time-step Monte Carlo method to simulate the SABR dynam-
ics has been developed. It is based on an efficient computational procedure for the
time-integrated variance. The technique employs a Fourier method to approximate the
marginal distribution of the time-integrated variance. Then, we have used a copula to
approximate the conditional distribution (time-integrated variance conditional on the
initial volatility). Resulting is a fast and accurate one time-step Monte Carlo method for
the SABR model, that may be used in the context of pricing European options with a
contract maturity of up to two years. By several numerical tests, we have shown that our
technique does not suffer from the well-known issues of the closed-form Hagan formula.
The latter formula is not accurate when small strike values and high volatilities are con-
sidered. As an alternative to this formula, our method may be suitable for calibration
purposes.
In Chapter 3, we have defined an accurate and robust multiple time-step Monte
Carlo method to simulate the SABR model with only a few time-steps, the so-called
mSABR method. The technique represents an extension of the method presented in
Chapter 2. Within the mSABR method we have incorporated several non-trivial method
components, like the use of a copula, the stochastic collocation interpolation technique
and a correlation approximation. The copula is employed to approximate the condi-
tional time-integrated variance process. The marginal distribution of the time-integrated
variance has been derived by means of Fourier techniques and numerical integration.
Due to the use of multiple time-steps, the straightforward generation of Monte Carlo
paths becomes computationally unaffordable. To deal with this computational cost is-
sue, we have applied an efficient sampling technique based on stochastic collocation
interpolation. The mSABR method appears an attractive SABR simulation technique
with an impressive ratio of accuracy and performance. The convergence and the sta-
115
116 6. C ONCLUSIONS AND O UTLOOK
bility of the method have been numerically assessed, whereby the mSABR method re-
quires only a few time-steps to achieve high accuracy. By the performed experiments,
we have shown that mSABR scheme reduces computational costs compared to other
Monte Carlo-based approaches when accurate results are required. As a multiple time-
step Monte Carlo method, it can be applied to pricing path-dependent options, like bar-
rier options. Furthermore, we have considered the extreme situation of negative interest
rates, for which the mSABR method (in combination with a shifted model) also performs
highly satisfactory.
In Chapter 4, we have presented the data-driven extension of the well-known COS

method, the so-called ddCOS method. This new technique generalizes the applicability
of the COS method towards cases where only data samples of the underlying risk neutral
asset dynamics are available. Therefore, a pre-defined stochastic model, with a corre-
sponding ChF, is not longer a requirement for this Fourier-based method. The method
relies on a closed-form expression in terms of Fourier cosine expansions for the density
estimation problem as it appears in statistical learning. With this Fourier based em-
pirical density approximation, the COS method machinery can be efficiently exploited,
resulting in a data-driven method which can be used for option valuation and also for
risk management. The expected rate of convergence of the ddCOS method has been nu-
merically confirmed. The method is particularly suitable for the accurate computation
6 of option sensitivities, specifically for the option ∆ and Γ. Therefore, it can be employed
within the Delta-Gamma approximation context for the calculation of risk measures,
solely based on asset samples. This fact has enabled us to explore more involved dy-
namics like those given by the SABR model. We have taken advantage of the simulation
schemes developed in Chapters 2 and 3 to test the ddCOS method. For monotonic data-
driven densities, the use of filters within the ddCOS method has appeared to be natural
and appropriate.
In Chapter 5, a parallel GPU implementation of the Stochastic Grid Bundling Method,

SGBM, has been presented. By means of the GPU parallelism, we have been able to
drastically reduce the computational effort for the valuation of early-exercise options.
The particular method components of the SGBM enables us to parallelize the nontrivial
backward stage, where parallelization has taken place according to the bundles that are
defined at each time step within SGBM. This represents a novelty with respect to other
GPU parallel implementations of early-exercise option pricing methods. It allowed us
to explore different configurations of bundles and to increase the option pricing prob-
lem dimensionality. We have introduced a different bundling technique, which is par-
ticularly efficient in terms of memory use and workload balancing. The convergence of
this technique has been numerically shown. We have developed a different approach
to computing the appearing conditional expectations, based on the ChF of the discrete
underlying process. This allowed us to use SGBM also for models for which the ChF is
not available. We have experienced with the CEV local volatility model (which is also a
part of the SABR model), with highly satisfactory results in terms of performance and
accuracy.
6.2. O UTLOOK 117
6.2. O UTLOOK
We think that the challenges in computational finance will necessarily require hybrid
and interdisciplinary computational approaches to come up with robust, accurate and
efficient solutions.
In Chapters 2 and 3, we have chosen components of the COS methodology as the
Fourier inversion technique to approximate the cumulative distribution function of the
time-integrated variance. Many alternatives can also be found in the literature. A par-
ticular interesting alternative is the so-called SWIFT method [102], which is based on
Shannon wavelets. The locality features of these basis functions make the technique
particularly suitable for the case of options with long maturities. The use of the SWIFT
method within the mSABR method should, for example, give fast and accurate approxi-
mations. By this, CPU time of the mSABR method may be further reduced.
The use of the clever interpolation based on stochastic collocation already gave us a
remarkable improvement in terms of computational cost in Chapter 3. In order to fur-
ther reduce execution times, the use of high-performance computing also appears to be
an interesting option in this context. In Chapter 5, GPU computing has been success-
fully employed. The parallelization may also be directly applied to the SCMC method or
to the sample generation within the mSABR method.
In the ddCOS method presented in Chapter 4, we have exploited the relation between
the Fourier cosine expansions and the density estimation from the statistical learning
framework. Again, the use of other basis functions, like wavelets, may form a possible ex-
tension. Specifically, Haar wavelets have particularly features like locality and compact
support, guaranteeing positive densities and allowing an efficient treatment of dynamic
data.
Another natural extension for the ddCOS method would be to consider density esti-
mation in higher dimensions. While in the context of the regularization approach, this
can be a challenging task, the connection between the ddCOS method and the empirical
ChF may form an interesting line of research to explore.
As mentioned, the ddCOS method has been derived in the framework of statistical
learning for density estimation. The density estimation problem has been intensively
studied from different points of view in the recent years. Particularly interesting is the
aspect of the machine learning approach and the algorithms originating from it, like
support vector machines, neural networks or deep learning. It would be highly interest-
ing to develop the ddCOS method further in the context of machine learning and data
science, especially because so much data is generated within the financial world.
In Chapter 5, finally, we have mainly worked with very basic dynamics for the under-
lying asset, i.e., geometric Brownian motion (GBM). In [29], the authors have considered
the Merton jump-diffusion model, deriving analytic formulas for the early-exercise com-
putation. Due to the increased asset path generation complexity, the gain provided by
a GPU parallel version should be even higher than in the GBM case. The inclusion of
several underlying models may give the opportunity of applying parallel SGBM towards
more complex problems, like the problem of Credit Valuation Adjustment of a financial
portfolio in the context of counterparty credit risk management.
The future for hybrid solution methods, data-driven technologies and parallel com-
puting seems bright within finance, insurance and economics.
References
[1] Lokman A. Abbas-Turki and Bernard Lapeyre. American options pricing on multi-
core graphic cards. In 2009 International Conference on Business Intelligence and
Financial Engineering, pages 307–311. IEEE, 2009.
[2] Leif B. G. Andersen. Efficient simulation of the Heston stochastic volatility model.
Journal of Computational Finance, 11(3):1–22, 2008.
[3] Theodore W. Anderson. On the Distribution of the Two-Sample Cramér-von Mises

Criterion. Annals of Mathematical Statistics, 33(3):1148–1159, 1962.
[4] Alexandre Antonov, Michael Konikov, and Michael Spector. SABR spreads its
wings. Risk Magazine, pages 58–63, August 2013.
[5] Alexandre Antonov, Michael Konikov, and Michael Spector. The free boundary
SABR: natural extension to negative rates. Risk Magazine, pages 58–63, August
2015.
[6] Philippe Barbe, Christian Genest, Kilani Ghoudi, and Bruno Rémillard. On
Kendall’s process. Journal of Multivariate Analysis, 58(2):197–229, 1996.
[7] Michael Benguigui and Françoise Baude. Fast American basket option pricing on a
multi-GPU cluster. In Proceedings of the 22nd High Performance Computing Sym-
posium, pages 1–8, Tampa, FL, USA, Apr 2014. 8 pages.
[8] Eric Benhamou. Fast Fourier Transform for discrete Asian options. Journal of
Computational Finance, 6(1):49–68, 2002.
[9] Daniel Berg. Copula goodness-of-fit testing: an overview and power comparison.
The European Journal of Finance, 15(7-8):675–701, 2009.
[10] Christopher M. Bishop. Pattern Recognition and Machine Learning. Information

Science and Statistics. Springer, 2006.
[11] Ljudmila A. Bordag. Geometrical properties of differential equations: applications

of Lie group analysis in financial mathematics. World Scientific Publishing Co,
2015.
[12] Anastasia Borovykh, Andrea Pascucci, and Cornelis W. Oosterlee. Pricing Bermu-
dan Options Under Local Lévy Models with Default. Journal of Mathematical
Analysis and Applications, 450(2):929–953, 2017.
[13] Svetlana Boyarchenko and Sergei Levendorskii. Efficient variations of the Fourier
transform in applications to option pricing. Journal of Computational Finance,
18(2):57–90, 2014.
119
120 R EFERENCES
[14] Mark Britten-Jones and Stephen M. Schaefer. Non-linear Value-at-Risk. Review of

Finance, 2(2):161–187, 1999.
[15] Mark Broadie and Özgür Kaya. Exact simulation of stochastic volatility and other
affine jump diffusion processes. Operations Research, 54(2):217–231, March 2006.
[16] Nicola Cantarutti, João Guerra, Manuel Guerra, and Maria do Rosário Grossinho.
Option pricing in exponential Lévy models with transaction costs, 2016. Available
at arXiv: https://arxiv.org/abs/1611.00389v3.
[17] Luca Capriotti. Fast Greeks by algorithmic differentiation. Journal of Computa-

tional Finance, 14(3):3–35, 2011.
[18] Luca Capriotti and Michael B. Giles. Algorithmic differentiation: adjoint Greeks
made easy. Risk Magazine, 25(10).
[19] Peter Carr and Dilip B. Madan. Option valuation using the Fast Fourier transform.
Journal of Computational Finance, 2:61–73, 1999.
[20] Cartesius webpage. https://www.surfsara.nl/systems/cartesius.
[21] Bin Chen, Cornelis W. Oosterlee, and Hans van der Weide. A low-bias simulation
scheme for the SABR stochastic volatility model. International Journal of Theoret-
ical and Applied Finance, 15(2):1250016–1 – 1250016–37, 2012.
[22] Nan Chen and Paul Glasserman. Malliavin Greeks without Malliavin calculus.
Stochastic Processes and their Applications, 117(11):1689–1723, 2007. Recent De-
velopments in Mathematical Finance: Special issue based on the CCCP Meeting,
April 2006, New York, NY.
[23] Rongda Chen and Lean Yu. A novel nonlinear Value-at-Risk method for modeling
risk of option portfolio with multivariate mixture of normal distributions. Eco-
nomic Modelling, 35:796–804, 2013.
[24] Umberto Cherubini and Elisa Luciano. Value-at-Risk trade-off and capital alloca-
tion with copulas. Economic notes, 30(2):235–256, 2001.
[25] Umberto Cherubini and Elisa Luciano. Bivariate option pricing with copulas. Ap-
plied Mathematical Finance, 9(2):69–85, 2002.
[26] Umberto Cherubini, Sabrina Mulinacci, Fabio Gobbi, and Silvia Romagnoli. Dy-
namic copula methods in finance, volume 625. John Wiley & Sons, 2011.
[27] Clogs webpage. http://sourceforge.net/projects/clogs/.
[28] Rafael Company, Vera Egorova, Lucas Jódar, and Carlos Vázquez. Computing
American option price under regime switching with rationality parameter. Com-
puters and Mathematics with Applications, 72(3):741 – 754, 2016.
121
[29] Fei Cong and Cornelis W. Oosterlee. Pricing Bermudan options under Merton
jump-diffusion asset dynamics. International Journal of Computer Mathematics,
92(12):2406–2432, 2015.
[30] Shane Cook. CUDA programming: a developer’s guide to parallel computing with
GPUs. Morgan Kaufmann Publishers Inc., San Francisco, CA, USA, 1st edition,
2013.
[31] John C. Cox. Note on option pricing I: Constant Elasticity of Variance diffusions.
Journal of Portfolio Management, 22:15–17, 1996.
[32] Harald Cramér. On the composition of elementary errors. Scandinavian Actuarial

Journal, 1928(1):13–74, 1928.
[33] CUB webpage. http://nvlabs.github.io/cub/.
[34] CUDA programming guide. http://docs.nvidia.com/cuda/cuda-c-programming-

guide/.
[35] CUDA webpage. http://www.nvidia.com/object/cuda_home_new.html.
[36] CUDPP webpage. http://cudpp.github.io/.
[37] cuRAND webpage. https://developer.nvidia.com/curand.
[38] Verche Cvetanoska and Toni Stojanovski. Using high performance computing and
Monte Carlo simulation for pricing American options. CoRR, abs/1205.0106, 2012.
[39] Duy-Minh Dang, Christina C. Christara, and Kenneth R. Jackson. An efficient

graphics processing unit-based parallel algorithm for pricing multi-asset Ameri-
can options. Concurrency and Computation: Practice and Experience, 24(8):849–
866, 2012.
[40] Duy-Minh Dang, Kenneth R. Jackson, and Mohammadreza Mohammadi. Dimen-

sion and variance reduction for Monte Carlo methods for high-dimensional mod-
els in finance. Applied Mathematical Finance, 22(6):522–552, 2015.
[41] Mark H.A. Davis and Martin P. Johansson. Malliavin Monte Carlo Greeks for jump
diffusions. Stochastic Processes and their Applications, 116(1):101–129, 2006.
[42] Paul Deheuvels. La fonction de dépendance empirique et ses propriétés, un test

non paramétrique d’indépendance. Bulletin de l’Académie Royale de Belgique,
Classe des Sciences (5), 65:274–292, 1979.
[43] Jacques du Toit and Uwe Naumann. Adjoint Algorithmic Differentiation Tool
Support for Typical Numerical Patterns in Computational Finance. To appear
in Journal of Computational Finance, 2017. Available at NAG Technical Reports:
https://www.nag.co.uk/doc/techrep/pdf/tr3_14.pdf.
[44] Daniel J. Duffy. Finite difference methods in financial engineering - a partial differ-
ential equation approach. John Wiley & Sons Ltd, 2006.
122 R EFERENCES
[45] Valdo Durrleman, Ashkan Nikeghbali, and Thierry Roncalli. Which copula is the
right one?, 2000. Available at SSRN: http://ssrn.com/abstract=1032545.
[46] Vera N. Egorova, Shi-Hau Tan, Choi-Hong Lai, Rafael Company, and Lucas Jó-
dar. Moving boundary transformation for American call options with transaction
cost: finite difference methods and computing. International Journal of Computer
Mathematics, 94(2):345–362, 2017.
[47] Paul Embrechts, Alexander McNeil, and Daniel Straumann. Correlation: pitfalls
and alternatives. Risk Magazine, 12:69–71, 1999.
[48] Bert Van Es, Peter Spreij, and Harry Van Zanten. Nonparametric volatility density
estimation. Bernoulli, 9(3):451–465, 2003.
[49] Fang Fang and Cornelis W. Oosterlee. A novel pricing method for European op-
tions based on Fourier-cosine series expansions. SIAM Journal on Scientific Com-
puting, 31:826–848, 2008.
[50] Fang Fang and Cornelis W. Oosterlee. Pricing early-exercise and discrete barrier
options by Fourier-cosine series expansions. Numerische Mathematik, 114(1):27–
62, 2009.
[51] Rob Farber. CUDA Application, design and development. Elsevier, 2011.
[52] Massimiliano Fatica and Everett Phillips. Pricing American options with least
squares Monte Carlo on GPUs. In Proceedings of the 6th Workshop on High Per-
formance Computational Finance, WHPCF ’13, pages 5:1–5:6, New York, NY, USA,
2013. ACM.
[53] Eric Fournié, Jean-Michel Lasry, Jérôme Lebuchoux, Pierre-Louis Lions, and Nizar
Touzi. Applications of Malliavin calculus to Monte Carlo methods in finance. Fi-
nance and Stochastics, 3(4):391–412, 1999.
[54] Beatrice Gaviraghi, Andreas Schindele, Mario Annunziato, and Alfio Borzì. On
optimal sparse-control problems governed by jump-diffusion processes. Applied
Mathematics, 7:1978–2004, 2016.
[55] Christian Genest and Louis-Paul Rivest. Statistical inference procedures for bi-
variate Archimedean copulas. Journal of the American Statistical Association,
88(423):1034–1043, 1993.
[56] Michael B. Giles. Smoking adjoints: fast Monte Carlo Greeks. Risk Magazine,
19(1):88–92.
[57] Michael B. Giles. Multi-level Monte Carlo path simulation. Operations Research,
56(3):607–617, 2008.
[58] Michael B. Giles. Vibrato Monte Carlo Sensitivities. In Pierre L’ Ecuyer and Art B.
Owen, editors, Monte Carlo and Quasi-Monte Carlo Methods 2008, pages 369–382.
Springer Berlin Heidelberg, Berlin, Heidelberg, 2009.
123
[59] Paul Glasserman. Monte Carlo Methods in Financial Engineering. Applications of

Mathematics : stochastic modelling and applied probability. Springer, 2004.
[60] Paul Glasserman and Zongjian Liu. Sensitivity estimates from characteristic func-
tions. Operations Research, 58(6):1611–1623, 2010.
[61] Paul Glasserman and Zongjian Liu. Estimating greeks in simulating Lévy-driven
models. Journal of Computational Finance, 14(2):3–56, 2011.
[62] Emmanuel Gobet, José G. López-Salas, Plamen Turkedjiev, and Carlos Vázquez.
Stratified regression Monte-Carlo scheme for semilinear PDEs and BSDEs with
large scale parallelization on GPUs. SIAM Journal on Scientific Computing,
38(6):C652–C677, 2016.
[63] Lech A. Grzelak and Cornelis W. Oosterlee. On the Heston model with stochastic
interest rates. SIAM Journal on Financial Mathematics, 2(1):255–286, 2011.
[64] Lech A. Grzelak and Cornelis W. Oosterlee. An equity-interest rate hybrid model
with stochastic volatility and the interest rate smile. Journal of Computational
Finance, 15(4):45–77, 2012.
[65] Lech A. Grzelak, Jeroen A. S. Witteveen, M. Suárez-Taboada, and Cornelis W.

Oosterlee. The Stochastic Collocation Monte Carlo sampler: highly effi-
cient sampling from “expensive” distributions, 2015. Available at SSRN:
https://ssrn.com/abstract=2529691.
[66] Tinne Haentjens and Karel J. In ’t Hout. Alternating direction implicit finite differ-
ence schemes for the Heston–Hull–White partial differential equation. Journal of
Computational Finance, 16:83–110, 2012.
[67] Patrick S. Hagan, Deep Kumar, Andrew S. Lesniewski, and Diana E. Woodward.
Managing smile risk. Wilmott Magazine, pages 84–108, 2002.
[68] Patrick S. Hagan, Deep Kumar, Andrew S. Lesniewski, and Diana E. Woodward.
Arbitrage-Free SABR. Wilmott Magazine, (69):60–75, 2014.
[69] Patrick S. Hagan, Andrew S. Lesniewski, and Diana E. Woodward. Probability

distribution in the SABR model of stochastic volatility. In Large Deviations and
Asymptotic Methods in Finance, pages 1–35, 2005.
[70] Martin B. Haugh and Leonid Kogan. Pricing American options: a duality approach.
Operations Research, 52(2):258–270, 2004.
[71] Christian Hendricks, Christof Heuer, Matthias Ehrhardt, and Michael Günther.
High-order ADI finite difference schemes for parabolic equations in the combina-
tion technique with application in finance. Journal of Computational and Applied
Mathematics, 316:175 – 194, 2017.
[72] Pierre Henry-Labordère. A general asymptotic implied volatility for stochas-

tic volatility models, 2005. Available at arXiv: http://arxiv.org/pdf/cond-
mat/0504317v2.pdf.
124 R EFERENCES
[73] Karel in ’t Hout and Radoslav Valkov. Numerical study of splitting

methods for American option valuation, 2016. Available at arXiv:
https://arxiv.org/abs/1610.09622.
[74] Kazuyuki Ishiyama. Methods for evaluating density functions of exponential func-
tionals represented as integrals of geometric Brownian motion. Methodology and
Computing in Applied Probability, 7(3):271–283, 2005.
[75] Othmane Islah. Solving SABR in exact form and unifying it with LIBOR market
model, 2009. Available at SSRN: http://ssrn.com/abstract=1489428.
[76] Shashi Jain and Cornelis W. Oosterlee. The Stochastic Grid Bundling Method: Ef-
ficient pricing of Bermudan options and their Greeks. Applied Mathematics and
Computation, 269:412–431, 2015.
[77] Bilal Jan, Bartolomeo Montrucchio, Carlo Ragusa, Fiaz Gul Khan, and Omar Khan.
Fast parallel sorting algorithms on GPUs. International Journal of Distributed and
Parallel systems, 14(1), 2012.
[78] Joanne E. Kennedy and Duy Pham. On the Approximation of the SABR with
mean reversion model: a Probabilistic Approach. Applied Mathematical Finance,
21(5):451–481, 2014.
[79] Kepler architecture webpage. http://www.nvidia.com/object/nvidia-kepler.html.
[80] Soňa Kilianova and Daniel Ševčovič. A transformation method for solving the
Hamilton-Jacobi-Bellman equation for a constrained dynamic stochastic optimal
allocation problem. The ANZIAM Journal, 55(01):14–38, 2013.
[81] Volodymyr Kindratenko, editor. Numerical computations with GPUs. Springer,

2014.
[82] David B. Kirk and Wen-mei W. Hwu. Programming massively parallel processors: a
hands-on approach. Elsevier, 2010.
[83] Peter E. Kloeden and Eckhard Platen. Numerical solution of stochastic differential
equations. Springer–Verlag, 1992.
[84] Donald E. Knuth. The art of computer programming, volume 3: (2nd Ed.) Sorting
and searching. Addison Wesley Longman Publishing Co., Inc., Redwood City, CA,
USA, 1998.
[85] Hisashi Kobayashi, Brian L. Mark, and William Turin. Probability, random pro-
cesses, and statistical analysis. Cambridge, 2012.
[86] Ralf Korn and Songyin Tang. Exact analytical solution for the normal SABR model.
Wilmott Magazine, 2013(66):64–69, 2013.
[87] R. A. Kronmal and M. E. Tarter. The estimation of probability densities and cumu-
latives by Fourier series methods. Journal of the American Statistical Association,
63(323):925–952, 1968.
125
[88] Álvaro Leitao, Lech A. Grzelak, and Cornelis W. Oosterlee. On a one time-step
Monte Carlo simulation approach of the SABR model: application to European
options. Applied Mathematics and Computation, 293:461–479, 2017.
[89] Álvaro Leitao, Lech A. Grzelak, and Cornelis W. Oosterlee. On an efficient multiple
time step Monte Carlo simulation of the SABR model. Quantitative Finance, 2017.
[90] Álvaro Leitao and Cornelis W. Oosterlee. GPU Acceleration of the Stochastic Grid
Bundling Method for Early-Exercise options. International Journal of Computer
Mathematics, 92(12):2433–2454, 2015.
[91] Álvaro Leitao, Cornelis W. Oosterlee, Luis Ortiz-Gracia, and Sander M. Bohte. On
the data-driven COS method. Submitted for publication, 2017.
[92] Alan L. Lewis. A Simple Option Formula for General Jump-Diffusion

and Other Exponential Levy Processes, 2001. Available at SSRN:
https://ssrn.com/abstract=282110.
[93] David X. Li. On default correlation: a copula function approach. Journal of Fixed
Income, 9:43–54, 2000.
[94] Francis A. Longstaff and Eduardo S. Schwartz. Valuing American options by simu-
lation: a simple least-squares approach. Review of Financial Studies, 14(1):113–47,
2001.
[95] Roger Lord and Adam Farebrother. Fifty shades of SABR simulation, 2014. Presen-
tation at 10th Fixed Income Conference, WBS Training, Barcelona, Spain. Available
at http://www.rogerlord.com/fiftyshadessabrwbs.pdf.
[96] K. Ma and Peter A. Forsyth. Numerical solution of the Hamilton-Jacobi-Bellman

formulation for continuous time mean variance asset allocation under stochastic
volatility. Journal of Computational Finance, 20(1):1–37, 2016.
[97] Paul Malliavin and Maria Elvira Mancino. A Fourier transform method for non-
parametric estimation of multivariate volatility. Annals of Statistics, 37(4):1983–
2010, 2009.
[98] Marko J. Misic and Milo V. Tomasevic. Data sorting using Graphics Processing
Units. Telfor Journal, 4(1), 2012.
[99] Modern GPU webpage. http://nvlabs.github.io/moderngpu/.
[100] Jan Obloj. Fine-tune your smile: Correction to Hagan et al, 2008. Available at arXiv:
http://arxiv.org/abs/0708.0998v3.
[101] Luis Ortiz-Gracia and Cornelis W. Oosterlee. Efficient VaR and Expected Shortfall
computations for nonlinear portfolios within the delta-gamma approach. Applied
Mathematics and Computation, 244:16–31, 2014.
126 R EFERENCES
[102] Luis Ortiz-Gracia and Cornelis W. Oosterlee. A highly efficient Shannon wavelet
inverse Fourier technique for pricing European options. SIAM Journal on Scientific
Computing, 38(1):118–143, 2016.
[103] Gilles Pagès and Benedikt Wilbertz. GPGPUs in computational finance: massive
parallel computing for American style options. Concurrency and Computation:
Practice and Experience, 24(8):837–848, 2012.
[104] Louis Paulot. Asymptotic implied volatility at the second order with application to
the SABR model, 2009. Available at SSRN: http://ssrn.com/abstract=1413649.
[105] Vladimir Piterbarg. A multi-currency model with FX volatility skew, 2005. Available
at SSRN: http://ssrn.com/abstract=685084.
[106] Vladimir Piterbarg. Smiling hybrids. Risk Magazine, 19(5):66–70, 2006.
[107] Nicolas Privault and Jiadong Yu. Stratified approximations for the pricing of op-
tions on average. Journal of Computational Finance, 19(4):95–113, 2016.
[108] L. C. Rogers. Monte Carlo valuation of American options. Mathematical Finance,

12(3):271–286, 2002.
[109] Joshua V. Rosenberg. Nonparametric pricing of multivariate contingent claims.

Journal of Derivatives, 10(3):9–26, 2003.
[110] Joshua V. Rosenberg and Til Schuermann. A general approach to integrated

risk management with skewed, fat-tailed risks. Journal of Financial Economics,
79(3):569–614, 2006.
[111] Maria J. Ruijter and Cornelis W. Oosterlee. Numerical Fourier Method and Second-
order Taylor Scheme for Backward SDEs in Finance. Applied Numerical Mathe-
matics, 103(C):1–26, May 2016.
[112] Maria J. Ruijter, Mark Versteegh, and Cornelis W. Oosterlee. On the application
of spectral filters in a Fourier option pricing technique. Journal of Computational
Finance, 19(1):75–106, 2014.
[113] Jason Sanders and Edward Kandrot. CUDA by example: an introduction to general-
purpose GPU programming. Addison-Wesley, 2011.
[114] Munuswamy Sankaran. Approximations to the non-central chi-square distribu-

tion. Biometrika, 50(1-2):199–204, 1963.
[115] Nadathur Satish, Mark Harris, and Michael Garland. Designing efficient sorting
algorithms for manycore GPUs. In Proceedings of the 23rd IEEE International Par-
allel and Distributed Processing Symposium, 2009.
[116] Sebastian Schlenkrich, André Miemiec, and Tilman Wolff-Siemssen. Low strike
extrapolation for SABR. Wilmott Magazine, 2014.
127
[117] Mark Schroder. Computing the Constant Elasticity of Variance option pricing for-
mula. Journal of Finance, 44(1):211–219, 1989.
[118] Rituparna Sen and Changie Ma. Forecasting density function: application in fi-
nance. Mathematical Finance, 5:433–447, 2015.
[119] Rüdiger U. Seydel. Tools for computational finance. Springer–Verlag, 2009.
[120] Bernard W. Silverman. Density estimation for statistics and data analysis. Chap-
man & Hall, London, 1986.
[121] Johannes V. Siven, Jeffrey T. Lins, and Anna Szymkowiak-Have. Value-at-Risk com-
putation by Fourier inversion with explicit error bounds. Finance Research Letters,
6(2):95–105, 2009.
[122] Abe Sklar. Fonctions de répartition à n dimensions et leurs marges. Publications

de l’Institut de Statistique de l’Université de Paris, 8:229–231, 1959.
[123] Nikolai V. Smirnov. Theory of Probability and Mathematical Statistics (selected

works). Nauka, Moscow, 1970.
[124] Robert D. Smith. An almost exact simulation method for the Heston model. Jour-
nal of Computational Finance, 11(1):115–125, 2007.
[125] Maria Suárez-Taboada and Carlos Vázquez. Numerical solution of a PDE model
for a ratchet-cap pricing with BGM interest rate dynamics. Applied Mathematics
and Computation, 218(9):5217–5230, 2012.
[126] Krishnahari Thouti and S.R. Sathe. An OpenCL method of parallel sorting algo-
rithms for GPU architecture. International Journal of Experimental Algorithms,
3(1), 2012.
[127] Thrust webpage. http://thrust.github.io/.
[128] John N. Tsitsiklis and Benjamin Van Roy. Regression methods for pricing complex
American-style options. IEEE transactions on Neural networks, 12(4):694–703, July
2001.
[129] Nikolai G. Ushakov. Selected topics in characteristic functions. Modern Probability

and Statistics. Mouton De Gruyter, 1999.
[130] Alexander van Haastrecht and Antoon A.J. Pelsser. Efficient, almost exact simula-
tion of the Heston stochastic volatility model. International Journal of Theoretical
and Applied Finance, 13(01):1–43, 2010.
[131] Vladimir N. Vapnik. Statistical learning theory. Wiley-Interscience, 1998.
[132] VexCL webpage. http://ddemidov.github.io/vexcl/.
[133] Paul Wilmott, Sam Howison, and Jeff Dewynne. The mathematics of financial
derivatives : a student introduction. Cambridge University Press, Cambridge, UK,
New York, 1995.
128 R EFERENCES
[134] Nicholas Wilt. The CUDA handbook: a comprehensive guide to GPU programming.
Addison-Wesley, 2013.
[135] Wolfram Mathematica webpage. http://www.wolfram.com/mathematica/.
[136] Qi Wu. Series expansion of the SABR joint density. Mathematical Finance,
22(2):310–345, 2012.
[137] Bowen Zhang and Cornelis W. Oosterlee. Efficient pricing of European-style Asian
options under exponential Lévy processes based on Fourier cosine expansions.
SIAM Journal on Financial Mathematics, 4(1):399–426, 2013.
Curriculum Vitæ
Álvaro L EITAO R ODRÍGUEZ
08-01-1985 Born in Xove, Spain.
E DUCATION
1997–2001 Secondary school
I.E.S. Illa de Sarón, Xove, Spain
2001–2003 Baccalaureate
I.E.S. María Sarmiento, Viveiro, Spain
2004–2010 Bachelor Degree in Technical Engineering of Computer Science

University of A Coruña, A Coruña, Spain
2009–2011 Master Degree in Mathematical Engineering

University of Vigo, Vigo, Spain
2011–2013 Researcher
M2NICA research group, Department of Mathematics
University of A Coruña, A Coruña, Spain
2013–2017 PhD Researcher

Scientific Computing group
Centrum Wiskunde & Informatica, Amsterdam, The Netherlands
2017 PhD. Applied Mathematics

Delft University of Technology, Delft, The Netherlands
Thesis: Hybrid Monte Carlo methods in computational fi-
nance
Promotor: Prof. dr. ir. Cornelis W. Oosterlee
129
List of Publications
Journal articles:
4. A. Leitao, C. W. Oosterlee, L. Ortiz-Gracia and S. M. Bohte, On the data-driven COS method,

Submitted for publication, 2017.
3. A. Leitao, L. A. Grzelak and C. W. Oosterlee, On an efficient multiple time-step Monte Carlo

simulation of the SABR model, Quantitative Finance, 2017.
2. A. Leitao, L. A. Grzelak and C. W. Oosterlee, On a one time-step Monte Carlo simulation

approach of the SABR model: Application to European options, Applied Mathematics and
Computation, 293:461–479, 2017.
1. A. Leitao and C. W. Oosterlee, GPU Acceleration of the Stochastic Grid Bundling Method for
Early-Exercise options, International Journal of Computer Mathematics, 92(12):2433–2454,
2015.
Chapters in books:
2. A. Leitao, L. A. Grzelak and C. W. Oosterlee, A highly efficient numerical method for the
SABR model, STRIKE - Novel Methods in Computational Finance. To be published in 2017.
1. A. Leitao and C. W. Oosterlee, Modern Monte Carlo methods and GPU computing, STRIKE
- Novel Methods in Computational Finance. To be published in 2017.
131
List of Attended Conferences
Presentations:
8. MathFinance conference, Frankfurt, Germany, April 2017.
7. Summer school on Quantitative methods for Risk management in Finance and Insurance,
A Coruña, Spain, July 2016.
6. The 19th European Conference on Mathematics for Industry, Santiago de Compostela, Spain,
June 2016.
5. SIAM conference on Uncertainty Quantification, Lausanne, Switzerland, April 2016.
4. International Conference on Computational Finance, Greenwich-London, United Kingdom,

December 2015.
3. Stochastics & Computational Finance: from academia to industry, Lisbon, Portugal, July
2015.
2. Python for finance: an Introduction, Lisbon, Portugal, December 2014.
1. The 18th European Conference on Mathematics for Industry, Taormina, Italy, June 2014.
Conference papers:
2. A. Leitao, L. A. Grzelak and C. W. Oosterlee, Efficient multiple time-step simulation of the

SABR Model, In Proceedings of the 19th European Conference on Mathematics for Industry,
2016, Springer Heidelberg.
1. A. Leitao and C. W. Oosterlee, On a GPU acceleration of the Stochastic Grid Bundling Method,
In Proceedings of the 18th European Conference on Mathematics for Industry, 2014, Springer
Heidelberg.
133
Acknowledgement
First and foremost, my most sincere gratitude goes to my supervisor and promotor prof.
Kees Oosterlee. His guidance and advice have been essential for the successful comple-
tion of this thesis. Kees never hesitates to share his knowledge and experience with his
students, keeping the motivation and the learning interest in the research process. Fur-
thermore, Kees directly offered me this opportunity, for which, I will always be grateful.
The last four years have been one of the nicest experience in my life which I had the
pleasure to share with lots of excellent people. The (former) students and collaborators
of Kees form a very friendly and wonderful group, not only cooperating and helping each
other but also having fun together. Many thanks to Andrea, Anton, Fei, Gemma, Ki Wai,
Marjon, Peiyao, Prashant, Qian, Sander, Shashi, Shuaiqiang and Zaza (dank u wel voor
de Nederlandse vertaling!), and the Spanish community, Luis, María, Marta and Paco.
The colleagues from the CWI, with whom, I have enjoyed the informal discussions about
several, sometimes crazy, topics during lunches and next to the coffee machine: Anne,
Barry, Bart, Benjamin, Bram, Chris, Daan, Daniel, Debarati, Folkert, Fredrik, Inti, Jan,
Jan Willem, Jeroen (×2), Joost, Keith, Krzysztof, Laurent, Nick, Sangeetika, Sirshendu,
Svetlana, Wander, Willem Jan, Xiaodong, Yous and Zhichao. With many of them, I have
also enjoyed practising basketball, football, squash or table-tennis. The help of the CWI
staff, particularly Nada and Duda, is very appreciated.
My PhD was partially carried out in the framework of a European Multi-ITN project
called STRIKE. I would like to specially thank the coordinators, Matthias Ehrhardt and
Jan ter Maten for their impressive work. I have spent a very nice time with my colleagues
in the project, the “STRIKErs” Zuzana, Zé, Pedro, Vera, Walter, Nicola, Ivan, Lara, Giang,
Shih-Hau, Beatrice, Radoslav, Christof, Silvie and Fazlollah, during the countless project
events. I also had a great time with Anastasia, Enrico, José Germán, Maarten, Matthieu,
Sidy and Zuzana in different conferences around Europe. It was really a pleasure to have
met all of them. I want to particularly express my gratitude to prof. Carlos Vázquez
Cendón who encouraged me to take this important step in my career.
I would also like to thank the members of my defence committee for reading my
dissertation and participating in my defence. Special thanks to Lech Grzelak for his con-
stant support and helpful comments during my PhD research.
Finally, I would like to express the most special gratitude to my family.
Gracias en primeiro lugar a meus pais pola educación que me deron, da que me
sinto moi orgulloso. Sempre me animaron e me apoiaron nos momentos importantes,
dándome os mellores consellos. Moitas gracias a miña nai por todo o seu cariño. E
moitas gracias a meu pai, a quen, despois de nove anos sen él, recordo cada día.
E por suposto, moitísimas gracias a Mela. Ela foi o meu gran apoio nos últimos catro
anos (e nos once anteriores), nos que soubemos desfrutar e aprender dunha situación
excepcional. A súa axuda, apoio e comprensión foron imprescindibles para acadar este
logro. Xuntos afrontaremos o que o futuro nos depare.
136 A CKNOWLEDGEMENT
Álvaro Leitao Rodríguez,

Amsterdam, June 2017

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Delft University of Technology

Hybrid Monte Carlo methods in computational finance

Leitao Rodriguez, Alvaro

This work is downloaded from Delft University of Technology.

ter verkrijging van de graad van doctor

Álvaro L EITAO R ODRÍGUEZ

Master of Science in Mathematical Engineering

This research was partially supported by the EU in the FP7-PEOPLE-2012-ITN Program

Hybrid Monte Carlo methods in computational finance

Two things are infinite:

cient Monte Carlo-based method for pricing multi-dimensional early-exercise deriva-

Monte-Carlomethoden worden zeer gewaardeerd en intensief gebruikt binnen de finan-

3.3 Components of the mSABR method. . . . . . . . . . . . . . . . . . . . . 48

5.7 Conclusions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

1.2. S TOCHASTIC PROCESSES

1.2.1. T HE SABR MODEL

dS(t ) = σ(t )S β (t )dWS (t ), S(0) = S 0 exp (r T ) ,

where the term B 1 , B 2 , B 3 and ω are given by

where V (t , S̄, σ) is the option price and D(t , T ) := exp(−r (T − t )).

1.3. P RICING TECHNIQUES

Theorem 1.3.1 (Feynman-Kac). Given the money-savings account, modelled by dB (t ) =

V (t 0 , S) = EQ e−r (T −t0 ) h(T, S)|F (t 0 ) ,

dS(t ) = µQ (t , S)dt + σ(t , S)dW Q (t ), for t > t 0 .

The Feynman-Kac formula can be generalized to more involved types of problems.

1.3.1. M ONTE C ARLO METHODS

and we have a technique to draw n independent and identically distributed samples in

If f is integrable over C then, by the strong law of the large numbers,

with probability one.

1.3.2. F OURIER INVERSION METHODS

process. They form the so-called Fourier pair, as

where fˆ denotes a Fourier transformed function, i.e., here, the ChF.

1.3.3. PDE METHODS

reduced applicability in higher problem dimensions due to the curse of dimensionality

1.3.4. H YBRID SOLUTION TECHNIQUES

1.3.5. D ATA IN COMPUTATIONAL FINANCE

1.4. GPU COMPUTING

1.5. M ULTI -ITN-STRIKE PROJECT

1.6. O UTLINE OF THE THESIS

the conditional distribution of the time-integrated variance is available in differential

2.2. T HE SABR MODEL

dS(t ) = σ(t )S β (t )dWS (t ), S(0) = S 0 exp (r T ) ,

2.2.1. SABR M ONTE C ARLO SIMULATION

where W̃σ (T ) is an independent Brownian motion.

• Simulation of SABR’s R Ttime-integrated variance, conditional on the terminal value

Algorithm 1: SABR’s one time-step Monte Carlo simulation.

2.3. CDF OF SABR’ S TIME - INTEGRATED VARIANCE

where R j is the logarithmic increment of σ2 (t ) between t j and t j −1 , j = 1, . . . , M . As

fˆR j (u) = fˆR (u) = exp(−i uα2 ∆t − 2u 2 α2 ∆t ), (2.7)

with α as in (2.1) and i the imaginary unit.

At this point, a backward recursion procedure in terms of R j will be set up by which

with Z j = log(1 + exp(Y j )).

From Equation (2.10), we determine fˆlog Ỹ (T ) , as follows

fˆlog Ỹ (T ) (u) = E[exp(i u log Ỹ (T ))]

We have reduced the computation of fˆlog Ỹ (T ) to the computation of fˆYM . As Y M is

fˆY1 (u) = fˆR M (u) = exp(−i uα2 ∆t − 2u 2 α2 ∆t ),

Now we can apply a cosine series expansion to f Y j −1 , so that fˆZ∗ becomes

We wish to compute fˆY∗M ( b−akπ

2.3.2. T REATMENT OF INTEGRAL I M

Analytic solution based on 2 F 1 . Integral I M in (2.20) can be rewritten as

with u k and u l as in (2.19).