Lifting Volterra Diffusions via Kernel Decomposition
Pages 481 - 489
Abstract
Rough volatility models have garnered considerable attention among practitioners due to their remarkable empirical fit. However, their non-Markovian nature arises from the presence of a kernel (leading to so-called Voltera diffusions), which complicates pricing and calibration tasks. In this paper, we present our novel contribution of employing machine learning techniques to either approximate or learn the kernel function in a manner that renders it Markovian, effectively “lifting” the non-Markovian Volterra diffusion. Through empirical investigations encompassing a diverse set of kernels, we demonstrate the efficacy of our approach, opening new avenues for improved modelling and analysis in rough volatility models.
References
[1]
Eduardo Abi Jaber. 2019. Lifting the Heston model. Quantitative finance 19, 12 (2019), 1995–2013.
[2]
Eduardo Abi Jaber and Omar El Euch. 2019. Multifactor approximation of rough volatility models. SIAM journal on financial mathematics 10, 2 (2019), 309–349.
[3]
Julia Ackermann, Thomas Kruse, and Ludger Overbeck. 2020. Inhomogeneous affine Volterra processes. arxiv:2012.10966 [math.PR]
[4]
Emmanuel Bacry, Iacopo Mastromatteo, and Jean-François Muzy. 2015. Hawkes processes in finance. Market Microstructure and Liquidity 1, 01 (2015), 1550005.
[5]
Christian Bayer, Peter Friz, and Jim Gatheral. 2016. Pricing under rough volatility. Quantitative Finance 16, 6 (2016), 887–904.
[6]
Marc A Berger and Victor J Mizel. 1980. Volterra equations with itô integrals—I. The Journal of Integral Equations (1980), 187–245.
[7]
Pierre Blanc, Jonathan Donier, and J-P Bouchaud. 2017. Quadratic Hawkes processes for financial prices. Quantitative Finance 17, 2 (2017), 171–188.
[8]
Peter Carr and Dilip Madan. 1999. Option valuation using the fast Fourier transform. Journal of computational finance 2, 4 (1999), 61–73.
[9]
Christa Cuchiero and Josef Teichmann. 2020. Generalized Feller processes and Markovian lifts of stochastic Volterra processes: the affine case. Journal of evolution equations 20, 4 (2020), 1301–1348.
[10]
David Duvenaud. 2014. The kernel cookbook: Advice on covariance functions. URL https://www. cs. toronto. edu/duvenaud/cookbook (2014).
[11]
Omar El Euch, Jim Gatheral, and Mathieu Rosenbaum. 2019. Roughening heston. Risk (2019), 84–89.
[12]
Eymen Errais, Kay Giesecke, and Lisa R Goldberg. 2010. Affine point processes and portfolio credit risk. SIAM Journal on Financial Mathematics 1, 1 (2010), 642–665.
[13]
Fang Fang and Cornelis W Oosterlee. 2009. A novel pricing method for European options based on Fourier-cosine series expansions. SIAM Journal on Scientific Computing 31, 2 (2009), 826–848.
[14]
Greg Fasshauer. 2025. Positive Definite and Completely Monotone Functions.
[15]
Gregory E. Fasshauer. 2010. Positive Definite Kernels : Past, Present and Future. https://api.semanticscholar.org/CorpusID:12880067
[16]
Damir Filipović. 2005. Time-inhomogeneous affine processes. Stochastic Processes and their Applications 115, 4 (2005), 639–659.
[17]
Jim Gatheral, Thibault Jaisson, and Mathieu Rosenbaum. 2018. Volatility Is Rough. Quantitative Finance 18, 6 (2018), 933–949.
[18]
Benyamin Ghojogh, Ali Ghodsi, Fakhri Karray, and Mark Crowley. 2021. Reproducing Kernel Hilbert Space, Mercer’s Theorem, Eigenfunctions, Nyström Method, and Use of Kernels in Machine Learning: Tutorial and Survey. arxiv:2106.08443 [stat.ML]
[19]
Alan G Hawkes. 2018. Hawkes processes and their applications to finance: a review. Quantitative Finance 18, 2 (2018), 193–198.
[20]
Satoshi Hayakawa, Harald Oberhauser, and Terry Lyons. 2023. Sampling-based Nyström Approximation and Kernel Quadrature. arxiv:2301.09517 [math.NA]
[21]
Steven L Heston. 1993. A closed-form solution for options with stochastic volatility with applications to bond and currency options. The review of financial studies 6, 2 (1993), 327–343.
[22]
Caroline Hillairet and Anthony Reveillac. 2023. Explicit correlations for the Hawkes processes. arxiv:2304.02376 [math.PR]
[23]
Eduardo Abi Jaber, Martin Larsson, and Sergio Pulido. 2019. Affine Volterra processes. arxiv:1708.08796 [math.PR]
[24]
Antoine Jacquier and Alexandre Pannier. 2022. Large and moderate deviations for stochastic Volterra systems. Stochastic Processes and their Applications 149 (2022), 142–187.
[25]
Zhu Li, Jean-Francois Ton, Dino Oglic, and Dino Sejdinovic. 2021. Towards A Unified Analysis of Random Fourier Features. arxiv:1806.09178 [stat.ML]
[26]
Fanghui Liu, Xiaolin Huang, Yudong Chen, and Johan A. K. Suykens. 2021. Random Features for Kernel Approximation: A Survey on Algorithms, Theory, and Beyond. arxiv:2004.11154 [stat.ML]
[27]
Maxime Morariu-Patrichi and Mikko S Pakkanen. 2022. State-dependent Hawkes processes and their application to limit order book modelling. Quantitative Finance 22, 3 (2022), 563–583.
[28]
David J. Prömel and David Scheffels. 2023. Stochastic Volterra equations with Hölder diffusion coefficients. arxiv:2204.02648 [math.PR]
[29]
Alexandre Richard, Xiaolu Tan, and Fan Yang. 2022. Discrete-time Simulation of Stochastic Volterra Equations. arxiv:2004.00340 [math.NA]
[30]
Mathieu Rosenbaum and Jianfei Zhang. 2022. Deep calibration of the quadratic rough Heston model. arxiv:2107.01611 [q-fin.CP]
[31]
Simo Särkkä and Arno Solin. 2019. Applied stochastic differential equations. Vol. 10. Cambridge University Press.
[32]
Matthias Seeger. 2004. Gaussian Processes for Machine Learning. International journal of neural systems 14, 02 (2004), 69–106.
[33]
John Shawe-Taylor and Nello Cristianini. 2004. Kernel methods for pattern analysis. Cambridge university press.
[34]
Vito Volterra. 1928. Variations and fluctuations of the number of individuals in animal species living together. ICES Journal of Marine Science 3, 1 (1928), 3–51.
[35]
Christopher KI Williams and Carl Edward Rasmussen. 2006. Gaussian processes for machine learning. Vol. 2. MIT press Cambridge, MA.
Index Terms
- Lifting Volterra Diffusions via Kernel Decomposition
Index terms have been assigned to the content through auto-classification.
Recommendations
Faster doubly stochastic functional gradient by gradient preconditioning for scalable kernel methods
AbstractThe doubly stochastic functional gradient descent algorithm (DSG) that is memory friendly and computationally efficient can effectively scale up kernel methods. However, in solving the highly ill-conditioned large-scale nonlinear machine learning ...
A centrosymmetric kernel decomposition for time-frequency distribution computation
Time-frequency distributions (TFDs) are bilinear transforms of the signal and, as such, suffer from a high computational complexity. Previous work has shown that one can decompose any TFD in Cohen's class into a weighted sum of spectrograms. This is ...
Clustering via kernel decomposition
Spectral clustering methods were proposed recently which rely on the eigenvalue decomposition of an affinity matrix. In this letter, the affinity matrix is created from the elements of a nonparametric density estimator and then decomposed to obtain ...
Comments
Please enable JavaScript to view thecomments powered by Disqus.Information & Contributors
Information
Published In
November 2023
697 pages
ISBN:9798400702402
DOI:10.1145/3604237
Copyright © 2023 ACM.
Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for components of this work owned by others than the author(s) must be honored. Abstracting with credit is permitted. To copy otherwise, or republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. Request permissions from [email protected].
Publisher
Association for Computing Machinery
New York, NY, United States
Publication History
Published: 25 November 2023
Check for updates
Author Tags
Qualifiers
- Research-article
- Research
- Refereed limited
Conference
ICAIF '23
Contributors
Other Metrics
Bibliometrics & Citations
Bibliometrics
Article Metrics
- 0Total Citations
- 140Total Downloads
- Downloads (Last 12 months)46
- Downloads (Last 6 weeks)1
Reflects downloads up to 14 Feb 2025
Other Metrics
Citations
View Options
Login options
Check if you have access through your login credentials or your institution to get full access on this article.
Sign inFull Access
View options
View or Download as a PDF file.
PDFeReader
View online with eReader.
eReaderHTML Format
View this article in HTML Format.
HTML Format