A Review of Volatility and Option

Pricing
by
Sovan Mitra
arXiv:0904.1292v1 [q-fin.PR] 8 Apr 2009

Abstract
The literature on volatility modelling and option pricing is a large and diverse area
due to its importance and applications. This paper provides a review of the most signif-
icant volatility models and option pricing methods, beginning with constant volatility
models up to stochastic volatility. We also survey less commonly known models e.g.
hybrid models. We explain various volatility types (e.g. realised and implied volatility)
and discuss the empirical properties.

Key words: Option pricing, volatility models, risk neutral valuation, empirical volatil-
ity.

1. Introduction and Outline

This paper provides a review of the most significant volatility models and their
related option pricing models, where we survey the development from constant up to
stochastic volatility. We define volatility, the volatility types and study the empirical
characteristics e.g. leverage effect. We discuss the key attributes of each volatility
modelling method, explaining how they capture theoretical and empirical character-
istics of implied and realised volatility e.g. time scale variance. We also discuss less
commonly known models.
The study of volatility has become a significant area of research within financial
mathematics. Firstly, volatility helps us understand price dynamics since it is one of
the key variables in a stochastic differential equation governing an asset price. Sec-
ondly, volatility is the only variable in the Black-Scholes option pricing equation that
is unobservable, hence the ability to model volatility is crucial to option pricing.
Thirdly, volatility is a crucial factor in a wide range of research areas. For example,
contagion effects involve the “transmission” of volatility from one country to another

Preprint submitted to Elsevier April 8, 2009

[Bau03]. Volatility can explain extreme events as Blake [Bla90] explains that the
October 1987 crash could have resulted from volatility changes.
Finally, volatility has a wide range of industrial applications from pricing exotic
derivatives to asset pricing models [Reb04]. Shiller [Shi89] argues the market’s volatility
dynamics can be applied to macroeconomic variables, particularly as the stock market
is a well known leading indicator of the economy. Shiller [Sch81] also claims volatility
can be used as a measure of market efficiency.
Option pricing in itself has become an important research area. Research interest
in options pricing began with the Black-Scholes option pricing paper [BS73]; since then
the derivatives market has grown into a multi-trillion dollar industry [Sto99]. Options
have become important to industry, particularly as they can be used to hedge out risk.
In fact in many situations it is more attractive to speculators and hedgers to trade an
option rather than an underlying due to the limited loss. Additionally, option trading
can normally be executed on a far higher level of leverage compared to trading stocks,
therefore offering potentially higher returns for the same initial deposit.
The outline of the paper is as follows. Firstly, we review basic financial mathematics
theory, which is essential for the study of volatility modelling and option pricing. Next,
we introduce the differing types of volatility and discuss their empirical behaviour e.g.
leverage effect. We then discuss the key models of volatility and their associated option
pricing methods. We finally end with a conclusion.

2. Review of Financial Mathematics Theory

This section provides a brief exposition of standard financial mathematics theory.

The recommended references on this area are Bjork [Bjö04], Øksendal [Øks03], Neftci
[Nef96] and Delbaen and Schachermayer [DS06].

2.1. Stochastic Calculus

2.1.1. Stochastic Differential Equations
From the time of Bachelier’s work [Bac00], the most popular method of modelling
asset prices has been using stochastic differential equations. Let X(t) denote a stochas-
tic process (such as a stock price) and we define it as a diffusion if it approximately
follows the stochastic difference equation:

X(t + ∆t) − X(t) = µ(t, X(t))∆t + σ(t, X(t))∆W (t), (1)

where:

• ∆W (t) = W (t + ∆t) − W (t);

2
 • µ(t, X(t)) denotes the drift of X(t);

• σ(t, X(t)) denotes the volatility (also known as the diffusion term).

The increment ∆W (t) is from W, which is called a Wiener process.

Definition 1. A stochastic process W(t) is called a Wiener process if it satisfies the

following conditions:
1. W (0) = 0;
2. W (t) has independent increments. In other words W (u)−W (t) and W (s)−W (r)
are independent for r < s ≤ t < u;
3. W (t) has continuous trajectories;
√
4. W (t) − W (s) ∼ N (0, t − s) for s < t.

The stochastic process is called a diffusion because the equation models diffusions in
physics. For continuous time modelling we let ∆t → 0 and so the stochastic difference
equation becomes:

dX(t) = µ(t, X(t))dt + σ(t, X(t))dW, (2)

for X(0) = a, (3)

where a is a constant. Alternatively, in integral form the equation becomes:

Z t Z t
X(t) = a + µ(s, X(s))ds + σ(s, X(s))dW. (4)
0 0

It is worth noting in equation (4) that the first integral is a standard Riemman integral
whereas the second integral is a stochastic integral, that is, an integral with respect to
a Wiener process.

2.1.2. Stochastic Integrals and Ito’s Lemma

To integrate a stochastic process X(t) we must know the conditions that guarantee
the existence of the stochastic integral. We can guarantee the integral’s existence if
the process X(t) belongs to the class L2 . To define the class L2 we must introduce the
idea of filtration.

Definition 2. Let {Ft } denote the set of information that is available to the observer.
If

Fs ⊆ Ft ⊆ FT , ∀s, t with s < t < T, (5)

then the set {Ft }, t ∈ [0, T ] is called a filtration.

3
 For a given stochastic process X(t), increasingly more information is revealed to
an observer as time progresses. Hence at time t=a (where a is a constant) some
information is revealed and this information is known with certainty at any future time
t > a. To keep track of the information flow that is revealed at time t we introduce
the filtration FtX .

Definition 3. The notation FtX denotes the information generated by process X(t)
on the interval [0,t]. If based upon the observations of the trajectory of X(s) over the
interval s ∈ [0, t] it is possible to determine if event A has occurred, then we say A is
FtX -measurable and write A ∈ FtX . If the stochastic process Z can be determined based
upon the trajectory of X(s) over the interval s ∈ [0, t] and Z(t) ∈ FtX , ∀t ≥ 0, then we
say Z is adapted to the filtration {FtX }t≥0 .

We now define the class L2 .

Definition 4. A stochastic process X(s) belongs to the class L2 [a, b] if the following
conditions are satisfied

• X(s) is adapted to the FtX filtration;

Rb
• a E[X 2 (s)]ds < ∞.

If we wish to obtain stochastic differential equations we may require the stochastic

version of the classical chain rule of differentiation. This is known as Ito’s lemma.

Theorem 1. (Ito’s Lemma) Assume that X(t) is a stochastic process with the stochas-
tic differential given by
dX(t) = µ(t)dt + σ(t)dW, (6)
where µ and σ are adapted processes. Let Z(t) be a new process defined by Z(t) =
f (X, t) and f is a twice differentiable function, then Z has the stochastic differential:
 
∂f ∂f 1 2 ∂2f ∂f
df (X(t), t) = +µ + σ 2
dt + σ dW. (7)
∂t ∂X 2 ∂X ∂X

2.1.3. Geometric Brownian Motion

To model asset prices it is tempting to apply Bachelier’s model. On studying the
empirical behaviour of stock prices over time, Bachelier in his doctoral thesis [Bac00]
first suggested modelling asset prices X(t) with the following equation:

dX(t) = µXdt + σdW, (8)

for X(0) = a, (9)

where a, µ and σ are constants. However, Bachelier’s equation was not satisfactory;
theoretically stock prices are always non-negative yet Bachelier’s equation allowed neg-
ative stock prices. We would expect percentage returns to be independent of stock price

4
X(t) yet Bachelier’s equation is not. Samuelson in 1965 [Sam65] introduced the Geo-
metric Brownian motion (GBM) of stock prices, which is the standard model for stock
prices to date:

dX/X = µdt + σdW, (10)

  
1 2
X(t) = X(0)exp µ − σ t + σW . (11)
2

In equation (10) X(t) is nonnegative with probability one and if X(t1 ) = 0 then

X(t) = 0, ∀t ≥ t1 . (12)

This property reflects financial “bankruptcy” since once X(t) equals 0 it will remain
permanently 0 thereafter.
If we have a portfolio of n assets X(t) = (X1 (t), X2 (t), ...., Xn (t)), governed by
n independent Wiener processes W = (W1 (t), W2 (t), ...., Wn (t)), we have the model
[Eth02]:
n
X
dXi /Xi = µi dt + σij dWj , i=1,..,n and j=1,..,n, (13)
j=1
n
! n
!
1X 2 X
Xi (t) = Xi (0)exp µi − σ t+ σij Wj , (14)
2 j=1 ij j=1

where

• σij is the volatility matrix

 
σ11 . . . σ1n
 . . 
 .. . . ...  ; (15)
 
σn1 . . . σnn

• the volatility of each stock Xi (t) is σi (t) where

v
uX
u n 2
σi = t σij ; (16)
j=1

• instantaneous correlation between Wi and Wk is given by

corr(dWi, dWk ) = ρik (t)dt (17)

n
X
1
= σij (t)σkj (t). (18)
σi (t)σk (t) j=1

5
2.2. Black-Scholes Option Pricing
The Black-Scholes analysis of European options [BS73] yields a closed form solution
to option pricing, only requiring observable variables (except for volatility). The key
insight into their analysis was to construct a dynamically replicating portfolio of a
European option to value it. Options have become one of the most important and
frequently traded derivatives in finance. We give a definition of a derivative security
as follows [Bin04].
Definition 5. A derivative security (also known as a contingent claim) is a financial
contract whose value at expiration time T is precisely determined by the price of an
underlying asset at time T.

As the name implies, an option gives the right (but not the obligation) to buy or sell
an asset at a pre-determined price and time. A call option gives the right to buy the
asset whereas a put gives the right to sell the asset at a predetermined price. One can
purchase European options, which are options that can only be exercised at expiration
T, or American options, which can be exercised any time during the life of the option.
The Black-Scholes portfolio replication argument for European call options begins
by considering a market consisting of two assets -a riskless bond B(t) and a stock X(t)
with equations respectively:

dB(t) = rB(t)dt, (19)

dX(t)/X(t) = µdt + σdW, (20)

where σ and µ are constants and r is the risk free rate of return. Black and Scholes man-
aged to determine the closed form value of a European call option C on the assumption
of no arbitrage (to be defined in section 2.4):

C(X(t), t, T, r, σ, K) = X(t)Ψ(d1 ) − Ke−r(T −t) Ψ(d2 ), (21)

ln(X(t)/K) + (r + 12 σ 2 )(T − t)
where d1 = p , (22)
σ (T − t)
ln(X(t)/K) + (r − 21 σ 2 )(T − t)
d2 = p , (23)
σ (T − t)
p
= d1 − σ (T − t). (24)

In C(X(t),t,T,r,σ,K) t is the time at which C is being priced, T is the expiration date,

Ψ(·) is the standard normal cumulative distribution function and K is the strike price.
For a more detailed discussion of the Black-Scholes option pricing equation the reader
is referred to [Duf01].

6
2.3. Risk Neutral Valuation
2.3.1. Martingales
In financial mathematics, martingale processes are important to understanding key
concepts, such as the idea of completeness. We therefore define martingales now.

Definition 6. An Ft − adapted process Mt is an Ft − martingale if

• E[| Mt |] < ∞, ∀t ≥ 0;

• E[Ms | Ft ] = Mt , for s ≥ t.

If we replace the equality with ≤ (or ≥) then Mt is a supermartingale (or submartingale

respectively).

Whereas a martingale can be considered a fair game, submartingales and super-

martingales can be considered as favourable and unfavorable games, respectively, from
the gambler’s perspective. A trivial example of a martingale is a constant k; let
X(t) = k then:

E[X(t + s)|Ft ] = X(t) = k. (25)

However in financial mathematics we are interested in stochastic processes since we

wish to model asset prices. We therefore define martingales for stochastic processes.

Proposition 1. A stochastic process X is a martingale if and only if the stochastic

differential is of the form

dX(t) = f (t)dW, (26)

where f is any process satisfying the condition:

Z t
E[f 2 (s)]ds ≤ ∞. (27)
0

Therefore for stochastic processes to be martingales they must be “driftless”.

2.3.2. Girsanov’s Theorem: Change of Measure

In addition to the Black-Scholes equation (equation (21)), there exists another
method for valuing European options, known as risk neutral valuation. Samuelson
showed that the price of a call option is equal to its discounted expected payoff:

C(X(t), K, t, T ) = e−r̃(T −t) E P [X(T ) − K]+ , (28)

where P is the probability measure and r̃ is the discount factor for risk. The difficulty
with this approach is that the discount factor r̃ and the probability measure P vary

7
according to the risk preference of an investor, so were unknown and typically arbi-
trarily chosen. For instance a risk neutral (or risk indifferent) investor does not require
an incentive or disincentive to take on a risky investment, therefore if the discounted
expected value remains constant he will pay the same price for an investment regardless
of the risk. Furthermore for a risk neutral investor we have r̃ equals the risk free rate of
return (which is observable), however P was unknown at the time Samuelson proposed
his option formula.
Cox et al. in [CRR79] realised that under a Black-Scholes model, option pricing is
independent of risk preferences. The investor’s risk aversion increases with the stock’s
drift µ; yet µ does not appear in the Black-Scholes option pricing equation (equation
(21)). By using equation (28), it was shown that all option prices would give the
same option price regardless of the risk preference chosen provided re and P are chosen
consistently. Therefore equation (28) and the Black-Scholes formula would always give
the same answer.
To value options as a risk neutral investor, we discount at the risk free rate and
take expectations under the risk neutral measure. To take expectations under the risk
neutral measure we need to change P in the stochastic differential equation, which
requires Girsanov’s Theorem. Girsanov’s Theorem tells us how a stochastic differential
equation (SDE) changes as probability measure P changes. Essentially, Girsanov’s
Theorem tells us a change in P corresponds to a change in drift µ and the rest of the
SDE remains unchanged. To explain how we change probability measures, let us define
probability spaces.
Definition 7. The triple {Ω, F , P} is called a probability space where Ω denotes the
sample space, the set of all possible events, F denotes a collection of subsets of Ω or
events and P is the probability measure on F or events. The quadruple {Ω, F , {Ft}t≥0 , P}
is called a filtered probability space.
Assume we have the probability space {Ω, F , P} then a change of measure from P
to Q means we have probability space {Ω, F , Q}. We now state Girsanov’s Theorem
for change of probability measures for SDEs.
Definition 8. (Girsanov’s Theorem) Suppose we have a family of information sets Ft
over a period [0,T] where T < ∞. Define over [0,T] the random process (known as the
Doleans exponential) ξt :
 Z t Z 
P 1 t 2
ξt = exp − λ(u)dW (u) − λ (u)du , (29)
0 2 0
where W P (t) is the Wiener process under probability measure P and λ(t) is an Ft -
measurable process that satisfies the Novikov condition
  Z t 
P 1 2
E exp λ (u)du < ∞, t ∈ [0, T ].
2 0

8
Then W Q is a Wiener process with respect to Ft under probability measure Q, where
W Q is defined by
Z t
Q P
W (t) = W (t) + λ(u)du, t ∈ [0, T ]. (30)
0

The probability measure Q is defined by

Z
Q[A] = ξt dP, A ∈ Ft , (31)
A

with A being an event in Ft .

Heuristically, Girsanov’s Theorem tells us that a change of measure is a subtraction

or addition of an Ft -adapted drift in the Wiener process [BR96]:

dW Q = dW P + λ(t)dt. (32)

Our discounted risk neutral process can be expressed as

dX/X = (µ − r)dt + σdW P , (33)

= (µ − r)dt + σ(dW Q − λ(t)dt). (34)

If we choose

λ(t) = (µ − r)/σ, (35)

we obtain equation

dX/X = σdW Q , (36)

which is a martingale (see section 2.3.1) since λ(t) cancels the drift. This choice of
λ(t) is also known as the market price of risk and the Sharpe ratio (discovered by
Sharpe [Sha66]). Note that only one choice of λ(t) in this SDE gives a martingale (or
alternatively eliminates the drift) hence Q is known as the unique equivalent martingale
measure. The equivalent martingale measure is also known as the risk neutral measure
and pricing under this measure is known as risk neutral valuation.

2.3.3. Multidimensional or Multifactor Girsanov’s Theorem

To change measures for multidimensional SDEs we require the multidimensional
Girsanov’s Theorem, which is very similar to the one dimensional version.
Definition 9. (Multidimensional Girsanov’s Theorem) Suppose we have a family of
information sets Ft . Let θ(t) = (λ1 (t), λ2 (t), ..., λn (t)) be an n-dimensional process that
is Ft -measurable and satisfies the Novikov condition:
" ( Z n )#
tX
1
E P exp λ2 (u)du < ∞. (37)
2 0 i=1 i

9
We define the random process ξt :
( n  Z Z t )
X t
1
ξt = exp − λi (u)dWiP(u) − λ2i (u)du , (38)
i=1 0 2 0

where dWiP for i=1,...,n is an n-dimensional Wiener process under probability measure
P. Then under the measure Q, WiQ is a multidimensional Wiener process defined by:
Z t
Q P
Wi = Wi + λi (u)du, for i = 1, 2, ...n. (39)
0

The probability measure Q is defined by

Z
Q[A] = ξt dP, A ∈ Ft . (40)
A

We wish to obtain an equivalent martingale measure Q related to the vector valued

Wiener process W = (W1 , W2 , ..., Wn ). Therefore each stock in the multiple stock
model must be a martingale with respect to Q. Since changing measure means changing
drift we have a vector θ(t) = (λ1 (t), ...., λn (t)) to give an n-dimensional Q-Wiener
process
Z t
WiQ = WiP + λi (s)ds. (41)
0
µ−r
In the one dimensional case we obtained a martingale by setting λ = σ
, in the
multiple stock model we apply a similar argument but in every dimension:
(µi − r)
λi (t) = P  , for i = 1, .., m. (42)
n
j=1 σij (t)

When the volatility matrix is invertible then unique λi solutions exist, admitting a
unique risk neutral process. Therefore the discounted risk neutral process for the
multiple stock model is:
n
X
dXi /Xi = σij dWiQ . (43)
j=1

2.4. Fundamental Theorems of Finance

Before stating the fundamental theorems of finance, we must define arbitrage and
a self-financing portfolio. A portfolio V (t) is self-financing if the change in value of
the portfolio is a result of the assets’ changing values and no external withdrawals or
additions to the portfolio are made. For instance a portfolio V(t) consisting of k1 shares
of X(t) and k2 bonds of B(t) is a self-financing portfolio if:

dV (t) = k1 dX(t) + k2 dB(t). (44)

We now define arbitrage.

10
Definition 10. An arbitrage possibility in a financial market is a self-financed portfolio
V (t) such that:

• V (0) ≤ 0;

• V (T ) ≥ 0 almost surely and

• E[V (T )] ≥ 0.

In words arbitrage is an event where it is possible to make a profit without the

possibility of incurring a loss. We say we have a “fair price” when an asset is free
from arbitrage. We are now in a position to state the two key theorems of financial
mathematics.

Theorem 2. (First Fundamental Theorem of Finance) The market is arbitrage free if

and only if there exists an equivalent martingale measure.

Theorem 3. (Second Fundamental Theorem of Finance) Assume that the market is

arbitrage free. The market is then complete if and only if there exists a unique equivalent
martingale measure.

In a no arbitrage market that is incomplete we have a variety of no arbitrage option

prices since we have a variety of martingale measures. Each martingale measure will
produce a different price, in general.
Completeness and arbitrage are separate properties and we can illustrate this follow-
ing the heuristic argument of Björk [Bjö04]. Let ̟2 denote the number of independent
sources of randomness and let ̟1 denote the number of tradable assets. Now the
market is arbitrage free if ̟1 ≤ ̟2 and complete if ̟1 ≥ ̟2 . Hence the market is
complete and arbitrage free if ̟1 = ̟2 .
For no arbitrage we must have ̟1 ≤ ̟2 since each new tradable asset gives an
opportunity to create an arbitrage portfolio. For example, let ̟2 =1 and ̟1 =2 and
each asset follows GBM that are identical (including the same Wiener process since
̟2 =1) but have different drifts:

dXi /Xi = µi dt + σdW. (45)

Then one can produce a riskless profit by shorting the lower drift asset X1 and using
the proceeds to purchase the higher drift asset X2 . For completeness we must have
̟1 ≥ ̟2 to enable us to trade or replicate every possible claim.
The no arbitrage definition enables us to state the put-call parity relationship for
European options, which does not require any assumptions other than the market is
arbitrage free.

11
Proposition 2. (Put-Call Parity)
Assume the market is arbitrage free. If a European call C(t,X) and a European put
P(t,X) with the same underlying asset X, strike K and expiration T exist then we have
the put-call parity relation:
P (t, X) = Ke−r(T −t) + C(t, X) − X(t) + D, (46)
where D is the cash dividend received from the underlying stock during the life of the
option.

2.5. Feynman-Kac Stochastic Representation Formula

The Feynman-Kac theorem provides a link between the partial differential equation
of a diffusion process and its expectation. This is useful because we can solve important
analytical problems either by taking expectations under a risk neutral measure or by
solving the partial differential equation.
Theorem 4. (Feynman-Kac Theorem)
Let X(t) satisfy the equation
dX(t) = µ(t, X(t))dt + σ(t, X(t))dW, (47)
with initial value at initial time s
X(s) = x, (48)
and let f (s, x) ∈ L2 be a function which satisifies
Z T " 2 #
∂f
E σ(t, X(t)) (t, X(t)) dt < ∞ (49)
0 ∂x
and boundary condition
f (T, x) = h(x). (50)
If the function f(s,x) is a solution to the boundary value problem
∂f 1 ∂2f ∂f
+ σ 2 (s, x) 2 + µ(s, x) − rf (s, x) = 0, (51)
∂s 2 ∂x ∂x
then f has the representation:
f (s, x) = e−r(T −s) E[h(X(T ))|X(s) = x]. (52)

The Feyman-Kac formula can be extended to the case of W = (W1 , W2 ) where

W1 , W2 are independent Wiener processes and we have two stochastic differentials in
X1 and X2 . Following Shreve [Shr04], let the stochastic processes follow

dX1 (t) = µ1 (t, X1 (t), X2 (t))dt + σ11 (t, X1 (t), X2 (t))dW1 + σ12 (t, X1 (t), X2 (t))dW2 ,
(53)
dX2 (t) = µ2 (t, X1 (t), X2 (t))dt + σ21 (t, X1 (t), X2 (t))dW1 + σ22 (t, X1 (t), X2 (t))dW2 ,
(54)

12
with initial values at initial time s

X1 (s) = x1 , (55)
X2 (s) = x2 , (56)

with conditions
RT ∂f
0
E[(σ11 (t, X1 (t), X2 (t)) (t, X1 (t), X2 (t)))2
∂x1
∂f
+ (σ12 (t, X1 (t), X2 (t)) (t, X1 (t), X2 (t)))2 ]dt < ∞,
∂x1
RT ∂f
0
E[(σ21 (t, X1 (t), X2 (t)) (t, X1 (t), X2 (t)))2
∂x2
∂f
+ (σ22 (t, X1 (t), X2 (t)) (t, X1 (t), X2 (t)))2 ]dt < ∞,
∂x2
with boundary condition

f (T, x1 , x2 ) = h(x1 , x2 ) (57)

and satisfy:

f (s, x1 , x2 ) = e−r(T −s) E[h(X1 (T ), X2(T ))|X1 (s) = x1 , X2 (s) = x2 ]. (58)

We then have the associated partial differential equation:

∂f ∂f ∂f
0 = + µ1 (s, x1 , x2 ) + µ2 (s, x1 , x2 )
∂s ∂x1 ∂x2
2

∂ f
1 2 2
+ σ11 (s, x1 , x2 ) + σ12 (s, x1 , x2 )
2 ∂x21
∂2f
+ (σ11 (s, x1 , x2 )σ21 (s, x1 , x2 ) + σ12 (s, x1 , x2 )σ22 (s, x1 , x2 ))
∂x1 ∂x2
1 2 2

∂2f
+ σ21 (s, x1 , x2 ) + σ22 (s, x1 , x2 ) − rf (s, x1 , x2 ).
2 ∂x22

3. An Introduction to Volatility
3.1. Different Types of Volatility
Wilmott in [Wil01] distinguishes between four different types of volatility, although
in practise little distinction is made between them. The four types of volatility are:

• volatility σ in equation (2). It is a measure of randomness in asset return and

since it exists at each moment in time it has no “timescale” associated with it.
For example volatility can be 20%,5%,60%.

13
 • historic volatility (also known as realised volatility): this is a measure of volatility
using past empirical price data and will be explained further in section 3.2.

• implied volatility: this is the volatility associated with empirical option prices
and will be explained further in section 3.2.

• forward volatility: this is the volatility obtained from some forward instrument.

We note that all four types of volatility in theory should not differ since they all refer
to the same variable σ, however in practise they may be different. For example some
researchers believe actual volatility and implied volatility are 2 separate variables and
treat them differently (see Schonbucher’s model in section 4.4 for instance).

3.2. Historic and Implied Volatility

In theory volatility should not depend on the method of measurement. However, in
practice this is not the case. Volatility σ can be empirically measured by two methods:
historic volatility and implied volatility. Historic volatility is calculated from empirical
(and therefore discrete) stock price data X(t0 ), ...., X(tn ) where ∆t = ti − ti−1 denotes
the chosen sampling interval. To estimate historic volatility σ
b we calculate the standard
deviation of an asset’s continuously compounded return per unit time:
p
Vχ
σb= √ , (59)
∆t
where

• Vχ is the sample variance

n
1 X
Vχ = (χi − χ). (60)
n − 1 i=1

Note that sample variance contains n-1 in the denominator, whereas variance of
a theoretical distribution contains n;

•
 
X(ti )
χi = ln ; (61)
X(ti−1 )

• χ is the sample mean

n
1X
χ= χi . (62)
n i=1

14
 In contrast to historic volatility we obtain implied volatility values from empirical
options data. Using Black-Scholes option pricing, call options C are a function of
C(X,t,T,r,σ,K), with all the independent variables observable except σ. Since the
quoted option price C obs is observable, using the Black-Scholes formula we can therefore
calculate or imply the volatility that is consistent with the quoted options prices and
observed variables. We can therefore define implied volatility I by:

CBS (X, t, T, r, I, K) = C obs , (63)

where CBS is the option price calculated by the Black-Scholes equation (equation (21)).
Implied volatility surfaces are graphs plotting I for each call option’s strike K and
expiration T. Theoretically options whose underlying is governed by GBM should have
a flat implied volatility surface, since volatility is a constant; however in practise the
implied volatility surface is not flat and I varies with K and T.
Implied volatility plotted against strike prices from empirical data tends to vary in a
“u-shaped” relationship, known as the volatility smile, with the lowest value normally
at X=K (called “at the money” options). The opposite graph shape to a volatility
smile is known as a volatility frown due to its shape. The smile curve has become a
prominent feature since the 1987 October crash (see for instance [Bat00] and [CP98]).
Various explanations have been proposed to account for volatility smiles. Firstly, it
has been suggested that options are priced containing information about future short
term volatility that is not already contained within past price information (see for
instance [Jor95]). Secondly, implied volatility is influenced more by market sentiment
rather than by pure fundamentals, for instance the VIX (a weighted average of implied
volatilities from S&P 500 index options) is used as a gauge of market sentiment [SW01].
Thirdly, the transaction costs involved in trading options is significantly higher
and more complicated compared to their associated underlyings, creating volatility
smiles (see for instance [PRS99]). Finally, Jarrow and Turnbull [JT96] discuss non-
simultaneous price observation; since option and stock prices are from two different
financial markets, there will always be observation time differences. Such time differ-
ences can cause substantial differences in the estimated implied volatility.

3.3. Empirical Characteristics of Volatility

Volatility has been observed to exhibit consistently some empirical characteristics,
which we will now discuss. Firstly, one of the most well known empirical characteristics
of volatility is the leverage effect, as first proposed by Black [Bla76]. Black observed
that volatility is negatively correlated with stock price and accounted for this through

15
the concept of leverage. Leverage (also known as the gearing or debt to equity ratio)
is defined by:
ι
Leverage= ,
MKT
where ι is the company’s total debt and MKT is the market capitalisation (number of
shares × share price). As the share price drops the company becomes riskier, since a
greater percentage of the company is debt financed, hence increasing volatility.
Black argued leverage could not entirely explain volatility since companies with
little or no debt still exhibit high volatility. Other explanations of volatility’s negative
corelation with stock price include portfolio rebalancing, where investors are forced to
liquidate their assets if they fall below a threshold price. Alternatively, threshold prices
may act as triggers for sales when interpretted in terms of prospect theory, as proposed
by Kahneman and Tversky [KT79]. Also there exists ownership concentration, where
an investor owns a substantial percentage of a company’s stock and sells all his holdings
at once (see for instance [CCW08]). In both cases a large volume of selling pushes prices
down further, increasing volatility.
Secondly, volatility (and return distributions) show dependency on the time scale
∆t chosen to measure it, as defined in equation (59). The return distribution becomes
increasingly more Gaussian as ∆t increases [Con01], known as “Gaussian aggregation”,
yet under GBM volatility is theoretically scale invariant. Such empirical observations
have motivated researchers to seek models that exhibit scale variation and is one of the
benefits of mean reverting stochastic volatility models.
Thirdly, Mandelbrot [Man63] and Fama [Fam65] were the first to observe volatility
clusters (positively autocorrelates) with time; large (small) price changes tend to fol-
low large (small) price changes. Such observations motivated GARCH and stochastic
volatility models (see section 4.3) and for this reason volatility clustering is sometimes
known as the “GARCH effect” [Con04]. The autocorrelation is significant over time
scales ∆t of days and weeks but insignificant over longer time scales. This is because
the autocorrelation strength decays following a power law with ∆t increasing.
The autocorrelation’s slow power law of decay has been cited as evidence of the
existence of long memory in volatility [Con01] and empirical evidence can be found
in [LGS99]. A stationary stochastic process X(t) has long memory if its covariance
υ(τ ) = cov(X(t), X(t + τ )) follows the power law of decay [Tan06]

υ(τ ) = τ d , for − 1 < d < 0, (64)

16
and
∞
X
υ(τ ) = ∞. (65)
τ =0

Fourthly, volatility tends to be correlated with significant financial or economic

variables. For instance Schwert [Sch89b] empirically shows that excessive volatility
increases during financial crises; during the Great Depression (1929-39) stock volatility
increased by a factor two or three times its typical values.
Finally, in commodities and other asset classes seasonal effects and calendar time
play a significant role in return and volatility distributions. For instance, French [Fre80]
noticed stock returns measured over the weekend on average are more negative than
any other day of the week. In the past decade the increasing availability of intraday
data has enabled researchers to show times of the day impact volatility, therefore
complicating intraday volatility estimation [GSW01].
Other volatility empirical characteristics include mean reversion, where volatility
tends to revert around some long term value [SS91]. This has motivated mean revert-
ing volatility models (see section 4.3 for more detail). Volatility tends to be correlated
with high trading volume [SX03] and company specific news (e.g. earnings announce-
ments) [Ros96]. Finally Shiller claims market volatility can be explained in terms of
psychological factors such as fashion, herding effects and overreaction to new informa-
tion [Shi89]. Shiller in [Shi89] also proposes a stock price model taking into account
psychological factors.

4. A Review of Volatility and Option Pricing Models

As described in section 2.1.3, Bachelier [Bac00] proposed a model for stock prices,
with constant volatility (see equation (8)). Bachelier reasoned that investing was the-
oretically a “fair game” in the sense that statistically one could neither profit nor lose
from it. Hence Bachelier included the Wiener process to incorporate the random na-
ture of stock prices. Osborne [Osb59] conducted empirical work supporting Bachelier’s
model. Samuelson [Sam65] continued the constant volatility model under the Geomet-
ric Brownian Motion stock price model (see equation (10)) on the basis of economic
justifications.
Over time, empirical data and theoretical arguments found constant volatility to
be inconsistent with the observed market behaviour (such as the leverage effect, as
discussed in section 3.3). A plot of the empirical daily volatility of the S&P 500 index
clearly shows volatility is far from constant. This consequently led to the develop-

17
ment of dynamic volatility modelling. Volatility modelling may be classified into four
categories:

1. Constant volatility σ;
2. Time dependent volatility σ(t);
3. Local volatility: volatility dependent on the stock price σ(X(t));
4. Stochastic volatility: volatility driven by an additional random process σ(ω).

These models will now be discussed in some detail in the subsequent sections.
It is worth mentioning that some models do not use Wiener processes to capture
price movements. For example, Madan and Seneta [MS90] propose a Variance-Gamma
model where price movements are purely governed by a discontinuous jump process.
Carr et al. discuss various jump processes in [CGMY02] such as the pure jump process
of Cox and Ross [CR76]. The advantages of pure jump models are: firstly, they real-
istically reflect the fact that trading occurs discontinuously, secondly price jumps are
consistent with information releases according to Fama’s “Efficient Market Hypothesis”
[Fam65] and finally, stock price processes jump even on an intraday time scale [CT04].

4.1. Time Dependent Volatility Model

As mentioned in section 3.3, it was empirically observed that implied volatility
varied with an option’s expiration date. Consequently, a straight forward extension
proposed to the constant volatility model was time dependent volatility modelling
[W+ 98]:

dX/X = µdt + σ(t)dW. (66)

Merton [Mer73] was the first to propose a formula for pricing options under time
dependent volatility. The option price associated with X is still calculated by the
standard Black-Scholes formula (equation (21)) except we set σ = σc where:
s
Z T
1
σc = σ 2 (τ )dτ , (67)
T −t t

i.e. d1 and d2 in the Black-Scholes equation become:

 
X 1 RT 2
log + µ(T − t) + σ (τ )dτ
K 2 t
d1 = qR , (68)
T 2
t
σ (τ )dτ
 
X 1 RT 2
log + µ(T − t) − σ (τ )dτ
K 2 t
d2 = qR . (69)
T 2
t
σ (τ )dτ

18
The equation (67) converts σ(t) to its constant volatility equivalent σc over time period
t to T. The distribution of X(t) is given by:
 
1 2 2
log(XT /Xt ) ∼ N (µ − σc )(T − t), σc (T − t) . (70)
2
Note that the constant volatility σc changes in value as t and T change. This property
enables time dependent volatility to account for empirically observed implied volatilities
increasing with time (for a given strike).

4.2. Local Volatility Models

4.2.1. Definition and Characteristics
In explaining the empirical characteristics of volatility, a time dependent volatility
model was found to be insufficient. For instance, time dependent volatility did not
explain the volatility smile, nor the leverage effect (see section 3.3), since volatility
cannot vary with price. Therefore volatility as a function of price (and optionally
time) was proposed, that is local volatility is σ = f (X, t):

dX/X = µdt + σ(X, t)dW. (71)

The term “local” arises from knowing volatility with certainty “locally” or when X is
known -for a stochastic volatility model we never know the volatility with certainty.
The advantages of local volatility models are that firstly, no additional (or un-
tradable) source of randomness is introduced into the model. Hence the models are
complete, unlike stochastic volatility models. It is theoretically possible to perfectly
hedge contingent claims. Secondly, local volatility models can be also calibrated to per-
fectly fit empirically observed implied volatility surfaces, enabling consistent pricing of
the derivatives (an example is given in [Dup97]). Thirdly, the local volatility model is
able to account for a greater degree of empirical observations and theoretical arguments
on volatility than time dependent volatility (for instance the leverage effect). We will
now look at some common local volatility models.

4.2.2. Constant Elasticity of Variance Model (CEV)

CEV was proposed by Cox and Ross [CR76]:
dX
= µdt + σ(X)dW, (72)
X
σ(X) = aX n−1 , (73)

where 0 ≤ n ≤ 1, a > 0 are constants. For n = 0 we obtain Bachelier’s model

(equation (8)) while n = 1 gives the Geometric Brownian Motion model (equation
(10)). Therefore n can be seen as a parameter for intuitively choosing between the two

19
extreme models. Additionally, n captures the level of leverage effect since volatility
increases as stock price decreases, as can be seen from equation (73).
The CEV model is analytically tractable, which is in contrast to the local volatility
model in section 4.2.4 where numerical solutions for derivatives become analytically
intractable [W+ 98]. Additionally, by appropriate choices of a and n we can fit CEV to
volatility smiles (see for instance Beckers [Bec80]).
The CEV model has been developed over the years giving various CEV modified
models. For instance the square root CEV model [CR76], which is similar to the
CIR interest rate model [CIJR85], Schroder [Sch89a] re-expresses the CEV model in
terms of a chi-squared distribution, enabling derivation of closed form solutions. Hsu
et al. [HLL08] determine the CEV model’s probability density function while Lo et al.
[LYH00] derive the option pricing formula for CEV with time dependent parameters.

4.2.3. Mixture Distribution Models

Mixture distributions have been applied to numerous statistical applications for
many years outside financial mathematics (Everitt [EH81] gives a survey of such ap-
plications). It had been known for some time that one could capture many empirical
characteristics of stock prices such as volatility smiles through mixture distributions
(see for instance Bingham and Kiesel [BK02] and Melick and Thomas [MT97]). How-
ever Brigo and Mercurio [BM00] were the first to prove the theoretical relation between
volatility and a mixture of lognormal distributions.
Let us assume the risk neutral probability density function p(t, X) at time t of a
stock price X(t) is a weighted sum of N lognormal probability densities pei (t, X):
N
X
p(t, X) = w̃i (t)e
pi (t, X), (74)
i=1
N
X
where w̃i (t) = 1, w̃i (t) ≥ 0, ∀t. (75)
i=1

The term w̃i (t) is a weighting for each component i at a point in time t, pei (t, X) denotes
the probability density of component i at a specific point in time for a specific stock
price X(t). We assume each pei (t, X) has the same mean µ but different variances σi2 (t).
Brigo and Mercurio then proved that the stock price follows the process:

dX/X = µdt + σ(X, t)dW, (76)

N
X
where σ 2 (X, t) = wi (t)σi2 (t). (77)
i=1

The mixture distribution model has been developed further by Brigo et al. (see
for instance [BM02],[BMS03], [BMRS04]). Alexander has also developed the mix-

20
ture distribution model; Alexander has applied it to the areas of stress testing portfo-
lios [AS08], combining it with GARCH processes [AL06] and bivariate option pricing
[AS04]. GARCH processes will be covered in more detail in section 4.4.
The mixture distribution models have been used to model more complicated volatil-
ity models due to their ability to capture a variety of distributions. For example, Leisen
[Lei05] shows how a mixture distribution model approximates Merton’s jump diffusion
model (to be covered in section 4.4) amongst others, Lewis [Lew02] shows how mixture
models can approximate stochastic volatility models.

4.2.4. Implied Local Volatility

Dupire [Dup94], Derman and Kani [DK94] proved for local volatility that a unique
risk-neutral process existed that was consistent with option data. The price of a Eu-
ropean call option by risk neutral valuation is given by:

C = e−rT E Q [X(T ) − K]+ , (78)

Z ∞
−rT
= e (X(T ) − K)p(X(T ))dX, (79)
K

where p(X(T)) is the risk neutral probability density function for X(T). Breeden and
Litzenberger [BL78] then showed from equation (79) that the risk neutral cumulative
distribution function F (·) at K is:
∂C
= −e−rT F (X(T ) ≥ K). (80)
∂K
Furthermore, the risk neutral probability density function p(X(T)=K) is

∂ 2 C rT
e = p(X(T ) = K). (81)
∂K 2
Hence we can recover the risk neutral density p(X(T)) from option data. This proba-
bility can be interpreted as the current view of the future outcome of the stock price.
Dupire [Dup94] then showed by applying p(X(T)) from equation (81) (obtained
from Breeden and Litzenberger’s work) to the Fokker-Planck equation, one could obtain
Dupire’s equation:

∂C X 2 ∂2C ∂C
= σ 2 (X, T ). . 2
− (r − D)X. − DC, (82)
∂T 2 ∂X ∂X
where D is the dividend. Rearranging equation (82) gives:
v
u ∂C ∂C
u + (r − D)X + DC
u
σ(X, T ) = u ∂T ∂X . (83)
t X 2 ∂2C
2 ∂X 2

21
Therefore the local volatility σ(X, T ) can be fully extracted from option data.
It can be seen from equation (83) that calculating σ requires partial differentials
with respect to T and K. We therefore require a continuous set of options data for
all K and T. This is highly unrealistic and quoted option prices tend to suffer from
significant illiquidity effects, affecting option bid-ask spreads [Nor03]. Furthermore,
Pinder [Pin03] shows that option bid-ask spreads are related to volatility, expiry and
trading volume. Since option data is discrete we require some interpolation method
to convert it to continuous data, for example Monteiro et al. [MTV08] apply a cubic
spline method. However, Wilmott [Wil00] states that local volatility computation is
highly sensitive to interpolation methods.
Dupire’s model implicitly assumes the options data contains all the information on
the underlying’s volatility if we calibrate to options data alone. However there is evi-
dence to show historic and implied volatility differ significantly [CP98]. Furthermore,
calibrating the local volatility surface to the options data tends to be unstable with
time, since the surface significantly changes from one week to another [DFW98].
Numerical computation of local volatility has been implemented by Andersen and
Brotherton-Ratcliffe using a finite difference method [ABR98]. Derman and Kani
([DKC96], [DKZ96]) and Rubinstein [Rub94] determine local volatilities by fitting a
unique binomial tree to the observed option prices. Tree fitting also has the computa-
tional advantage of not being affected by different interpolation methods.

4.3. Stochastic Volatility Models

4.3.1. Definition and Characteristics
Although local volatility models were an improvement on time dependent volatil-
ity, they possessed certain undesirable properties. For example, volatility is perfectly
correlated (positively or negatively) with stock price yet empirical observations suggest
no perfect correlation exists. Stock prices empirically exhibit volatility clustering but
under local volatility this does not necessarily occur. Consequently after local volatility
development, models were proposed that allowed volatility to be governed by its own
stochastic process. We now define stochastic volatility.

Definition 11. Assume X follows the stochastic differential equation

dX/X = µdt + σ(ω)dW1. (84)

Volatility is stochastic if σ(ω) is governed by a stochastic process that is driven by an-

other (but possibly correlated) random process, typically another Wiener process dW2 .
The probability space (Ω, F , P) is Ω = C([0, ∞) : R2 ), with filtration {Ft }t≥0 represent-
ing information on two Wiener processes (W1 , W2 ).
The process governing σ(ω) must always be positive for all values since volatility can

22
only be positive. The Wiener processes have instantaneous correlation ρ ∈ [−1, 1] de-
fined by:
corr(dW1(t), dW2 (t)) = ρdt.

Empirically ρ tends to be negative in equity markets due to the leverage effect (see
section 3.3) but close to 0 in the currency markets. Although ρ can be a function of
time we assume it is a constant throughout this thesis.
The key difference between local and stochastic volatility is that local volatility is
not driven by a random process of its own; there exists only one source of randomness
(dW1 ). In stochastic volatility models, volatility has its own source of randomness
(dW2 ) making volatility intrinsically stochastic. We can therefore never definitely de-
termine the volatility’s value, unlike in local volatility.
The key advantages of stochastic volatility models are that they capture a richer
set of empirical characteristics compared to other volatility models [MR05]. Firstly,
stochastic volatility models generate return distributions similar to what is empirically
observed. For example, the return distribution has a fatter left tail and peakedness
compared to normal distributions, with tail asymmetry controlled by ρ [Dur07]. Sec-
ondly, Renault and Touzi [RT96] proved volatility that is stochastic and ρ=0 always
produces implied volatilities that smile (note that volatility smiles do not necessarily
imply volatility is stochastic).
Thirdly, historic volatility shows significantly higher variability than would be ex-
pected from local or time dependent volatility, which could be better explained by
a stochastic process. A particular case in point is the dramatic change in volatility
during the 1987 October crash (Schwert [Sch90] gives an empirical study on this).
Finally, stochastic volatility accounts for the volatility’s empirical dependence on the
time scale measured (as discussed in section 3.3), which should not occur under local
or time dependent volatility.
The disadvantages of stochastic volatility are firstly that these models introduce
a non-tradable source of randomness, hence the market is no longer complete and
we can no longer uniquely price options or perfectly hedge. Therefore the practical
applications of stochastic volatility are limited. Secondly stochastic volatility models
tend to be analytically less tractable. In fact, it is common for stochastic volatility
models to have no closed form solutions for option prices. Consequently option prices
can only be calculated by simulation (for example Scott’s model in section 4.3.3).

4.3.2. Stochastic Volatility and the Driving Process

Stochastic volatility models fundamentally differ according to their driving mech-
anisms for their volatility process. Different driving mechanisms maybe favoured due

23
to their tractability, theoretical or empirical appeal and we can categorise stochastic
volatility models according to them. Many stochastic volatility models favour a mean
reverting driving process. A mean reverting stochastic volatility process is of the form
[FPS00a]:

σ = f (Y ), (85)
dY = α(m − Y )dt + βdW2, (86)

where:

• β ≥ 0 and β is a constant;

• m is the long run mean value of σ;

• α is the rate of mean reversion.

Mean reversion is the tendency for a process to revert around its long run mean value.
We can economically account for the existence of mean reversion through the cob-
web theorem, which claims prices mean revert due to lags in supply and demand
[LH92]. The inclusion of mean reversion (α) within volatility is important, in particu-
lar, it controls the degree of volatility clustering (burstiness) if all other parameters are
unchanged. Volatility clustering is an important empirical characteristic of many eco-
nomic or financial time series [Eng82], which neither local nor time dependent volatility
1
models necessarily capture. Additionally, a high α
can be thought of as the time re-
quired to decorrelate or “forget” its previous value.
The equation (86) is an Ornstein-Uhlenbeck process in Y with known solution:
Z t
−αt
Y (t) = m + (Y (0) − m)e +β e−αt dW2 , (87)
0

where Y(t) has the distribution

 2

−αt β −2αt
Y (t) ∼ N m + (Y (0) − m)e , (1 − e ) . (88)
2α

Note that alternative processes to equation (86) could have been proposed to define
volatility as a mean reverting stochastic volatility model, for example the Feller or
Cox-Ingersoll-Ross (CIR) process [FPS00b]:
√
dY = α(m − Y )dt + β Y dW2 . (89)

However with σ = f (Y ) in equation (86) we can represent a broad range of mean

reverting stochastic volatility models in terms of a function of Y.

24
4.3.3. Significant Stochastic Volatility Models
There is no generally accepted canonical stochastic volatility model and a large
number of them exist, therefore we review here the most significant ones.

1. Johnson and Shanno Model

Johnson and Shanno in 1987 [JS87] proposed one of the first stochastic volatility
models:

dX = µ1 Xdt + σX n dW1 , n ≥ 0, (90)

dσ = µ2 σdt + σ k βdW2 , k ≥ 0, (91)

where corr(dW1(t), dW2 (t)) = ρdt. A Monte Carlo method is proposed to deter-
mine the price of options under the stochastic volatility process by risk neutral
valuation. Johnson’s and Shanno’s computational results show that their option
prices are consistent with what is empirically observed (see section 3.3), that is
they exhibit a volatility smile and an increase in value with expiry.
2. Scott Model
Scott in 1987 [Sco87] considered the case where one assumes a geometric process
for stock prices and an Ornstein-Uhlenbeck process for the volatility:

dX/X = µdt + σdW1 , (92)

dσ = α(m − σ)dt + βdW2 . (93)

Scott proposed a stochastic mean reverting process based on empirical stock

price returns and describes a parameter estimation method based on moments
matching. Scott assumes ρ=0 to facilitate computation of option prices. The
option prices for Scott’s process are calculated by Monte Carlo simulation and
he observes a marginal improvement in option pricing accuracy compared to
standard Black-Scholes option pricing.
3. Hull-White Model
Hull and White modelled volatility as follows [HW87]:

dX/X = µ1 dt + σdW1 , (94)

dσ 2 /σ 2 = µ2 dt + βdW2 . (95)

The Hull-White model is an important contribution since it provides a closed

form solution to European option prices when ρ=0 and for a given risk neutral
measure γ. Let us define
Z T
1
σe2 = σ 2 (s)ds, (96)
T −t t

25
 where σe2 is a random variable. The option price is computed using the standard
p
Black-Scholes formula, under a risk neutral measure γ, with volatility σe2 . In
other words:
p
C(t, X, K, T, σ(ω)) = E [CBS (t, X, K, T, σe2 )].
γ
(97)

The option pricing equation (97) provides results consistent with empirically
observed currency options, where empirically ρ ≃ 0. Furthermore equation (97)
is still valid for any stochastic volatility process provided ρ = 0 [FPS00a]. For
correlated volatility, option prices are obtained using Monte Carlo simulation.
4. Stein and Stein Model
Stein and Stein in 1991 [SS91] proposed an Ornstein-Uhlenbeck process for volatil-
ity based on tractability and empirical considerations:

dX/X = µdt + σdW1 , (98)

dσ = −α(σ − m)dt + βdW2 , (99)

where ρ = 0. Note that although equation (99) implies volatility can be negative,
Stein and Stein state that only σ 2 is ever applied in calculation [SS91]. A closed
form solution to option prices is derived for particular choices of risk neutral
measures, which is in contrast to Johnson and Shanno [JS87] and Scott [Sco87],
who only provide numerical methods to option pricing.
5. Heston model
Heston’s model [Hes93] created in 1993 stands out from other stochastic volatility
models because there exists an analytical solution for European options that takes
in account correlation between dW1 and dW2 (although this requires assumptions
on the risk neutral measure):

dX/X = µdt + σdW1 , (100)

dσ 2 = α(m − σ 2 )dt + βσdW2 . (101)

The dσ 2 modelling process originated from CIR interest rate model [CIJR85].
To price an option under risk neutral valuation, we must specify a risk neutral
measure (due to market incompleteness) and this is chosen on economic justifica-
tions [Hes93]. Heston then finds an analytical solution for options using Fourier
transforms; the reader is referred to Heston [Hes93] and Musiela and Rutkowski
[MR05] for a derivation.
Due to the existence of an analytic solution to option pricing the Heston model
has been subsequently developed by various researchers. For example, Scott in-

26
 cludes stochastic interest rates [Sco97], Pan includes stochastic dividends [Pan02]
and Bates adds jumps to the stochastic process [Bat96].

4.4. Other Volatility Models

Apart from Bates [Bat96], the models discussed so far all have continuous sample
paths yet empirical stock prices appear to exhibit “jumps”. Additionally, the frequency
and magnitude of the jumps are too large to be explained by lognormal distributions,
which would be obtained under a GBM process. Merton therefore proposed the jump
diffusion model [Mer76], which added a random jump component to the GBM process:

dX/X = (µ − θk)dt + σdW + dP, (102)

where

• θ is the average number of jumps/year;

• k is the average jump size measured as a percentage of the asset price X;

• P is a Poisson process and independent of dW.

A Poisson process counts the number of events that occur in a given period. We define
a Poisson process P(ν) as [LM66]:

e−νt (νt)k
p(X = k) = , k = 0, 1, 2, .... (103)
k!
where

• ν is called the rate parameter, where ν is the expected number of events per unit
time;

• X is a random variable denoting the number of events;

• p(X=k) is the probability that the random number of events from 0 to time t
equals k.

In the case of the jump diffusion model the events are the jumps themselves. To obtain
a closed form solution for option prices under the jump diffusion model, Merton makes
a key argument as follows. We know that a replicating portfolio constructed in the
derivation of the original Black-Scholes equation ([BS73]) will eliminate the risk arising
from the GBM and so must earn the risk free rate of return. Next Merton assumes the
jump process represents nonsystematic risk and risk that is not priced into the market.
Therefore the same replicating portfolio applied in the Black-Scholes equation must

27
earn the risk free rate of return. Merton then proved that the option price is the sum
of an infinite series of options, valued by the standard Black-Scholes option pricing
equation without the jumps.
∞
X ′
e−θ τ (θ′ τ )n
C(X(t), K, t, T, r, σ) = CBS (X(t), K, t, T, rn , σn ),
n=0
n!
where
τ = T − t,
θ′ = θ(1 + k),
nln(1 + k)
rn = r − θk + ,
τ
nς 2
σn2 = σ 2 + .
τ
The nth option in the series is valued by the standard Black-Scholes equation, CBS ,
assuming n jumps have occurred before expiry. Since the series converges exponentially
it can be implemented computationally [CT04]. The term ς 2 arises from the fact that
a Poisson process can be approximated by a normal distribution. Therefore we can
assume the logarithm of the jumps are normally distributed with variance ς 2 . The
jumps fatten the return distribution’s tail, therefore the model is more consistent with
empirically observed distributions compared to GBM. Merton accounts for jumps as the
arrival of new information that have more than a marginal effect on price movements.
Another class of volatility models is to use a stochastic process to model the evo-
lution of the implied volatility directly. In all the models discussed so far, we have as-
sumed implied volatility and the underyling’s volatility are the same. Implied volatility
methods model an option’s (or any derivative’s) implied volatility σ ∗ separately from
the underlying’s volatility σ. One significant implied volatility model is Schonbucher’s
stochastic implied volatility model [Sch99]. Schonbucher models the implied volatility
σ ∗ of a vanilla option as a SDE and the underlying’s SDE separately:

dσ ∗ = µ2 dt + σ2 dW1 + βdW2 . (104)

The underlying’s SDE is:

dX/X = µ1 dt + σ1 dW1 . (105)

Another class of volatility models is the lattice approach, with its origins from bi-
nomial trees by Cox et al. [CRR79]. Britten-Jones and Neuberger [BJN00] model
stochastic volatility by fitting a lattice model that is consistent with observed option
prices. The model is parameterised by up and down price movements and their asso-
ciated probabilities for each branch, using empirical option data as its input. This is

28
an improvement on Derman’s and Kani’s model, who fit a tree to local volatility only.
Britten-Jones and Neuberger also show how a variety of stochastic volatility models
(such as regime switching volatility) can be fitted to be consistent with observed option
price data.
Discrete time volatility models are another class of models that exist, such as
GARCH(p,q) [Bol86], the generalised autoregressive conditional heteroscedasticity model:
q p
X X
σt 2 = a0 + ai ε2t−i + 2
bi σt−i , for εt ∼ N (0, σt2), (106)
i=1 i=1

where ai and bi are weighting constants. We obtain the ARCH(q) [Eng82] model by
setting p=0. Due to the success of GARCH in econometrics it has been substan-
tially extended by various researchers, for example Nelson [Nel91], Sentana [Sen95]
and Zakoian [Zak94]. Whereas with continuous time modelling one is able to derive
closed form solutions, which can reduce computation and provide new insights, this is
generally not possible with discrete time models.
Finally, one class of volatility models are “hybrid” models; combining various
volatility models into one. For instance Alexander [Ale04] extends Brigo’s and Mer-
curio’s mixture model [BM00] by combining it with the binomial tree of Cox et al.
[CRR79] to incorporate stochastic volatility. The SABR model [HKLW02] is a stochas-
tic extension of the CEV model [CR76], where SABR is an abbreviation for stochastic
alpha, beta and rho in its equations. The SABR model captures the dynamics of a
forward price F of some asset (e.g. stock) under stochastic volatility; its risk neutral
dynamics under measure Q is:

dF (t) = σ(t)F β (t)dW1Q , (107)

dσ(t) = ασ(t)dW2Q , (108)
where corr(dW1.dW2 ) = ρ ∈ [−1, 1], 0 ≤ β ≤ 1, α ≥ 0. (109)

A common shortfall in all the volatility models reviewed so far has been that all
these models are short term volatility models. The models implicitly ignore any long
term or broader economic factors influencing the volatility model, which is empirically
unrealistic and theoretically inconsistent. Furthermore, although some models may
specify a closed form solution for option pricing, they provide no method or recom-
mendation for calibration, which is important to modelling and option pricing.

5. Conclusions

This paper has surveyed the key volatility models and developments, highlighting
the innovation associated with each new class of volatility models. In conclusion it can

29
be seen from our review of volatility models that the development of has progressed in
a logical order to address key shortcomings of previous models.
Time dependent models addressed option prices varying with expiration dates, local
volatility also addressed volatility smiles and the leverage effect, whereas stochastic
volatility could incorporate all the effects captured by local volatility and a range of
other empirical effects e.g. greater variability in observed volatility. However the trade-
off associated with improved volatility modelling has been at the expense of analytical
tractability.

30
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