computation of greeks using malliavin calculus · t)π] for some weight function π, in the jump...

Computation of Greeks using

Malliavin calculus

Thesis Presented for the Degree of

DOCTOR OF PHILOSOPHY

in the Department of

Mathematics and Applied Mathematics

UNIVERSITY OF CAPE TOWN

By

Farai Julius Mhlanga

Supervisor: Professor Ronald I. Becker

April 1, 2011

Copyright c© University of Cape Town

Abstract

The valuation of derivatives (in the classical sense) of financial options with respect to

different parameters appearing in the underlying asset requires differentiating real-valued

functions f of the random price process Xt, namely f(Xt). In many cases, f is not differen-

tiable. This thesis includes an introduction to the Malliavin calculus machinery which is used

in the valuation. The main purpose is to derive explicit formulae for Greeks of a wider class

of options using Malliavin calculus. The Malliavin calculus deals with the differentiation and

integration of fairly general random variables and by using the integration by parts formula

it avoids the need to differentiate payoff functions. It also does not require explicit knowl-

edge of the density of the underlying asset. We first review the calculations of Greeks using

Malliavin calculus in the Brownian motion case. Then we derive explicit formulae for Greeks

in the form of expectations, under the risk neutral probability measure, of the option payoff

multiplied by a weight function, namely E[f(Xt)π] for some weight function π, in the jump

diffusion case. We also derive some explicit formulae in the case of stochastic volatility and

some variants of it which include jumps in the price and variance processes. Furthermore,

we obtain an expression for the Malliavin derivative of pure jump Levy stochastic differential

equations in terms of its first variation process. Then we give the necessary and sufficient

conditions for a function to serve as a weight function in the pure jump case. Working in

the white noise setting, we review the extension of the domain of the Malliavin derivative

to the whole L2 in both the pure diffusion and pure jump cases. Using the Donsker delta

function of a pure diffusion process and a pure jump process we derive explicit formulae for

∆. In this way, we can compute Greeks in great generality. All the formulae obtained can

then be used to evaluate the Greeks by Monte Carlo methods which are well-established.

i

Declaration

PhD Thesis Title: Computation of Greeks using Malliavin calculus.

I Farai Julius Mhlanga hereby:

• grant the University of Cape Town free licence to reproduce the above thesis in whole

or in part, for the purpose of research,

• declare that:

1. the above work is my own unaided work, both in concept and execution, and that

apart from the normal guidance from my supervisor, I have received no assistance.

2. neither the substance nor any part of the above thesis has been submitted in the

past, or is being, or is to be submitted for another degree or qualification at this

or any other University or Institution of higher learning.

I am now presenting the thesis for examination for the degree of PhD.

Signature:.......................

Date:.........................

ii

Dedication

To my late father.

iii

Acknowledgements

I wish to express my sincere gratitude to Professor Ronald I. Becker, my supervisor, for

his guidance, inspiration and invaluable support throughout this study. He read over the

manuscript at every step of the process and provided truly innovative ideas, and continu-

ally encouraged me to add value here, to update there and to clarify this concept. This

has improved the manuscript tremendously. I also thank Professor Giulia Di Nunno for her

suggestions and encouragement.

I would like to thank Dr Diane Wilcox for her financial support through the National Re-

search Foundation Grant Holder Bursary scholarship. I would like also to thank the NUFU,

DAAD and AMMSI whose funding made my study possible. I would also like to thank

AIMS (African Institute for Mathematics Sciences) for administering my DAAD scholarship

and for allowing me to use its facilities during consultation times with my supervisor. The

pleasant atmosphere at AIMS has been much appreciated. Special thanks go to the Depart-

ment of Mathematics and Applied Mathematics for their financial assistance through the

Chisnal award, tutorship and part time lectureship which they offered during the period of

my study.

The staff and my fellow students at the Department of Mathematics and Applied Mathe-

matics at the University of Cape Town also deserve recognition. Without their intellectual

stimulation and support, my progress would have suffered greatly. I would like also to thank

the people I have met at conferences who have provided me with useful comments and sug-

gestions. These include Professor Bernt Øksendal, Professor Frank Proske, Professor M.

Zervos, Professor Yaozhong Hu, Dr Raouf Ghomrasni and several others.

I am thankful to my beloved wife, Moleen, for her constant encouragement and prayers for

my success. I thank my mother for her prayers and words of encouragement too. I thank

my friends Hersinia Marwa, Evans Mangwiro, Kumbirai Gaza, Stanley Mtetwa, Phillimon

Mlambo and several others for their support.

Finally, I would like to thank God for awarding me a precious time and good health.

iv

Contents

Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . i

Declaration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii

Dedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii

Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iv

1 Introduction 1

1.1 Approaches to evaluating Greeks . . . . . . . . . . . . . . . . . . . . . . . . 3

1.2 Background on the Malliavin calculus . . . . . . . . . . . . . . . . . . . . . . 7

1.3 Aims and structure of the thesis . . . . . . . . . . . . . . . . . . . . . . . . . 8

2 Basic Properties of the Malliavin calculus 11

2.1 Wiener Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.2 Malliavin derivative . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.3 Skorohod integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.4 SDEs and Malliavin calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

2.5 The integration by parts formula . . . . . . . . . . . . . . . . . . . . . . . . 24

2.6 Iterated Wiener-Ito integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

2.7 Malliavin derivative via chaos expansion . . . . . . . . . . . . . . . . . . . . 30

2.8 Skorohod integral via chaos expansion . . . . . . . . . . . . . . . . . . . . . 32

2.9 The Clark-Haussmann-Ocone formula . . . . . . . . . . . . . . . . . . . . . . 34

v

3 Application of Malliavin calculus to the Calculations of Greeks for Con-

tinuous Processes 36

3.1 Generalized Greeks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

3.2 Greeks for European Options . . . . . . . . . . . . . . . . . . . . . . . . . . 48

3.3 Greeks for Exotic options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

3.4 Greeks for Barriers and Look-back options . . . . . . . . . . . . . . . . . . . 56

3.5 Greeks for the Heston model . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

4 Application of white noise calculus for Gaussian Processes to the Calcula-

tion of Greeks 67

4.1 Basic concepts of Gaussian white noise analysis . . . . . . . . . . . . . . . . 68

4.2 Stochastic test functions and stochastic distribution functions . . . . . . . . 74

4.3 The Wick product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

4.4 The Hermite Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

4.5 Hida-Malliavin derivative . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

4.6 Conditional expectation on (S)∗ . . . . . . . . . . . . . . . . . . . . . . . . . 93

4.7 The Donsker delta function . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

4.8 Financial Application: Calculating Greeks . . . . . . . . . . . . . . . . . . . 98

5 Malliavin calculus for Pure Jump Levy SDEs 104

5.1 Basic definitions and results for Levy processes . . . . . . . . . . . . . . . . . 105

5.2 Chaos expansion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108


5.4 Stochastic derivative . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

5.5 Differentiability of pure jump Levy stochastic differential equation . . . . . . 120

5.6 The necessary and sufficient condition for a function to serve as a weighting

function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126

6 Calculations of Greeks for Jump Diffusion Processes 131

vi

6.1 Basic elements of a Levy chaotic calculus . . . . . . . . . . . . . . . . . . . . 132

6.2 Chaos expansion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135

6.2.1 Directional derivative . . . . . . . . . . . . . . . . . . . . . . . . . . . 136

6.2.2 Wiener-Poisson space . . . . . . . . . . . . . . . . . . . . . . . . . . . 139


6.4 Greeks for jump diffusion models . . . . . . . . . . . . . . . . . . . . . . . . 145

6.5 Greeks for the Heston model with jumps . . . . . . . . . . . . . . . . . . . . 150

6.6 Greeks for Levy process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152

7 White noise calculus for Levy Processes and its Application to the Calcu-

lations of Greeks 158

7.1 Basic concepts of Levy white noise analysis . . . . . . . . . . . . . . . . . . . 158

7.2 Chaos expansion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160

7.3 The Hida/Kondratiev spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . 163

7.4 Levy Wick product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165

7.5 Levy Hermite transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167

7.6 Levy stochastic derivative . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168

7.7 Donsker delta function of a Levy process . . . . . . . . . . . . . . . . . . . . 170

7.8 Application: Computing Greeks . . . . . . . . . . . . . . . . . . . . . . . . . 171

References 174

vii

Chapter 1

Introduction

Investors make predictions on how option prices vary over a certain period of time basing

on the past market events. In general, the vast number of different market events makes

it a difficult task. It is therefore, important to understand which factors contribute to the

movement of prices and with what effect. This sensitivity analysis is carried over parameters

appearing in models for price dynamics and the so-called Greeks represent a form of measure

for price sensitivity to some factors. The name Greek is used because of market practice

of using names of Greek letters (real and invented) to represent these risk parameters (see

[49]).

A Greek is a derivative (in the classical sense) of a financial quantity, usually an option price,

with respect to one of the parameters of the model. Hence, the Greeks measure the stability

of the financial quantity under variations of the parameters. Practitioners need to have a

well developed intuition of the dependence of their position on the movements and events

in the market. Greeks are also useful for hedging and risk-management purposes (see [49]).

Thus, there is a need to efficiently compute Greeks.

We define the Greek in mathematical terms as follows. We consider a general Ito diffusion

process Xt, 0 ≤ t ≤ T given by

dXt = b(Xt)dt+ σ(Xt)dWt (1.1)

where Wt, 0 ≤ t ≤ T is a standard Brownian motion with values in Rn. The coefficients

b(x) and σ(x) are deterministic and are assumed to satisfy the usual conditions, to be stated

later on, to ensure the existence and uniqueness of the solution of equation (1.1). Moreover,

the diffusion coefficient σ(x) is assumed to satisfy a uniform ellipticity condition that will

be stated later on.

1

We first state the pricing formula. We consider a payoff Φ of some financial quantity

depending on the prices at a finite number of times, that is,

Φ = Φ(Xt1 , · · · , Xtn)

where Φ : Rn → R is infinitely differentiable and Φ and all its partial derivatives have

polynomial growth. Given 0 < t1 < . . . < tn = T, the option price u(x) is computed, under

the risk neutral probability measure, as

u(x) = E[e−rTΦ(Xt1 , · · · , Xtn) | X0 = x] (1.2)

where E denotes the expectation, r is the risk free interest rate assumed to be constant

and Xti , i = 1, . . . , n are the underlying assets depending on some model parameters. We

mention that the option price can be numerically computed by using Monte Carlo methods

(see [19] Chapter 8).

Then, the Greek is calculated as follows

Greek :=∂

∂αE[e−rTΦ(Xt1(α), · · · , Xtn(α)) | X0 = x] (1.3)

where the parameter α could be the initial price x, the drift coefficient b (if constant), the

volatility coefficient σ (if constant) or any of the constant parameters appearing in the model.

Xti(α), i = 1, . . . , n emphasize the dependence of Xti on α. Specifically, the most interesting

Greeks (in the standard Black-Scholes model) include:

• Delta, denoted by ∆, defined as the derivative of the option price with respect to the

initial price

∆ =∂

∂xu(x). (1.4)

∆ plays a key role in the computation of other Greeks as well.

• Gamma, denoted by Γ, defined as the second derivative of the option price with respect

to the initial price

Γ =∂2

∂x2u(x). (1.5)

Γ measures the sensitivity of ∆.

• Vega, denoted by V , defined as the derivative of the option price with respect to the

volatility

V =∂

∂σu(x). (1.6)

V is not a Greek letter.

2

Other Greeks include Rho, denoted by ρ, which is defined as the derivative of the option

price with respect to interest rate

ρ =∂

∂ru(x), (1.7)

and Theta, denoted by Θ, which is defined as the derivative of the option price with respect

to terminal time

Θ = − ∂

∂Tu(x). (1.8)

Θ is expressed as a negative derivative to represent the sensitivity of the price with decreasing

maturity. Unlike other factors, however, the movement in remaining maturity is perfectly

predictable. Hence, time is not a risk factor.

Remark

If the drift b and the volatility σ are functions of the underlying asset price then ∂∂bu(x) and

∂∂σu(x) are not defined. We will see later on how this situation is handled (see Chapter 3

Section 3.1).

We want to express the Greek (equation (1.3)) as an expectation without derivatives in order

to enable efficient computation. Such a representation is crucial for Monte Carlo evaluation

(see [53] page 146). The mathematical challenge arises from payoff functions which tend to be

discontinuous, non-differentiable or even more complicated. We are particularly interested

in the case where the payoff function Φ is a discontinuous function. A typical example is the

digital option, that is, Φ(x) = 1x≥K for some constant K > 0.

1.1 Approaches to evaluating Greeks

In this section we present different approaches, with their limitations, that have been used

traditionally to compute the Greeks. A detailed review of these approaches can be found

in [41] on Chapter 7. We illustrate the different approaches to the case of the ∆ of the

Black-Scholes price of a call option. The ideas can then be easily extended to other Greeks

as well.

1. Finite difference

This method involves generating, independently, two estimate option prices u(x) and

u(x+ε) from the initial price x and x+ε, respectively, for small ε, so that an estimate

3

∆ of ∆ can be computed as follows:

∆ =u(x+ ε)− u(x)

ε. (1.9)

Repeating this several times and averaging we obtain an estimator converging to

∆ =u(x+ ε)− u(x)

ε(1.10)

where u(x) and u(x+ε) are the option prices at x and x+ε respectively. Assuming that

we simulate u(x + ε) and u(x) using common random numbers and that for (almost)

all values of the random numbers the output u(·) is continuous in the input x, we have

Var(∆) =Var(u(x+ ε)) + Var(u(x))

ε2= O(ε−2). (1.11)

This is because the underlying assets X(ε) and X are generated independently of each

other. Therefore, the variance of ∆ becomes very large if ε is made small. To get an

estimator that converges to ∆ we must let ε decrease slowly as n increases resulting

in slow overall convergence. A detailed review of finite difference approach is found in

[41] on page 378. Glynn [42] has shown that the best possible convergence rate using

this approach is n−14 . Replacing the forward difference with the central difference,

∆ =u(x+ ε)− u(x− ε)

2ε,

improves the convergence rate to n−13 . However, using common random numbers, it has

been shown in [40] and [42] that one can achieve the convergence rate of n−12 , reported

to be the best that can be expected from Monte Carlo methods. The disadvantage

of the common random number finite difference method is that it may perform very

poorly when the payoff function is discontinuous, as in the case of a digital option and

a barrier option (see [53] page 140).

2. Pathwise method

This approach assumes that the payoff function Φ is a continuously differentiable with

bounded derivatives and the underlying variable Xti , i = 1, . . . , n, is differentiable

with respect to underlying parameters. This allows the interchanging of the derivative

operator and the expectation operator. Thus, we have

∂

∂xE[e−rTΦ(Xt1 , · · · , Xtn)] = E[e−rT

n∑i=1

∂

∂xiΦ(Xti)

∂Xti

∂x]. (1.12)

4

The pathwise method computes the derivative of the payoff function with respect to

the parameter of interest. This method only works for specific payoff functions, hence

we cannot generalize the implementation of this approach. However, the method gives

unbiased results when applicable (see [41] page 386). This approach cannot be applied

to non-differentiable payoff functions as in the case of barrier and digital options, for

example.

3. Likelihood ratio method

This method was first introduced by Broadie and Glasserman [20]. The method

assumes that the law of Xt1 , . . . , Xtn is explicitly known and is given by a density, say,

p(x1, . . . , xn) which is also a function of x. Thus, we can write equation (1.2) as

E[e−rTΦ(Xt1 , · · · , Xtn)] =

∫e−rTΦ(xt1 , · · · , xtn)p(x1, · · · , xn)dx1 · · · dxn. (1.13)

Computing the derivative of the option price with respect to the initial price x we

obtain

∂

∂xE[e−rTΦ(Xt1 , · · · , Xtn)] =

∫e−rTΦ(xt1 , · · · , xtn)

∂

∂xp(x1, · · · , xn)dx1 · · · dxn.

(1.14)

If this indeed holds then multiplying and dividing the integrand by p(x1, . . . , xn) yields

∂

∂xE[e−rTΦ(Xt1 , · · · , Xtn)]

=

∫e−rTΦ(xt1 , · · · , xtn)

∂∂xp(x1, · · · , xn)p(x1, · · · , xn)

p(x1, · · · , xn)dx1 · · · dxn

=

∫e−rTΦ(xt1 , · · · , xtn)

∂

∂x[ln p(x1, · · · , xn)]p(x1, · · · , xn)dx1 · · · dxn

:= E[e−rTΦ(Xt1 , · · · , Xtn) · π] (1.15)

where π is the weighting function defined by

π =∂

∂xln p(Xt1 , · · · , Xtn).

The importance of this approach is that it provides us with an efficient way of avoiding

the derivative of the payoff function Φ. We compute the derivative of the probability

density of the underlying variables rather than the derivative of the payoff function.

The likelihood ratio method gives a single weight function. It has been proven that

when applicable the likelihood ratio method gives a weight function with minimal

variance (see [7], [9] and [36]). The expectation in equation (1.15) can now be computed

5

using Monte Carlo methods. However, this analysis is theoretical since, in general, the

density function p(x1, . . . , xn) is not explicitly known as in the case of Asian options,

for example.

4. Malliavin calculus

To circumvent the drawbacks outlined in the other approaches that we have

mentioned above we use Malliavin calculus recently introduced in [35] and [36]. Use

of the Malliavin calculus avoids the need to differentiate payoff functions and does not

require explicit knowledge of the density of the underlying asset. The main tool we

use is the integration by parts formula. Using the Malliavin calculus we can show that

all Greeks mentioned above can be represented as the expected value of the payoff

function multiplied by a weight function

Greek := E[e−rTΦ(Xt1 , · · · , Xtn) · π] (1.16)

where π is a weight function to be determined (see [8], [9], [15], [35], [36], [43], [44],

[57], [67] and the references therein). An important advantage is that the weight

function π, which is usually a function of the underlying variable, is independent of

the payoff function Φ. In the case of discontinuous payoff functions, Malliavin calculus

improves the efficiency of Greek computation. We can, therefore, construct a Monte

Carlo algorithm for general options and not specifically for each option.

The Malliavin calculus approach gives several weight functions (see [8], [9], [57] and

[67]). If the density of the underlying variable is known, both the Malliavin calculus

and the likelihood method give the same weight function (see [67]). Thus, the Malliavin

calculus can be seen as an extension of the likelihood method. The real advantage of

using Malliavin calculus is that it is applicable to both complicated and discontinuous

payoff functions such as look-back options and digital options. It also works well when

dealing with underlying random variables whose density is not explicitly known, as

in the case of Asian options, for example. We mention that the Malliavin calculus

approach is not reported to have lower variance than the finite difference method for

smooth functions like the vanilla options (see [35]).

We will focus (in this thesis) on the Malliavin calculus approach. Our motivation to use

Malliavin calculus is: we want a method which is applicable to a wide class of option prices.

The Malliavin calculus approach enables us to obtain tractable formulae for Greeks which

can be simulated using Monte Carlo methods.

6

1.2 Background on the Malliavin calculus

The Malliavin calculus was introduced by Paul Malliavin in his celebrated paper “Stochastic

calculus of variation and hypoelliptic operators” that was published in 1976 (see [65]). It

is an infinite dimensional calculus defined on the Wiener space. The initial application of

Malliavin calculus was to prove the results about the smoothness of densities of solutions

of stochastic differential equations driven by Brownian motion (see [38], [50], [65], [70] and

[83]). This remained the only known application for Malliavin calculus for several years.

In 1984 Ocone [72] obtained an explicit interpretation of the Clark-Haussmann-Ocone

formula in terms of the Malliavin derivative. The formula is usually abbreviated as the CHO

formula. This result was later applied to finance by Ocone and Karatzas in 1991 (see [55]).

The authors prove that the Malliavin derivative can be used to obtain explicit formulae for

the replicating portfolios of contingent claims in markets driven by Brownian motion. This

led to a huge increase in the interest in the Malliavin calculus both among mathematicians

and finance researchers (see [17], [24], [27], [31], [60], [66], [74] and the references therein).

In 1999 Fournie et al. [35] use the Malliavin calculus to compute Greeks. Following their

work, much attention has been dedicated to finding more efficient and more general methods

of computing the Greeks using Monte Carlo methods (see [7], [8], [9], [10], [11], [15], [43],

[44], [57], [58], [67] and the references therein) and also to extending the computations to

a wider class of options and models. The above cited papers compute the Greeks, using

Malliavin calculus, with price dynamics driven by Brownian motion only.

Over the last decade, there has been an increasing interest in jump type diffusion models

for modelling in stochastic finance (see [23] and [81]). This is because there is a growing

evidence that models driven by jump processes may be more realistic than those that insist

on continuous sample paths. The continuous time models use the normal distribution to

fit the log returns of the underlying asset prices whereas the data suggest that log returns

of stocks/indices are not normally distributed. The log returns of most financial assets are

skewed and have kurtosis higher than that of normal distribution (see [81]). It is therefore

natural to want to explore possible extensions of the work of Fournie et al. [35] to cases

where the underlying asset prices are modelled by processes with jumps.

Although markets driven by processes with jumps are not in general complete, the Greeks

remain important for financial applications. For example, Greeks are useful in model

calibration which involves minimizing over model parameters to bring model prices as close

as possible to market prices (see [14]).

Malliavin calculus was first extended to jump diffusion models by Bichteler et al. [18]. The

7

main focus in this monograph is on the existence and smoothness of the density of the

solution of a stochastic differential equation with jumps, a study which is not relevant to the

study of the Greeks. Carlein and Pardoux [21] define an analogous calculus to the Malliavin

calculus on the Poisson space (see also [68]). Nualart and Schoutens [71] develop a theory

of chaos expansion for functionals of Levy processes when the Levy measure satisfies an

exponential moment condition. This has been used as a basis to define Malliavin calculus

similar to the Brownian motion case (see [27], [62], [70] and [84]).

The extension of the work of Fournie et al. [35] to models with jumps already exists in the

literature (see [6], [25], [26], [32], [34], [48], [66], [76], [78] and the references therein). In [25]

the authors use Malliavin calculus for simple Levy processes to calculate Greeks for a class of

“separable” jump diffusions. The general idea in [25] is to take the Malliavin derivative in the

direction of the Wiener process on the Wiener-Poisson space (to be defined later on). This

enables the authors to stay in the framework of a Malliavin calculus for the Brownian motion

case without major changes. El-Khatib and Privault [32] consider a market driven by jumps

alone. The authors use a Malliavin calculus defined on a Poisson space to compute Greeks

for Asian options but imposing a regularity condition on the payoff. Their approach cannot

be used to compute Greeks for European options. Forster et al. [34] use the hypoelliptic

condition and the standard Malliavin calculus to obtain Greeks with the weight function

given as a Skorohod integral. In this thesis we deal with discontinuous payoff functions and

also cases where the volatility is stochastic.

The extension of the work of Fournie et al. [35] to models with jumps still introduces many

challenges prompting further research, for example the Malliavin derivative of the pure jump

case is not a derivative; it is, instead, a difference operator.

1.3 Aims and structure of the thesis

The aim of this thesis is to provide a broad coverage of the many applications of the Malli-

avin calculus to the calculations of the Greeks, and to provide the reader with a summary

of the necessary background to be able to understand this.

The thesis is organized as follows. Chapter 2 is devoted to a review of some basic defini-

tions and results related to Malliavin calculus as well as to give some important remarks.

In Chapter 3 we demonstrate, using some results in Chapter 2, how the Malliavin calculus

techniques are applied to calculate Greeks for continuous processes. The chain rule and the

integration by parts formula are used extensively to compute ∆, Γ and V for different types

8

of payoff functions. In Chapter 4 we review white noise analysis for continuous processes

(see [1] and [27]). In particular, we consider the Wick product, the Hermite transform and

the Donsker delta function. Then we apply these concepts to calculate the Greeks for pure

diffusion processes. In Chapter 5 we define a first variation process of a pure jump Levy

stochastic differential equation and give a representation formula for the stochastic

derivative of a pure jump Levy stochastic differential equation. We also give the necessary

and sufficient conditions for a function to serve as a weighting function in the pure jump

case. This is an extension to the pure jump case of the work in [10] where the author gives

the necessary and sufficient conditions for a function to serve as a weighting function in the

pure diffusion case.

Chapter 6 deals with the jump diffusion case. We calculate Greeks for different models. Here

we are inspired by ideas in [25] where the authors calculate weight functions for processes

driven by a Brownian motion and a Poisson process with deterministic jump sizes. We

extend the work to more general Levy processes. We also approximate a Levy process by

Brownian motion and compute its corresponding Greeks. Greeks for original Levy processes

are then obtained by a limiting argument. In Chapter 7 we review white noise analysis for

Levy processes. As in Chapter 4 we consider the Wick product, the Hermite transform and

the Donsker delta function for Levy processes (see [27]). Using the Donsker delta

representation formula, we are able to express the option price in terms of the Donsker delta

function. Then we calculate ∆ for pure jump Levy processes in the same way as in Chapter 4.

In addition, we will present some new results as indicated below. The new results are

contained in Chapter 3 Section 3.5, Chapter 4 Section 4.8, Chapter 5 Sections 5.5 and 5.6,

Chapter 6 Sections 6.5 and 6.6 and in Chapter 7 Section 7.8. We also give interesting

examples and proofs that are missing in the literature. We mention that the Greeks for the

Heston model are known in the literature. The full derivation of the Greeks was not available.

We provide this in Section 3.5. The use of white noise is common but its extension to the

computation of Greeks via Malliavin calculus was not available. This is done in Section 4.8.

We derive a new representation for the Malliavin derivative of a pure jump Levy stochastic

differential equation (see Section 5.5) in terms of its first variation process and then give

the necessary and sufficient conditions for a function to serve as a weighting function (see

Section 5.6). In Section 6.5 we compute the Greeks for the Heston model with jumps. This

is an extension of results in Section 3.5. The extension to the additional jump case is new at

the best of my knowledge. The idea of approximating the Levy process appears in several

9

papers in the literature (see [4]). In Section 6.6, we compute Greeks of an approximation

of a Levy process and then pass through limit arguments to obtain Greeks for the original

Levy process. This approach can be applied to a wide class of Levy processes. In Section

7.8 we only compute ∆ for a pure jump Levy process using the Donsker delta function

representation defined in the white noise setting. This is a new result. Other Greeks are left

for future work.

10

Chapter 2

Basic Properties of the Malliavin

calculus

This chapter is an introduction and a survey of the Malliavin calculus machinery (see [70]).

The original construction of Malliavin calculus was given on the Wiener space Ω = C0([0, T ])

(to be defined later on). We review this construction of the Malliavin derivative in the first

half of this chapter. We also review the concept of a chain rule as well as an integration by

parts formula that will be used in the following chapters to compute Greeks. In the second

half of this chapter we review the construction of Malliavin calculus based on the chaos ex-

pansion. The definition of Malliavin calculus in this way allows for a useful combination with

Hida’s white noise calculus. We conclude the chapter by considering the Clark-Haussmann-

Ocone formula. The formula is useful in mathematical finance but is not used in this thesis.

The material discussed here was taken from [5], [38], [54], [60], [66], [69], [70] and [74].

2.1 Wiener Space

We work on a filtered probability space (Ω,F ,F = (Ft)0≤t≤T, P ) on which a Brownian

motion W (t, ω) is defined (t ∈ [0, T ], ω ∈ Ω). F = (Ft)0≤t≤T is assumed to be the filtration

generated by W (t, ω). Note that W (t, ω) is a function of two variables. If we fix ω then

this becomes a continuous function of one variable t, that is, W (t, ω) ∈ C([0, T ]). So at

each state ω the Brownian motion W (t, ω) associates a continuous function. We write this

function as ω(t), that is,

ω(t) = W (t, ω). (2.1)

11

The Ito construction of the stochastic integral yields∫ T

0

h(t)dW (t, ω), h ∈ L2([0, T ]) (2.2)

(or more generally, h ∈ L2([0, T ]× Ω)). This integral can also be written as∫ T

0

h(t)dω(t)

where we have ω(t) ∈ C([0, T ]) and the integral is with respect to Wiener measure on

C([0, T ]) (see [54] page 125). We note that whereas there is no a priori additive structure

on Ω, the identification

Ω 3 ω ↔ ω(t) ∈ C([0, T ])

enables us to use the Brownian motion to convert Ω naturally into a vector space. It thus

becomes possible to define derivatives with respect to ω ∈ Ω in a similar way to the usual

vector space definitions as we will see.

Let H = L2([0, T ]) and let

W (H) = ∫ T

0

hdW | h ∈ H.

Let Sn be the space of all polynomials of degree n in elements of W (H) and let S be the

space of all such polynomials. If F ∈ S then there exists n, a polynomial f of degree n and

h1, . . . , hn ∈ H such that

F (ω) = f(W (h1), · · · ,W (hn)) (2.3)

where W (h) :=∫ T

0h(t)dWt. We can orthonormalize the hi, i = 1, 2, . . . , n by using the

Gram-Schmidt procedure, that is, letting

ξ1 =h1

‖ h1 ‖, ξ2 =

(h2 − (ξ1, h2)ξ1)

‖ h2 − (ξ1, h2)ξ1 ‖, ...

and then write the hi, i = 1, 2, . . . , n in terms of the ξi, i = 1, 2, . . . , n and multiply out. This

gives

F (ω) = f(W (ξ1), · · · ,W (ξn))

where f is another polynomial of degree n and ξ1 = h1

‖h1‖ . Hence, we can assume where

appropriate that the hi, i = 1, 2, . . . , n are orthonormal in H.

Define

γn :=1

(2π)n2

∫ ∞

−∞e−

|x|22 dx.

12

Remark

We define

E[f(W (ξ1), · · · ,W (ξn))] :=

∫Rn

f(x1, · · · , xn)dγn(x)

where dγn(x) is given by

dγn(x) =1

(2π)n2

e−|x|2

2 dx. (2.4)

This remark is very useful in the setup of the theory.

We note that if ξ ∈ L2([0, T ]) is normal then

1 =‖ ξ ‖2=

∫ T

0

| ξ(t) |2 dt = var(W (ξ))

so

W (ξ) ∼ N(0, 1)

and

E[f(W (ξ))] =

∫Rf(x)dγ1(x).

2.2 Malliavin derivative

Since our state space is a vector space we can define the derivative of a random variable

F (ω) = W (h) =

∫ T

0

h(t)dW (t, ω) (2.5)

in the direction γ with

γ(t) =

∫ t

0

g(s)ds, g ∈ H

as follows.

Definition 2.2.1 Let F : Ω → R be a random variable of the form (2.5) and let γ ∈ Ω be

of the form

γ(t) =

∫ t

0

g(s)ds, g ∈ H. (2.6)

Then the directional derivative DγF of a random variable F at the point ω ∈ Ω in the

direction γ ∈ Ω is given by

DγF (ω) :=d

dε[F (ω + εγ)]ε=0 = lim

ε→0

F (ω + εγ)− F (ω)

ε(2.7)

if the limit exists.

13

The set of γ ∈ Ω which can be written in the form (2.6) for some g ∈ H is called the

Cameron-Martin space and we denote it by H. Note that it is not possible to proceed in the

manner we are about to with functions that are not Cameron-Martin functions. However,

there are ways of getting around this.

Let

F (ω) =

∫ T

0

h(t)dω(t).

Then

1

ε(F (ω + εγ)− F (ω)) =

1

ε

(∫ T

0

h(t)d(ω + εγ)(t)−∫ T

0

h(t)dω(t)

)=

1

ε

(ε

∫ T

0

h(t)dγ(t)

).

Since dγ(t) = g(t)dt we have

1

ε(F (ω + εγ)− F (ω)) =

∫ T

0

h(t)g(t)dt = 〈h, g〉H .

Here 〈·, ·〉H is the inner product on H. We will write

D

(∫ T

0

h(t)dW (t, ω)

)= h.

We note that this is only valid for a deterministic integrand h. For a general case we refer

to Proposition 2.4.2 (to be given later on). From these considerations we give the following

definition.

Definition 2.2.2 The derivative D : S → L2([0, T ]× Ω) of a random variable

F (ω) = f(W (h1), . . . ,W (hn)) is defined by

DF =n∑i=1

∂f

∂xi(W (h1), · · · ,W (hn))hi(t). (2.8)

D is called the Malliavin derivative on S.

Since D(∫ T

0h(t)dWt

)= D(W (h)) = h,

DF =n∑i=1

∂f

∂xi(W (h1), · · · ,W (hn))D(W (hi)). (2.9)

14

As f has only polynomial growth we have DF ∈ L2(Ω× [0, T ]). Sometimes it is convenient

to write

DtF =n∑i=1

∂f

∂xi(W (h1), · · · ,W (hn))hi(t). (2.10)

However, because hi ∈ H, Dt is not really well-defined, since hi is only defined up to a set

of Lebesgue measure 0. The notation is nevertheless suggestive. We will use both Dt and D

(which are operators with different domain and range) where there can be no confusion.

We have the following product rule.

Lemma 2.2.3 Suppose that F,G ∈ S. Then

D(FG) = (DF )G+ F (DG). (2.11)

We have the following integration by parts formula which is found in [70] page 25.

Lemma 2.2.4 Suppose that F ∈ S and h ∈ H. Then

E [〈DF, h〉H ] = E [FW (h)] . (2.12)

Proof

By first dividing by ‖ h ‖ if necessary, we may assume that the norm of h is one. There exist

orthonormal elements of H, ξ1, . . . , ξn, such that ξ1 = h and F is a smooth random variable

of the form

F = f(W (ξ1), . . . ,W (ξn)). (2.13)

Let γn(x) denote the density of the standard normal distribution on Rn. Then, by integration

by parts, we have

E [〈DF, h〉H ] = E [〈DF, ξ1〉H ] = E

[n∑i=1

∂

∂xif〈ξi, ξ1〉H

]= E

[∂

∂x1

f(W (ξ1), · · · ,W (ξn))

]=

∫Rn

∂

∂x1

f(x)dγn(x) = − 1

(2π)n2

∫Rn

f(x)∂

∂x1

(e−

|x|22

)dx

=1

(2π)n2

∫Rn

f(x)x1e− |x|2

2 dx = E [FW (ξ1)] = E [FW (h)] . 2

Applying Lemma 2.2.4 to the product FG where F and G are random variables and using

Lemma 2.2.3, we get following result.

15

Corollary 2.2.5 Suppose that F,G ∈ S and h ∈ H. Then we have

E [G〈DF, h〉H ] = E [−F 〈DG, h〉H + FGW (h)] . (2.14)

Define

‖ F ‖1,2:=‖ F ‖L2(Ω) + ‖ DtF ‖L2([0,T ]×Ω) . (2.15)

We recall that random variables are elements of L2 and hence only defined almost surely

with respect to the Wiener measure. Given a random variable F , define

G(ω) = F (ω + k)

where k is a fixed continuous function. In which direction can we shift the argument ω of a

functional while keeping it well-defined? If k lies in the Cameron-Martin space H then G(ω)

is well-defined. However, if k lies in the complement, it can be shown (see [38]) that G is

not a well-defined function in L2 and cannot be used to define a directional derivative in the

direction k.

Let D1,2 be the closure of S in the norm ‖ · ‖1,2. We will show that the derivative on S can

be extended to a closed operator on D1,2.

Definition 2.2.6 A : H → K is a closable operator on a normed complete linear space if

Fn → F, AFn → G1, Gn → F, AGn → G2 implies G1 = G2.

If A is closable and we know F on a subset S of the space then we can extend A to an

operator A defined on the closure of S called the closure of A by defining AF = G whenever

there exists a sequence Fn → F such that AFn → G. The closability implies that G is

uniquely defined by the later two conditions. To prove closability, we need only to show that

Fn → 0 and AFn converges implies AFn → 0. The following lemma was taken from [70]

page 26.

Lemma 2.2.7 The Malliavin derivative D : S → L2(Ω× [0, T ]) is closable. It has a closed

extension to D1,2 (which is also denoted by D).

Proof

Let Fn, n ≥ 1 be a sequence of random variables such that Fn converges to 0 in L2(Ω) and

16

the sequence of derivatives DFn converge to η in L2(Ω × [0, T ]). Our aim is to show that

η = 0. For any h ∈ H and for any bounded random variable F ∈ S such that FW (h) is

bounded in L2 we have

E [〈η, h〉HF ] = limn→∞

E [〈DFn, h〉HF ]

= limn→∞

E [−Fn〈DF, h〉H + FnFW (h)] by (2.14)

= 0.

This is because Fn converges to 0 in L2(Ω) as n tends to infinity and the random vari-

ables 〈DF, h〉H and FW (h) are bounded. This implies that η = 0 and the proof is com-

plete. 2

Define

D : D1,2 → L2([0, T ]× Ω)

as the closure of our previously defined operator. In general it will not be defined on the

whole of L2(Ω) and will not be continuous. However, Fn converges in D1,2, if and only if

both Fn and DFn converge and if Fn → F , DFn → G then G = DF (see [70] page 26).

Example

DeW (h) = eW (h)h, h ∈ H.

Let

Fn =n∑i=0

1

i!(W (h))i.

Then

DFn =n∑i=1

i

i!(W (h))i−1D(W (h)) =

n∑i=1

1

(i− 1)!(W (h))i−1D(W (h)) = Fn−1D(W (h)) = Fn−1h.

Note that Fn → eW (h) in L2(Ω). Since D is a closed operator it follows that

DFn → eW (h)h in L2([0, T ]× Ω).

Since both Fn and DFn converge it follows from the fact that D is closed that

limn→∞ Fn = eW (h) is in the domain of D and

DeW (h) = limn→∞

DFn = eW (h)h.

The chain rule holds for the Malliavin derivative in the following form.

17

Proposition 2.2.8 Let F = (F1, . . . , Fn) ∈ D1,2 and let ϕ : Rn → R be a continuously

differentiable function with bounded partial derivatives. Then ϕ(F ) ∈ D1,2 and

Dtϕ(F ) =n∑i=1

∂ϕ

∂xi(F )DtFi, t ≥ 0 a.s. (2.16)

Proof

The proof follows easily by approximating the random variable F by a sequence of smooth

random variables and ρ by ρ ·γε where γε is an approximation of the density (see [70] page

29). We omit the details. 2

Example

For some fixed s ∈ (0, T ] put

ϕ(W ) = W 2s =

(∫ T

0

1[0,s](t)dWt

)2

. (2.17)

Then

Dtϕ(W ) = 2Wt1[0,s](t) =

2Wt if t ≤ s

0 otherwise.

Corollary 2.2.9 Let u(s, ω) be Fs-adapted and assume that u(s, ·) ∈ D1,2. Then Dtu(s, ω) =

0 for t > s.

Proof

We will show later on (see Lemma 2.7.3) that Dt(E[u(s, ω) | Fs]) = E[Dtu(s, ω) | Fs]1[0,s](t).

Therefore we have

Dtu(s, ω) = DtE[u(s, ω) | Fs] = E[Dtu(s, ω) | Fs]1[0,s](t).

Since s < t the result follows. 2

The intuition of the corollary is that if u(s, ω) only depends on the early parts of the paths

up to time s then perturbing the paths later on, that is, on the region t > s makes no

difference to u(s, ω).

The following proposition is useful.

Proposition 2.2.10 Let Fn, n ≥ 1 be a sequence of random variables in D1,2. Assume

that Fn converges to F in L2(Ω) and

supn

E(‖ DFn ‖2

L2(Ω)

)<∞.

18

Then F belongs to D1,2 and the sequence of derivatives DFn, n ≥ 1 converges to DF in

L2(Ω× [0, T ], L2(Ω)).

Proof

The proof can be found in [70] page 29. We omit the details. 2

2.3 Skorohod integral

The Malliavin derivative D is a closed linear operator defined on D1,2 and

D : D1,2 → L2(Ω× [0, T ]). (2.18)

D is an unbounded operator with domain dense in L2(Ω) whose domain and range belong to

different Hilbert spaces. In standard theory, adjoints of unbounded operators have the same

domain and range. Here we show one way of defining an adjoint operator δ of D such that

δ : L2(Ω× [0, T ], L2(Ω)) → L2(Ω× [0, T ]). (2.19)

We shall denote the domain of the adjoint operator δ by Dom(δ).

Definition 2.3.1 Let u ∈ L2(Ω× [0, T ]). Then u belongs to the domain Dom(δ) of δ if for

all F ∈ D1,2 we have

| E[〈DF, u〉L2(Ω)

]|=| E

[∫ T

0

DtFu(t)dt

]|≤ c ‖ F ‖L2(Ω) (2.20)

where c is some constant depending on u. If u belongs to Dom(δ), then

δ(u) =

∫ T

0

utδWt (2.21)

is the element of L2(Ω) such that the integration by parts formula holds:

E[

(∫ T

0

DtFutdt

)] = E[Fδ(u)] for all F ∈ D1,2. (2.22)

δ : L2(Ω × [0, T ], L2(Ω)) → L2(Ω × [0, T ]). is linear and densely defined. We call δ the

Skorohod integral.

Remark

Definition 2.3.1 gives the relationship between the derivative operator and the Skorohod

integral.

19

Lemma 2.3.2 Suppose F ∈ S and h ∈ H. Then

δ(Fh) = FW (h)− 〈h,DF 〉H . (2.23)

In addition, setting F = 1 one obtains

δ(h) = W (h), h ∈ H.

Proof

The proof can be found in [38] page 11. We recall the integration by parts formula

E[G〈DF, h〉H)

]= E [〈−FDG, h〉H + FGW (h)] .

This can be extended to all F,G ∈ D1,2, h ∈ H. Let F ∈ S, G ∈ S0 and h ∈ H. Then

E [δ(Fh)G] = E [〈Fh,DG〉H ] = E [F 〈h,DG〉H ] = E [−G〈h,DF 〉H + FGW (h)] .

Hence, since S is dense in L2(Ω), we have

δ(Fh) = FW (h)− 〈h,DF 〉L2(Ω). 2

The lemma can be extended to all F ∈ D1,2. An important property of the Skorohod

integral δ is that its domain Dom(δ) contains all adapted stochastic processes which belong

to L2(Ω× [0, T ]). For such processes the Skorohod integral δ coincides with the Ito stochastic

integral (see [47] page 48). This is given in the following proposition.

Proposition 2.3.3 If u is an adapted process belonging to L2(Ω× [0, T ]), then

δ(u) =

∫ T

0

u(t)dWt. (2.24)

Proof

The proof follows from the following proposition (Proposition 2.3.4) by setting F = 1. 2

Further, if the random variable F is FT -adapted and belongs to D1,2 then, for any u in

Dom(δ), the random variable Fu will be Skorohod integrable. This yields the following

proposition (see [70] page 39).

Proposition 2.3.4 Let F belongs to D1,2 and u ∈ Dom(δ) such that E[∫ T

0F 2u2

tdt] < ∞.

Then Fu ∈ Dom(δ) and

δ(Fu) = Fδ(u)−∫ T

0

DtFutdt, (2.25)

20

whenever the right hand side belongs to L2(Ω). In particular, if u is moreover adapted, we

have

δ(Fu) = F

∫ T

0

utdWt −∫ T

0

DtFutdt. (2.26)

Proof

For any smooth random variable G ∈ S0, using Equation (2.22), it holds that

E [〈Fu,DG〉H ] = E [〈u, FDG〉H ]

= E [〈u,D(FG)−GDF 〉H ]

= E [〈u,D(FG)〉H − 〈u,GDF 〉H ]

= E [(Fδ(u)− 〈u,DF 〉H)G] .

This implies the desired result. 2

Proposition 2.3.4 is a product rule for Skorohod integrals. The rule, however, differs from

the corresponding rule in ordinary calculus by the minus sign. We will apply this proposition

several times when calculating Greeks.

2.4 SDEs and Malliavin calculus

We consider a 1-dimensional stochastic differential equation

dXt = b(Xt)dt+ σ(Xt)dWt, X0 = x. (2.27)

where b : R → R and σ : R → R are bounded with bounded partial derivatives and σ(x) 6= 0

for all x ∈ R. Its integral form is given by

Xt = x+

∫ t

0

b(Xs)ds+

∫ t

0

σ(Xs)dWs. (2.28)

We want to bring Dt under the integral signs and give a representation for DrXt. We do

this in the following.

Proposition 2.4.1 Let u(t, ω) be an Ft-adapted process and let t < T . Then

Dt

(∫ T

0

u(s, ω)ds

)=

∫ T

t

Dtu(s, ω)ds. (2.29)

21

Proof

The proposition is an immediate consequence of Corollary 2.2.9. 2

For the stochastic integral, we have the following proposition.

Proposition 2.4.2 Let u(s, ω) be a stochastic process such that

E[

∫ T

0

u2(s, ω)ds] <∞

and assume that u(s, ·) ∈ D1,2 for all s ∈ [0, T ], that Dtu ∈ Dom(δ) for all t ∈ [0, T ] and that

E[

∫ T

0

(δ(Dtu))2dt] <∞.

Then δ(u) ∈ D1,2 and

Dt(δ(u)) = u(t, ω) +

∫ T

0

Dtu(s, ω)dWs. (2.30)

If in addition u(t, ω) is adapted we can write Equation (2.30) as

Dt

(∫ T

0

u(s, ω)dWs

)= u(t, ω) +

∫ T

t

Dtu(s)dWs. (2.31)

Proof

The proof is given in [74] page 5.6. Here consider a special case where we restrict ourselves

to an adapted process of the form

u(t, ω) = F (ω)h(t)

with h(t) = h1[t,T ] and Ft-measurable F . Let t ≤ T . Then

Dt

(∫ T

0

Fh(s)dWs

)= Dt

(∫ t

0

Fh(s)dWs +

∫ T

t

Fh(s)dWs

)= 0 +Dt

∫ T

t

Fh(s)dWs by Corollary 2.2.9

= Dt(FW (h1[t,T ]))

= (DtF )W (h1[t,T ]) + FDt(W (h1[t,T ]) by (2.11)

=

∫ T

t

DtFh(s)dWs + Fh(t)

= u(t) +

∫ T

t

Dtu(s)dWs. 2

Remark

The limits of the integral on the left hand side reduce from∫ T

0to∫ Tt

at the end because the

22

integrand is adapted. This follows by Corollary 2.2.9.

We now assume that the solution Xt belongs to D1,2. If we take the Malliavin derivative on

both sides of Equation (2.28) we obtain (using Propositions 2.4.1 and 2.4.2), for r < t,

DrXt = Dr

(∫ t

0

b(Xs)ds

)+Dr

(∫ t

0

σ(Xs)dWs

)=

∫ t

r

Drb(Xs)ds+ σ(Xr) +

∫ t

r

Drσ(Xs)dWs.

Then applying Proposition 2.2.8 we obtain

DrXt =

∫ t

r

b′(Xs)DrXsds+ σ(Xr) +

∫ t

r

σ′(Xs)DrXsdWs.

The primes denote the derivative with respect to the space variable. Fix r and set Zt := DrXt

for r < t. We obtain, using the Ito formula, the following stochastic differential equation

dZt = b′(Xt)Ztdt+ σ′(Xt)ZtdWt, t > r (2.32)

with initial condition Zr = σ(Xr). The solution of Equation (2.32) is given by

Zt = σ(Xs) exp∫ t

s

[b′(Xu)−1

2(σ′(Xu))

2]du+

∫ t

s

σ′(Xu)dWu (2.33)

A fundamental matrix of a linear system of equations is an n × n matrix whose columns

form a linearly independent set of solutions, and so span the solution space. A fundamental

matrix is nonsingular for all t. Now let Yt be a fundamental matrix of the equation called

the Variational Equation

dYt = b′(Xt)Ytdt+ σ′(Xt)YtdWt, Y (0) = I. (2.34)

Using the Ito formula the exact solution is given by

Yt = exp∫ t

0

[b′(Xs)−1

2(σ′(Xs))

2]ds+

∫ t

0

σ′(Xs)dWs. (2.35)

Given any fundamental matrix Yt, the solution of the equation with initial constant c at

t = r is

YtY−1r c. (2.36)

But Zt = DrXt is such a solution and so

DrXt = YtY−1r σ(Xr)1r≤t (2.37)

23

which is equivalent to

Yt = DrXtσ−1(Xr)Yr1r≤t (2.38)

We remark that in one dimension Ut = ∂∂xXt is a solution of the Variational Equation

satisfying U0 = 1. Hence in that case we can take

Yt :=∂

∂xXt. (2.39)

Yt, 0 ≤ t ≤ T is commonly known as the first variation process of Xt, 0 ≤ t ≤ T.

2.5 The integration by parts formula

We use the Malliavin derivative and the relation between the Malliavin derivative and the

Skorohod integral to obtain an integration by parts formula which plays an important role

in the calculation of Greeks. We give this in the following proposition (see [70] page 330).

Proposition 2.5.1 Let F, G be two random variables such that F,G ∈ D1,2. Consider a

random variable u(t, ω) for fixed ω, u(t, ·) ∈ H such that 〈DF, u〉H 6= 0 a.s. and

Gu(〈DF, u〉H) ∈ Dom(δ). Then for any continuously differentiable f with polynomial growth

we have

E [f ′(F )G] = E [f(F )H(F,G)] . (2.40)

where H(F,G) = δ(Gu(〈DF, u〉H)−1).

Proof

An application of the chain rule (see Proposition 2.2.8) gives

〈Df(F ), u〉H = 〈f ′(F )DF, u〉H = f ′(F )〈DF, u〉H .

Since 〈DF, u〉H 6= 0 we have

f ′(F ) = 〈Df(F ), u〉H(〈DF, u〉H)−1.

Consequently, we have

E [f ′(F )G] = E[〈Df(F ), u〉HG(〈DF, u〉H)−1

]= E

[〈Df(F ), Gu(〈DF, u〉H)−1〉H

].

Since Gu(〈DF, u〉H) ∈ Dom(δ), an application of Equation (2.22) gives

E [f ′(F )G] = E[f(F )δ(Gu(〈DF, u〉H)−1)

]24

which proves Equation (2.40). 2

Remark

1. If u = DF , then Equation (2.40) becomes

E [f ′(F )G] = E[f(F )δ

(GDF

‖ DF ‖2H

)].

2. If u is a deterministic process then it suffices to assume that G(〈DF, u〉H)−1 ∈ D1,2 as

this implies that Gu(〈DF, u〉H)−1 ∈ Dom(δ).

2.6 Iterated Wiener-Ito integrals

In this section we define the Malliavin derivative via the Wiener-Ito chaos decomposition.

See [27], [60] and [74] for more details about the Wiener-Ito chaos decomposition. We

mention that one can construct the Malliavin calculus as in the previous sections and use

it in concrete applications without mentioning the chaos decomposition. Thus, we restrict

ourselves to a short discussion of this topic. We first briefly review the construction of the

multiple Wiener-Ito integral with respect to the Brownian motion. Our main reference is

[60].

We use T for [0, T ] and H = L2(T,B, µ) where µ is a σ-finite measure without atoms and Bis a Borel σ field. For any set A ∈ B with µ(A) <∞, we define

W (A) = W (1A).

Then A→ W (A) is a Gaussian measure with independent increments, that is, if A1, . . . , An

are disjoint sets with finite measure, the random variables W (A1), . . . ,W (An) are indepen-

dent and for any A ∈ B with measure µ(A) < ∞, W (A) has the distribution N(0, µ(A)).

We will say that W is an L2(Ω)-valued Gaussian measure on (T,B). Thus, W (h) can be

regarded as the stochastic integral of the function h ∈ L2(T ) with respect to W . We will

write

W (h) =

∫T

hdW, h ∈ L2(T ).

Let

Sn = (t1, t2, . . . , tn) ∈ T n : ∃i 6= j such that ti = tj (2.41)

25

be the diagonal set of T n. We consider the set En of elementary functions of the form

f(t1, . . . , tn) =k∑

i1,i2,...,in=1

ai1,i2,...,in1Ai1×···×Ain

(t1, . . . , tn) (2.42)

where k ≥ 1 is finite, ai1,i2,...,in ∈ R and A1, A2, . . . , Ak are pairwise disjoint sets of finite

measure. The coefficients ai1,i2,...,in satisfy the condition

ai1,i2,...,in = 0 if ip = iq for some p 6= q. (2.43)

The condition (2.43) implies that the function f vanishes on the set Sn. The collection of

elementary functions of the form (2.42)-(2.43) is a vector space. For any elementary function

f of the form (2.42)-(2.43), we define

In(f) =k∑

i1,i2,...,in=1

ai1,i2,...,inW (Ai1) · · ·W (Ain). (2.44)

The value of In(f) is well-defined, that is, its definition does not depend on how f is repre-

sented by Equations (2.42)-(2.43). In addition, the mapping In is linear on the vector space

of elementary functions of the form (2.42)-(2.43).

The connection between Wiener-Ito chaos expansion and Malliavin calculus is best explained

through the symmetric expansion.

Definition 2.6.1 The symmetrization f(t1, t2, . . . , tn) of a function f(t1, t2, . . . , tn) is de-

fined by

f(t1, t2, . . . , tn) =1

n!

∑σ

f(tσ(1), tσ(2), . . . , tσ(n)) (2.45)

where the summation is taken over all the permutations σ of the set 1, 2, . . . , n.

We say that f is symmetric if f = f . Since Lebesgue measure is symmetric we have∫Tn

| f(tσ(1), tσ(2), . . . , tσ(n)) |2 dt1dt2 · · · dtn =

∫Tn

| f(t1, t2, . . . , tn) |2 dt1dt2 · · · dtn (2.46)

for any permutation σ. Therefore, by the triangle inequality, we have

‖ f ‖L2(Tn)≤1

n!

∑σ

‖ f ‖L2(Tn)=1

n!n! ‖ f ‖L2(Tn)=‖ f ‖L2(Tn) . (2.47)

Thus , we have

‖ f ‖L2(Tn)≤‖ f ‖L2(Tn) . (2.48)

This leads to the following lemma.

26

Lemma 2.6.2 If f is an elementary function of the form (2.42)-(2.43) then In(f) = In(f).

Proof

The proof can be found in [60] on page 170. We omit the details. 2

Lemma 2.6.3 If f is an elementary function of the form (2.42)-(2.43) then E[In(f)] = 0

and

E[In(f)2] = n!

∫Tn

| f(t1, t2, . . . , tn) |2 dt1dt2 · · · dtn. (2.49)

Proof

Let f be defined by Equations (2.42)-(2.43). Then In(f) is given in Equation (2.44). Since

the sets A1, A2, . . . , An are pairwise disjoint the corresponding product has expectation 0.

Hence, we have E[In(f)] = 0.

By Lemma 2.6.2 we have In(f) = In(f). Hence, we may assume that f is symmetric, that

is,

aiσ(1),iσ(2),...,iσ(n)= ai1,i2,...,in

for any permutation σ. Thus, we can write In(f) as

In(f) = n!k∑

i1,...,in=1

ai1,i2,...,inW (Ai1) · · ·W (Ain)

and

E[In(f)2] = (n!)2

k∑i1,...,in

k∑j1,...,jn

ai1,...,inaj1,...,jnE[W (Ai1) · · ·W (Ain)W (Aj1) · · ·W (Ajn)].

We note that, for fixed set of indices i1 < . . . < in, we have, by Ito isometry,

E[W (Ai1) · · ·W (Ain)W (Aj1) · · ·W (Ajn)] =

∏np=1 µ(Aip), if j1 = i1, . . . , jn = in

0 otherwise.

It follows that

E[In(f)2] = (n!)2∑

i1<...<in

a2i1,...,in

n∏p=1

µ(Aip) = n!

∫Tn

f(t1, . . . , tn)2dt1 · · · dtn, (2.50)

which proves Equation (2.49) since f is assumed to be symmetric. 2

Lemma 2.6.4 Let f be a function in L2(T n). Then there exists a sequence fk∞k=1 of

elementary functions satisfying (2.42)-(2.43) such that

limk→∞

∫Tn

| f(t1, t2, . . . , tn)− fk(t1, t2, . . . , tn) |2 dt1dt2 · · · dtn = 0. (2.51)

27

Proof

The proof is immediate because the usual set of elementary functions is dense in L2(T n) and

the set Sn has Lebesgue measure zero. 2

Now suppose f ∈ L2(T n). Choose a sequence fk∞k=1 of elementary functions converging to

f in L2(T n) which exists by Lemma 2.6.4. Then, by the linearity of In and Lemma 2.6.3 we

have

E[(In(fk)− In(fl))2] = n! ‖ fk − fl ‖2

L2(Tn)≤ n! ‖ fk − fl ‖2L2(Tn)→ 0, as k, l→∞. (2.52)

The inequality in (2.52) is justified by (2.48). Hence, the sequence In(fk)∞k=1 is Cauchy in

L2(Ω). Define

In(f) := limk→∞

In(fk) in L2(Ω). (2.53)

The value of In(f) is well-defined, namely, it does not depend on the choice of the sequence

fk∞k=1 used in Equation (2.53).

Definition 2.6.5 Let f ∈ L2(T n). The limit In(f) in Equation (2.53) is called the multiple

Wiener-Ito integral of f and is denoted by∫Tn

f(t1, t2, . . . , tn)dW (t1)dW (t2) · · · dW (tn).

The Lemmas 2.6.2 and 2.6.3 can be extended to functions in L2(T n) using symmetry, the

approximations in Lemma 2.6.4 and the definition of the multiple Wiener-Ito integral. We

state this in the following theorem (see [60] page 172).

Theorem 2.6.6 Let f ∈ L2(T n), n ≥ 1. Then we have

1. In(f) = In(f) where f is the symmetrization of f .

2. E[In(f)] = 0.

3. E[In(f)2] = n! ‖ f ‖2 where ‖ · ‖ is the norm on L2(T n).

Next we give the relationship between a multiple Wiener-Ito integral and an iterated Ito

integral.

28

Theorem 2.6.7 Let f ∈ L2(T n), n ≥ 2. Then

In(fn) :=

∫Tn

f(t1, t2, . . . , tn)dW (t1)dW (t2) · · · dW (tn)

= n!

∫ T

0

· · ·∫ tn−2

0

(∫ tn−1

0

f(t1, t2, . . . , tn)dW (tn)

)dW (tn−1) · · · dW (t1) := n!Jn(fn)

where f is the symmetrization of f and

Jn(fn) =∫ T

0· · ·∫ tn−2

0

(∫ tn−1

0f(t1, t2, . . . , tn)dW (tn)

)dW (tn−1) · · · dW (t1).

Note that in each step the corresponding integrand is adapted because of the limits of the

preceding integrals.

Proof

The equality is clear if f is any elementary function of the form (2.42)-(2.43). In the general

case, the equality follows by a density argument taking into account that the iterated stochas-

tic Ito integral satisfies the same Ito isometry property as the multiple stochastic integral.

Details can be found in [60] on page 173. 2

Finally, we give a result on the orthogonality of In(f) and Im(g) in the Hilbert space L2(Ω)

when n 6= m (see [60] page 175).

Theorem 2.6.8 Let n 6= m. Then

E[In(f)Im(g)] = 0 (2.54)

for any f ∈ L2(T n) and g ∈ L2(Tm).

We now state the following theorem without proof. A detailed proof can be found in [70]

page 6.

Theorem 2.6.9 The space L2(Ω) can be decomposed into the orthogonal direct sum

L2(Ω) = K0 ⊕K1 ⊕K2 ⊕ · · · ⊕Kn ⊕ · · · ,

where ⊕ denotes the direct sum, Kn consists of linear combinations of multiple Wiener-Ito

integrals of order n. Each function F in L2(Ω) can be uniquely represented by

F =∞∑n=0

In(fn), fn ∈ L2(T n) (2.55)

29

where L2(T n) denotes the symmetrization of the space L2(T n) and the following equality

holds:

‖ F ‖2L2(Ω)=

∞∑n=0

n! ‖ fn ‖2L2([0,T ])n . (2.56)

For n = 0 we adopt the convention that I0(f0) = f0 when f0 is constant.

Corollary 2.6.10 For each F ∈ L2(Ω) there exist fn ∈ L2(T n) such that

F =∞∑n=0

In(fn).

Moreover, the functions fn are uniquely defined on L2(T n).

2.7 Malliavin derivative via chaos expansion

The operators Dt and δ can be represented in terms of the Wiener-Ito chaos expansion. The

following theorem is found in [70] page 33.

Theorem 2.7.1 Let F ∈ D1,2 and let F =∑∞

n=0 In(fn), fn ∈ L2(T n). Write

fn(·, t) = fn(x1, . . . , xn−1, t) so that fn is a symmetric function of the (x1, x2, . . . , xn). Then

DtF =∞∑n=1

nIn−1(fn(·, t)). (2.57)

Also

DtIn(fn) = nIn−1(fn(·, t)). (2.58)

In addition, F ∈ D1,2 if and only if

∞∑n=1

nn! ‖ fn ‖2L2(Tn)<∞. (2.59)

Remark

DtF is obtained simply by removing one of the stochastic integrals, letting the variable t to

be free and multiplying by the factor n.

Proof

Suppose first that F = In(fn) where fn is a symmetric and elementary function of the form

(2.42)-(2.43). Then

DtF =n∑j=1

k∑i1,...,in=1

ai1,...,inW (Ai1) · · · 1Aij(t) · · ·W (Ain) = nIn−1(fn(·, t)) (2.60)

30

where we have used the symmetry in the second equality. Let F ∈ D1,2, F =∑∞

n=0 In(fn),

fn ∈ L2(T n). We consider the sequence of partial sums FN =∑N

n=0 In(fn), N ≥ 0. Then

FN converges to F in L2(Ω) as N → ∞. This implies that the sequence DFN converges to

DF in L2(Ω× T n) as N →∞.

On the other hand, for each n ≥ 1, In(fn) is the limit in L2(Ω) of the sequence In(fkn)k≥1,

where fkn is an elementary function and the sequence fkn converges to fn in L2(T n) as k →∞.

Hence, the sequence D(In(fkn)) converges to D(In(fn)) in L2(Ω× T n) as k →∞. Moreover,

the sequence In−1(fkn(·, t)) converges to In−1(fn(·, t)) in L2(Ω×T n) as k →∞. Finally, using

the closability of the operator D and Equation (2.60) the result follows. 2

Next we will compute the derivative of a conditional expectation with respect to a σ-field

generated by Gaussian stochastic integrals. Let A ∈ B. We will denote by FA the σ-field

(completed with respect to the probability P ) generated by random variables W (B), B ⊂A,B ∈ B0. We need the following lemma (see [70] page 33).

Lemma 2.7.2 Suppose F is a sequence of integrable random variable with a representation

in (2.55). Let A ∈ B. Then

E[F | FA] =∞∑n=0

In(fn1⊗nA ). (2.61)

where 1⊗nA = 1A ⊗ · · · ⊗ 1A.

Proof

It suffices to assume that F = In(fn) where fn is a function in L2(Ω). By density arguments

we set

fn = 1B1×···×Bn

where B1, . . . , Bn are pairwise disjoint sets of finite measure. Then we have

E[F | FA] = E[W (B1) · · ·W (Bn) | FA] = E[n∏i=1

(W (Bi ∩ A) +W (Bi ∩ Ac)) | FA]

= In(1B1∩A×···×Bn∩A) 2

Lemma 2.7.3 Suppose F ∈ D1,2 and let A ∈ B. Then the conditional expectation E[F | FA]

also belongs to the space D1,2 and we have

Dt(E[F | FA]) = E[DtF | FA]1A(t) a.e in [0, T ]× Ω. (2.62)

31

Proof

By Theorem 2.7.1 and Lemma 2.7.2 we have

DtE[F | FA] =∞∑n=1

nIn−1(fn(·, t)1⊗(n−1)A )1A(t) = E[DtF | FA]1A(t). 2

We note that DtI(f) = f(t). Suppose that F ∈ D1,2 with a Wiener chaos expansion

F =∑∞

n=0 In(fn). Then applying Theorem 2.7.1 k times we obtain

Dkt1,...,tk

F =∞∑n=k

n(n− 1) · · · (n− k + 1)In−k(fn(·, t1, . . . , tn))

= k!fk(t1, . . . , tk) + (k + 1)!I1(fk+1(t1, . . . , tk, ·) + . . . (2.63)

We can write Equation (2.63) as

Dkt1,...,tk

F =∞∑n=k

n!

(n− k)!In−k(fn(·, t1, . . . , tn)).

Using Equation (2.56) we see that the L2-norm of this is given by

‖ Dkt1,...,tk

F ‖2L2([0,T ]k)=

∞∑n=k

(n!)2

((n− k)!)2(n− k)! ‖ fn ‖2

L2([0,T ]k)=∞∑n=k

(n!)2

(n− k)!‖ fn ‖2

L2(Tn) .

We recall that

E[In(fn)] = 0 for all n ≥ 1.

Hence, by taking the expectation on both sides of Equation (2.63) we obtain the following

result:

fk =1

k!E[Dk

t1,...,tkF ] for every k ≥ 0. (2.64)

This is called the Stroock formula (see [30]). The formula is given in [70] page 35 as an

exercise.

2.8 Skorohod integral via chaos expansion

In this section we define the Skorohod integral in terms of the Wiener-Ito chaos expansion.

Let u(t, ω), ω ∈ Ω, t ∈ [0, T ] be a stochastic process such that

u(t, ·) is Ft measurable for all t ∈ [0, T ] (2.65)

32

and

E[u2(t, ω)] <∞ for all t ∈ [0, T ]. (2.66)

Then, for each t ∈ [0, T ], we can apply the Wiener Ito chaos expansion to the random

variable ω → u(t, ω) and obtain

u(t, ω) =∞∑n=0

In(fn(·, t)) (2.67)

where for each n ≥ 1, fn ∈ L2(Sn+1) is a symmetric function in the first n variables. The

following theorem was taken from [70] page 41.

Theorem 2.8.1 Let u(t, ω) =∑∞

n=0 In(fn(·, t)). Then u(t, ω) belongs to Dom(δ) if and only

if the series

δ(u) =∞∑n=0

In+1(fn) (2.68)

converges in L2(Ω).

We can write Equation (2.68) without symmetrization because, for each n,

In+1(fn) = In+1(fn). However, we need the symmetrization in order to compute the L2-norm

of the stochastic integrals.

Proof

Let n ≥ 1 and g ∈ L2(T n) a symmetric function. We have that

E [〈u,D (In(g))〉H ] =∞∑m=0

∫Tn

E [Im(fm(·, t))nIn−1(g(·, t))] dt

= n

∫Tn

E [In−1(fn−1(·, t))In−1(g(·, t))] dt

= n(n− 1)!

∫Tn

〈fn−1(·, t), g(·, t)〉L2(Tn−1)dt

= n!〈fn, g〉L2(Tn)

= n!〈fn−1, g〉L2(Tn)

= E[In(fn−1In(g)

].

Suppose first that u ∈ Dom(δ). Then using Equation (2.22) and the above calculation we

obtain that for all n ≥ 1 and g ∈ L2(T n) symmetric,

E [δ(u)In(g)] = E [〈u,D(In(g))〉H ] = E[〈In(fn−1), In(g)〉H

].

33

This implies that In(g) is the projection of δ(u) in the n-th Wiener chaos. Hence, the series

(2.68) converges in L2(Ω) to its sum which is equal to δ(u).

Conversely, we assume that the series (2.68) converges in L2(Ω) and we denote its sum by

V . Let FN =∑N

n=0 In(gn), where gn ∈ L2(T n) are symmetric and N ≥ 1. Using the above

calculation we obtain that for all N ≥ 1

E[∫

Tn

utDtFNdt

]=

N∑n=1

E[In(fn−1)In(gn)

].

In particular,

| E[∫

Tn

utDtFNdt

]|≤‖ V ‖L2(Ω)‖ FN ‖L2(Ω) .

Let F ∈ D1,2, F =∑∞

n=0 In(gn), where gn ∈ L2(T n) are symmetric. Then FN converges to

F in L2(Ω) as N →∞ and DFN converges to DF in L2(T n × Ω) as N →∞. Therefore

| E[∫

Tn

utDtFdt

]|≤‖ V ‖L2(Ω)‖ FN ‖L2(Ω),

which implies that u ∈ Dom(δ). 2

2.9 The Clark-Haussmann-Ocone formula

Suppose that W = Wt, t ∈ [0, T ] is a 1-dimensional Brownian motion. The Ito represen-

tation theorem states that any F ∈ L2(Ω) can be written as

F = E[F ] +

∫ T

0

φ(t)dWt (2.69)

where φ is an adapted process in L2([0, T ] × Ω). If, in addition, F ∈ D1,2 it turns out that

the process φ can be expressed as a Malliavin derivative of F . This is called the Clark-Ocone

representation formula (see [70] page 46).

Theorem 2.9.1 Let F ∈ D1,2. Then we have

F = E[F ] +

∫ T

0

E[DtF | Ft]dWt a.s. (2.70)

Proof

Suppose that F can be written in the form (2.69) with φ ∈ L2([0, T ] × Ω). Then, for any

34

ϕ ∈ L2([0, T ]×Ω), using Ito isometry and that the expected value of an Ito integral is 0, we

have

E [δ(ϕ)F ] = E[∫ T

0

ϕ(t)dWt

(E[F ] +

∫ T

0

φ(t)dWt

)]=

∫ T

0

E[ϕ(t)φ(t)]dt. (2.71)

On the other hand, using integration by parts formula and taking into account that ϕ is

adapted, we obtain

E [δ(ϕ)F ] = E[∫ T

0

ϕ(t)DtFdt

]=

∫ T

0

E [ϕ(t)E [DtF | Ft]] dt. (2.72)

Comparing Equations (2.71) and (2.72) we get

φ(t) = E [DtF | Ft]

which proves Equation (2.70). 2

Theorem 2.9.1 shows that the Malliavin derivative Dt provides an identification of the

integrator in the martingale representation theorem in a Brownian motion framework, which

plays a central role in financial mathematics, in particular, to obtain replicating hedging

strategies for options (see [14]). Therefore, the hedging portfolio is naturally related to the

Malliavin derivative Dt of the terminal payoff.

35

Chapter 3

Application of Malliavin calculus to

the Calculations of Greeks for

Continuous Processes

The main focus of this thesis is to prove some extensions of the work of Fournie et al. [35]

and [36] to the case where the market is driven by a Levy process. Before proceeding to this,

we briefly review the applications of Malliavin calculus to compute Greeks of different types

of option prices in the Brownian motion case. We use the chain rule and the integration

by parts formula. Most of the results we obtain are known in the literature but the results

are sometimes only quoted (see [8] and [37]). Here, we give explicit calculations of these

results. In particular we calculate ∆, Γ and V (V=Vega) for path independent (European

call option) and path dependent (Asian option) options. We also compute ∆ for barrier and

look-back options and for the Heston model.

It turns out that the Greeks we obtain are expressed as expectations of the payoff function

multiplied by a random variable which is a function of the underlying process. This is crucial

for simulation by the Monte Carlo method. We mention that all the results below follow

from results in [35] and [36]. Instead of considering the payoff as the element of the L2 space

as in [35], we will consider it as a function of polynomial growth. This allows us to handle

more cases, for example the digital option.

Let Xt ∈ Rn be an Ito-diffusion process given by the stochastic differential equation

dXt = b(Xt)dt+ σ(Xt)dWt, X0 = x (3.1)

36

where Wt, 0 ≤ t ≤ T is a n-dimensional standard Brownian motion. The coefficients b(x)

and σ(x) are deterministic and are assumed to be bounded with bounded partial derivatives.

In addition, the coefficients b(x) and σ(x) satisfy Lipschitz condition and linear growth

condition, that is, there exists a constant C <∞ such that

| b(x)− b(y) | + | σ(x)− σ(y) |≤ C | x− y |, (3.2)

| b(x) | + | σ(x) |≤ C(1+ | x |). (3.3)

These conditions ensure the existence of a unique strong solution to Equation (3.1).

Let Φ denote the payoff function of some financial quantity. We consider a payoff depending

on the prices at a finite number of times, that is,

Φ = Φ(Xt1 , · · · , Xtn)

where Φ : Rn → R is of polynomial growth (to be defined later on) with bounded derivatives

and 0 < t1 < · · · < tn = T . Examples of payoff functions include the European call option

Φ(x) = (x−K)+ where K is a strike price and the digital option Φ(x) = 1x≥K. Here n = 1

and t1 = T the expiry date of the option.

Given 0 < t1 < . . . < tn = T we will assume that the option price u(x) is computed, under

the risk neutral probability measure, as

u(x) = E[e−rTΦ(Xt1 , · · · , Xtn) | X0 = x] (3.4)

where r is the risk free interest rate assumed to be constant and x is the initial stock price.

In particular, when n = 1 and t1 = T we have u(x) = E[e−rTΦ(XT ) | X0 = x]. We want to

compute the following expression

∂

∂αE[e−rTΦ(Xt1(α), · · · , Xtn(α)) | X0 = x], α = x, σ. (3.5)

There are several ways of doing this (see Chapter 1). Here we present the Malliavin calculus

approach. At several places, we will require the matrix σ to satisfy the following condition:

∃η > 0 ξTσT (x)σ(x)ξ > η | ξ |2 for all ξ, x ∈ Rn with ξ 6= 0. (3.6)

where ξT denotes the transpose of ξ. This is called the uniform ellipticity condition.

As stated in [35], the condition (3.6) ensures that σ−1(Xt)Yt belongs to L2(Ω× [0, T ]) where

Yt, 0 ≤ t ≤ T is the first variation process (see Chapter 2 Section 2.4) of Xt, 0 ≤ t ≤ T.In addition, if b is a bounded function then σ−1(Xt)b(Xt) belongs to L2(Ω× [0, T ]) and σ−1b

37

is a bounded function.

The weight function obtained when computing Greeks using the integration by parts formula

should not degenerate with probability one, otherwise the computation will not be valid. To

avoid this degeneracy we introduce the set Υn (see [35]) defined by

Υn = a ∈ L2([0, T ]) |∫ ti

0

a(t)dt = 1 for all i = 1, . . . , n. (3.7)

We need the following lemma.

Lemma 3.0.2 Let a ∈ Υn and Xti ∈ D1,2. Then∫ T

0

DtXtia(t)σ−1(t)Y (t)dt = Y (ti), i = 1, . . . , n.

Proof

We have∫ T

0

DtXtia(t)σ−1(t)Y (t)dt =

∫ T

0

Y (ti)Y (t)−1σ(t)1t≤tia(t)σ−1(t)Y (t)dt by (2.37)

=

∫ T

0

Y (ti)Y (t)−1σ(t)σ−1(t)Y (t)a(t)1t≤tidt

=

∫ T

0

Y (ti)a(t)1t≤tidt

=

∫ ti

0

Y (ti)a(t)dt

= Y (ti)

∫ ti

0

a(t)dt

= Y (ti) 2

We also need the following definition.

Definition 3.0.3 A function Φ : Rn → R is of polynomial growth if there exist constants

C > 0, β ≥ 0 such that

| Φ(x) |≤ C(1+ | x |)β for all x ∈ Rn. (3.8)

The following proposition is a modification of one of the main results in [35].

38

Proposition 3.0.4 Assume that b and σ (in Equation (3.1)) are continuously differentiable

with bounded partial derivatives and that the matrix σ satisfies the uniform ellipticity condi-

tion (see (3.6)). Then, for any Φ : R → R of polynomial growth and a ∈ Υn, we have

∆ = E[e−rTΦ(Xt1 , · · · , Xtn)π | X0 = x] (3.9)

where π =∫ T

0a(t)(σ−1(Xt)Yt)

TdWt.

This is called the Bismut-Elworthy-Li formula.

Proof

We first assume that Φ is continuously differentiable with bounded partial derivatives. We

show that we can compute ∆ by calculating the derivative of the payoff function with respect

to x inside the the expectation operator. We have

e−rTΦ(Xx+ht1 , · · · , Xx+h

tn )− e−rTΦ(Xxt1, · · · , Xx

tn)

‖ h ‖−〈e−rT

∑ni=1

∂∂xi

Φ(Xt1 , · · · , Xtn)Yti , h〉‖ h ‖

→ 0 a.s.

(3.10)

as ‖ h ‖→ 0 where Yti =∂Xti

∂xand Xx+h

t denote the solution Xt starting from x + h, that

is, Xx+h0 = x+ h. The term

〈e−rT∑n

i=1∂

∂xiΦ(Xt1 ,··· ,Xtn )Yti ,h〉‖h‖ is bounded above by some number

independent of h since the payoff function Φ has bounded partial derivatives. Let M be the

bound of the partial derivatives of Φ. Then, we can show that

‖ e−rTΦ(Xx+ht1 , · · · , Xx+h

tn )− e−rTΦ(Xxt1, · · · , Xx

tn) ‖‖ h ‖

≤Mn∑i=1

‖ Xx+hti −Xx

ti‖

‖ h ‖. (3.11)

Using the result that∑n

i=1

‖Xx+hti

−Xxti‖

‖h‖ is bounded (see [79] page 256) leads to the boundedness

ofe−rT Φ(Xx+h

t1,··· ,Xx+h

tn)−e−rT Φ(Xx

t1,...,Xx

tn)

‖h‖ . This, in turn, tells us that

‖ e−rTΦ(Xx+ht1 , · · · , Xx+h

tn )− e−rTΦ(Xxt1, · · · , Xx

tn) ‖‖ h ‖

−〈e−rT

∑ni=1

∂∂xi

Φ(Xt1 , · · · , Xtn)Yti , h〉‖ h ‖

is bounded. Since it converges to zero a.s., the dominated convergence theorem says that it

converges also to zero in L1(Ω). We therefore conclude that

∂

∂xE[e−rTΦ(Xt1 , · · · , Xtn) | X0 = x] = E[e−rT

n∑i=1

∂

∂xiΦ(Xt1 , · · · , Xtn)Yti | X0 = x]. (3.12)

We now compute ∆ as follows:

∆ =∂


n∑i=1

∂

∂xiΦ(Xt1 , · · · , Xtn)

∂Xti

∂x| X0 = x]

= E[e−rTn∑i=1

∂

∂xiΦ(Xt1 , · · · , Xtn)Yti | X0 = x] (3.13)

39

where Yti is the first variation process associated with Xti . Assume that Xti belongs to D1,2.

From Lemma 3.0.2 we have

Yti =

∫ T

0

DtXtia(t)σ−1(Xt)Ytdt

for any a(t) ∈ Υn. Therefore, we have

∆ = E[

∫ T

0

e−rTn∑i=1

∂

∂xiΦ(Xt1 , · · · , Xtn)DtXtia(t)σ

−1(Xt)Ytdt | X0 = x]

= E[

∫ T

0

Dt(e−rTΦ(Xt1 , · · · , Xtn))a(t)σ−1(Xt)Ytdt | X0 = x] by (2.16)

= E[e−rTΦ(Xt1 , · · · , Xtn)

∫ T

0

a(t)(σ−1(Xt)Yt)TdWt | X0 = x] (3.14)

by (2.22) and adaptedness of a(t)(σ−1(Xt)Yt)T .

We now consider the general case. We approximate Φ by a sequence Φkk∈N of functions

each with bounded derivatives and compact support. Let uk(x) = E[e−rTΦk(Xt1 , · · · , Xtn) |X0 = x] be the price associated with the payoff function Φn. We need to first show that uk(x)

converges to u(x). Since Xt1 , . . . , Xtn satisfy Equation (3.1), we let pt1,...,tn(x, x1, . . . , xn) be

the transition density of Xt1 , . . . , Xtn . Then

| uk(x)− u(x) | = | e−rT∫

Rn

[Φk(x1, . . . , xn)− Φ(x1, . . . , xn)]pt1,...,tn(x, x1, . . . , xn)dx1 · · · dxn |

≤(e−2rT

∫Rn

| Φk(x1, . . . , xn)− Φ(x1, . . . , xn) |2 dx1 · · · dxn) 1

2

(∫Rn

pt1,...,tn(x, x1, . . . , xn)2dx1 . . . dxn

) 12

The dominated convergence theorem implies that | Φk(x1, . . . , xn) − Φ(x1, . . . , xn) |2 con-

verges to 0 as k → ∞. Since b and σ are continuously differentiable with bounded partial

derivatives, the transition density is bounded by

| pt(x, y) |≤1√

2πσ2te−

(x−y)2

2σ2t . (3.15)

We note that

| pt1,...,tn(x, x1, . . . , xn) | = pt1(x, x1)pt2−t1(x1, x2) · · · ptn−tn−1(xn−1, xn) <∞.

Therefore we have

| uk(x)− u(x) |→ 0 for all x. (3.16)

40

Hence uk(x) converges to u(x).

From the above calculations we have

∂

∂xuk(x) = E[e−rTΦk(Xt1 , · · · , Xtn)π | X0 = x]

where π =∫ T

0a(s)(σ−1(Xs)Ys)

TdWs. Furthermore, let

h(x) := E[e−rTΦ(Xt1 , · · · , Xtn)π | X0 = x].

By the Cauchy-Schartz inequality we have

| ∂∂xuk(x)− h(x) |≤ εk(x)ψ(x)

where

ξk(x) =(E[e−rTΦk(Xt1 , · · · , Xtn)− e−rtΦ(Xt1 , · · · , Xtn)

]2) 12

and

ψ(x) =

(E[∫ T

0

a(t)(σ−1(Xt)Yt)TdWt

]2) 1

2

.

By a continuity argument of the expectation operator, this proves that

supx∈K

| ∂∂xuk(x)− h(x) |≤ εk(x)ψ(x) for some x ∈ K

where K is an arbitrary compact subset of Rn which provides

∂

∂xuk(x) → h(x) uniformly on compact subset of Rn. (3.17)

From (3.16) and (3.17) we conclude that the function u(x) is continuously differentiable and

that

∂

∂xE[e−rTΦ(Xt1 , · · · , Xtn) | X0 = x] = E[e−rTΦ(Xt1 , · · · , Xtn)π | X0 = x]. 2

Remark

The application of formula (3.9) does not require the payoff function to be differentiable

nor to know the density function of Xt but we do need to know Xt. The weight function π

in formula (3.9) is independent of the payoff function and is not unique (it depends on the

choice of a(t)).

Next we consider Γ which is defined as the second derivative of the option price with respect

to the initial price x. It is actually the derivative of ∆ with respect to the initial price x.

41

Proposition 3.0.5 Assume that the matrix σ is uniformly elliptic. Let u = a(t)(σ−1(Xt)Yt)T

and δ(u) =∫ T

0a(t)(σ−1(Xt)Yt)

TdWt for any a ∈ Υn. Let Φ : R → R be a function of poly-

nomial growth. Then we have

Γ = E[e−rTΦ(Xt1 , · · · , Xtn)

(δ(u)δ(u) +

∂

∂x(δ(u))

)| X0 = x]. (3.18)

Proof

We first assume that Φ is a continuously differentiable function with bounded derivatives.

By definition of Γ we have

Γ =∂2

∂x2E[e−rTΦ(Xt1 , · · · , Xtn) | X0 = x] =

∂

∂xE[e−rTΦ(Xt1 , · · · , Xtn)δ(u) | X0 = x].

(3.19)

We note that Φ(Xt1 , · · · , Xtn)δ(u) is a function of both Xt1 , . . . , Xtn and x. Thus, we have

Γ =∂

∂xE[e−rTΦ(Xt1 , · · · , Xtn)δ(u) | X0 = x]

= E[e−rTn∑i=1

∂

∂xi(Φ(Xt1 , · · · , Xtn)) δ(u)

∂Xti

∂x| X0 = x]

+ E[e−rTΦ(Xt1 , · · · , Xtn)∂

∂x(δ(u)) | X0 = x]

= E[e−rTn∑i=1

∂

∂xi(Φ(Xt1 , · · · , Xtn)δ(u))Yti | X0 = x]

+ E[e−rTΦ(Xt1 , · · · , Xtn)∂

∂x(δ(u)) | X0 = x].

The first term on the right hand side is similar to the one in the computation of ∆. We pro-

ceed in the same way as from Equation (3.13) to Equation (3.14) in the proof of Proposition

3.0.4 and we obtain

Γ = E[e−rTΦ(Xt1 , · · · , Xtn)δ(u)δ(u) | X0 = x] + E[e−rTΦ(Xt1 , · · · , Xtn)∂

∂x(δ(u)) | X0 = x]

= E[e−rTΦ(Xt1 , · · · , Xtn)

(δ(u)δ(u) +

∂

∂x(δ(u))

)| X0 = x].

The result can be extended to the general case by a density argument. We omit the de-

tails. 2

Remark

We can interchange the partial derivative with respect to x and the Skorohod integration to

get another expression for Γ.

42

3.1 Generalized Greeks

The definitions of ρ and V given in Chapter 1 are the common ones for the Black-Scholes

model. We, however, need to develop a more robust framework for the definitions of ρ and

V since the drift and the volatility terms are functions of the underlying asset price. ρ and

V quantify the impact of small perturbation in a specified direction on the drift and the

volatility respectively (see [9]).

We consider the stochastic differential equation given in Equation (3.1). Let b : [0, T ]×Rn →Rn be a bounded function and let σ : Rn → Rn×n be another bounded function. We

assume, for every ε ∈ [0, 1], that b(·), (b + εb)(·), σ(·) and (σ + εσ)(·) are continuously

differentiable with bounded derivatives. Moreover, we assume that (σ + εσ)(·) satisfies the

uniform ellipticity condition:

∃η > 0, ξT (σ + εσ)T (x)(σ + εσ)(x)ξ > η | ξ |2 for all ξ, x ∈ Rn with ξ 6= 0.

As in [35] we introduce the following set.

Υn = a ∈ L2([0, T ]) |∫ ti

ti−1

a(t)dt = 1 for i = 1, . . . , n. (3.20)

We need the following theorem (see [56]).

Theorem 3.1.1 Let Wt, 0 ≤ t ≤ T be a Brownian motion under probability measure P ,

Ft0≤t≤T be the filtration generated by Wt, 0 ≤ t ≤ T, θt, 0 ≤ t ≤ T be an Ft-adaptedprocess satisfying

E[exp

(1

2

∫ T

0

θ2sds

)] <∞,

and let

W θt = Wt +

∫ t

0

θsds

and

Zθt = exp(

∫ t

0

θsdWs −1

2

∫ t

0

θ2sds), t ∈ [0, T ]

and define the probability measure Q by

Q(F ) =

∫F

ZθTdP, F ∈ Ft.

Then the process W θt is a Brownian motion under the probability measure Q and more

generally we have

E[F (W θ)] = E[F (W )ZθT ] F ∈ L1(Ω, P ).

43

This is called the Girsanov Theorem. We can show that E[ZθT ] = 1 (see [60] page 137).

We define a perturbed process Xεt , 0 ≤ t ≤ T with respect to the property under investi-

gation. For ρ we define the perturbed process Xεt , 0 ≤ t ≤ T by

dXεt = [b(Xε

t ) + εb(Xεt )]dt+ σ(Xε

t )dWt, Xε0 = x (3.21)

where ε is a small parameter. We also define the perturbed option price uε(x):

uε(x) = E[e−rTΦ(Xε(·)) | Xε0 = x]. (3.22)

Equations (3.21) and (3.22) imply the impact of a structural change in the drift and the

price. We, therefore, define a generalized ρ as follows.

Definition 3.1.2 ρ is defined as the partial derivative of the perturbed option price with

respect to ε in the direction b:

ρ :=∂

∂εuε(x) |ε=0 . (3.23)

The following proposition gives the derivative of the perturbed option price uε(x) with respect

to ε at ε = 0 [35].

Proposition 3.1.3 Assume that the matrix σ is uniformly elliptic. Then, for any Φ : R →R of polynomial growth, the function ε→ uε(x) is differentiable at ε = 0 for any x ∈ Rn and

∂

∂εuε(x) |ε=0= E[e−rTΦ(Xt1 , · · · , Xtn)

∫ T

0

(σ−1(Xt)b(Xt))TdWt | X0 = x]. (3.24)

Proof

Let

M εT = expε

∫ T

0

(σ−1(Xεt )b(X

εt ))

TdW εt −

ε2

2

∫ T

0

‖ σ−1(Xεt )b(X

εt ) ‖2 dt

and define the probability measure Q, equivalent to P , by

dQ

dP= M ε

T .

Therefore we can write Equation (3.21) as

dXεt = b(Xε

t )dt+ σ(Xεt )dW

εt , Xε

0 = x (3.25)

with dW εt = dWt + εσ−1(Xε

t )b(Xεt )dt. By Theorem 3.1.1 W ε

t is a standard Brownian motion

under the probability measure Q. The joint distribution of (Xε,W ε) under Qε coincide with

the joint distribution of (X,W ) under P . Therefore, we have

uε(x) = EQ[e−rTM εTΦ(Xε

t1, · · · , Xε

tn) | Xε0 = x] = EP [e−rTM ε

TΦ(Xt1 , · · · , Xtn) | X0 = x]

44

where M εT = expε

∫ T0

(σ−1(Xt)b(Xt))TdWt − ε2

2

∫ T0‖ σ−1(Xt)b(Xt) ‖2 dt. The upper index

Q in EQ indicates that the expectation is taken with respect to the measure Q.

Let

Yt = ε

∫ t

0

(σ−1(Xs)b(Xs))TdWs −

ε2

2

∫ t

0

‖ σ−1(Xs)b(Xs) ‖2 ds

so that

Mt = eYt .

Using Ito formula we have

M εt = 1 + ε

∫ t

0

M εsσ

−1(Xs)b(Xs)dWs. (3.26)

Rearranging this and setting t = T we have, for ε 6= 0,

M εT − 1

ε=

∫ T

0

M εt (σ

−1(Xt)b(Xt))TdWt.

As ε→ 0, we have

M εT − 1

ε→∫ T

0

(σ−1(Xt)b(Xt))TdWt in L2

which follows by the dominated convergence theorem. By an application of Cauchy-Schwartz

inequality we have

| uε(x)− u(x)

ε− E[e−rTΦ(Xt1 , · · · , Xtn)

∫ T

0

(σ−1(Xt)b(Xt))TdWt | Xε

0 = x] |

= | E[e−rTM εTΦ(Xt1 , · · · , Xtn)]− E[e−rTΦ(Xt1 , · · · , Xtn)]

ε

− E[e−rTΦ(Xt1 , · · · , Xtn)

∫ T

0

(σ−1(Xt)b(Xt))TdWt | Xε

0 = x] |

≤ KE[

(M ε

T − 1

ε−∫ T

0

(σ−1(Xt)b(Xt))TdWt

)2

]

where K is a constant independent of ε. Therefore, we have

∂

∂εuε(x) |ε=0= lim

ε→0

uε(x)− u(x)

ε= E[e−rTΦ(Xt1 , · · · , Xtn)

∫ T

0

(σ−1(Xt)b(Xt))TdWt | X0 = x]. 2

Next we consider the derivative of the option price with respect to the volatility σ. We define

the perturbed process Xεt , 0 ≤ t ≤ T by

dXεt = b(Xε

t )dt+ [σ(Xεt ) + εσ(Xε

t )]dWt Xε0 = x. (3.27)

45

The first variation process Zεt associated with the process Xε

t , that is, the derivative of Xεt

with respect to the parameter ε (Zεt := ∂

∂εXεt ) is given by

dZεt = b′(Xε

t )Zεt dt+ σ(Xε

t )dWt +n∑i=1

(∂

∂εiσ + ε

∂

∂εiσ)(Xε

t )Zεt dW

it , Zε

0 = 0

where 0 is the column vector in Rn. We will use the notation Xt, Yt and Zt for X0t , Y

0t and

Z0t respectively. The generalized V is defined as follows.

Definition 3.1.4 V is defined as the partial derivative of the perturbed option price with

respect to ε in the direction σ:

V :=∂

∂εuε(x) |ε=0 . (3.28)

The following proposition is found in [35].

Proposition 3.1.5 Assume that the matrix σ is uniformly elliptic. Let βti = ZtiY−1ti ,

0 ≤ ti ≤ T such that σ−1(Xt)Ytβa(T ) ∈ Dom(δ) for all t ∈ [0, T ]. Then, for any Φ : R → Rof polynomial growth, we have

∂

∂εuε(x) |ε=0= E[e−rTΦ(Xt1 , · · · , Xtn)δ

(σ−1(X)Y βa(T )

)| X0 = x] (3.29)

where βa(t) =∑n

i=1 a(t)(βti −βti−1)1ti−1≤t≤ti for any a ∈ Υn and δ

(σ−1(X)Y βa(T )

)is the

Skorohod integral of σ−1(Xt)Ytβa(T ), 0 ≤ t ≤ T .

Proof

We proceed as in the proof of Proposition 3.0.4. We first assume that Φ is continuously

differentiable function with bounded derivatives. We first prove that the derivative of uε(x)

with respect to ε is obtained by differentiating inside the expectation operator. Considering

ε as a degenerate process, we can apply Theorem 39 page 250 in [79] which ensures that we

can choose versions of Xεt , 0 ≤ t ≤ T which are continuously differentiable with respect to

ε for each (t, ω) ∈ [0, T ]×Ω. Since Φ is assumed to be continuously differentiable, we prove

by the same arguments that we have in the sense of L1 norm:

∂

∂εuε(x) = E[e−rT

n∑i=1

∂

∂εiΦ(Xε

t1, · · · , Xε

tn)∂

∂εXεti| Xε

0 = x]

= E[e−rTn∑i=1

∂

∂εiΦ(Xε

t1, · · · , Xε

tn)Zti | Xε0 = x]. (3.30)

46

Using

DtXti = YtiY−1t σ(Xt)1t≤ti (3.31)

we get

∫ T

0

DtXtiσ−1(Xt)Yt1t≤ti βa(T )dt =

∫ T

0

YtiY−1t σ(Xt)σ

−1(Xt)Ytβa(T )1t≤tidt

=

∫ T

0

Yti βa(T )1t≤tidt = Yti

∫ ti

0

βa(T )dt

= Yti

i∑j=1

(∫ tj

0

a(t)(βtj − βtj−1)1tj−1≤t≤tjdt

)

= Yti

i∑j=1

(∫ tj

tj−1

a(t)(βtj − βtj−1)dt

)

= Yti

i∑j=1

∫ tj

tj−1

a(t)dt(βtj − βtj−1)

= Ytiβti

= Zti (3.32)

where the second last equality follows because∫ tjtj−1

a(t)dt = 1 and βt0 = 0.

Hence

∂

∂εuε(x) = E[e−rT

n∑i=1

∂

∂εiΦ(Xε

t1, · · · , Xε

tn)Zti | Xε0 = x]

= E[e−rT∫ T

0

n∑i=1

∂

∂εiΦ(Xε

t1, · · · , Xε

tn)DtXtiσ−1(Xt)Ytβtdt | Xε

0 = x]

= E[

∫ T

0

Dt

(e−rTΦ(Xε

t1, · · · , Xε

tn))σ−1(Xt)Ytβtdt | Xε

0 = x]

= E[e−rTΦ(Xt1 , · · · , Xtn)δ(σ−1(Xt)Y βt

)| X0 = x]

where the third and last equalities follow by Equation (2.16) and the integration by parts

formula, respectively.

The general case follows by using approximations. 2

Remark

The weight function is independent of the payoff function.

47

3.2 Greeks for European Options

In the following sections we explicitly calculate Greeks for different options. We will consider

an underlying asset S described by a geometric Brownian motion, under the risk neutral

probability,

St = S0 +

∫ t

0

rSτdτ +

∫ t

0

σSτdWτ , S0 = x (3.33)

where St is the price of the underlying asset, S0 is the initial value, r is the riskless interest

rate, σ is the volatility and Wt, 0 ≤ t ≤ T is the standard Brownian motion. This model

describes the stock prices or stock indices. We will assume that the coefficients r and σ > 0

are constants. The solution at time t of the differential equation (3.33) is given by

St = x exp((r − σ2

2)t+ σWt). (3.34)

Then∂St∂x

=Stx. (3.35)

We consider a path independent payoff function of the form Φ(ST ) for some T where ST is

the stock price at time T . The payoff function Φ only depends on the price of the stock at

final time T . Then, the option price u(x) of a European option at time t = 0 is given by

u(x) = E[e−rTΦ(ST ) | S0 = x]. (3.36)

We note that the density of ST is known.

Lemma 3.2.1 Let Sτ be given by Equation (3.34). Then we have

DtSτ = σSτDtWτ = σSτ1t<τ . (3.37)

Proof

We have

DtSτ = Dt

(x exp((r − σ2

2)τ + σWτ )

)= x exp((r − σ2

2)τ)Dt(exp(σWτ ))

= x exp((r − σ2

2)τ)Dt(exp

(∫ τ

0

σdWu

))

= x exp((r − σ2

2)τ) exp(σWτ )Dt

∫ τ

0

σdWu

= σSτ1t<τ 2

48

Proposition 3.2.2 Let Φ : R → R be a function of polynomial growth. Then ∆ is given by

∆ =e−rT

σxTE[Φ(ST )WT ]. (3.38)

Proof

Assuming first that Φ is continuously differentiable with bounded derivatives, we have

∆ =∂

∂xE[e−rTΦ(ST )] = E[e−rTΦ′(ST )

∂ST∂x

]

= E[e−rTΦ′(ST )STx

] =e−rT

xE[Φ′(ST )ST ]. (3.39)

From Lemma 3.2.1 we have

DtST = σST1t<T

where Dt is the Malliavin derivative. We can write∫ T

0

DtSTdt =

∫ T

0

σST1t<Tdt = σSTT. (3.40)

Rearranging this we have

ST =1

σT

∫ T

0

DtSTdt, σ > 0. (3.41)

Substituting for ST in Equation (3.39) we obtain

∆ =e−rT

xE[Φ′(ST )

1

σT

∫ T

0

DtSTdt]

=e−rT

σxTE[

∫ T

0

Dt(Φ(ST ))dt] by (2.16)

=e−rT

σxTE[Φ(ST )δ(1)] by (2.22)

where δ(1) is the Skorohod integral of 1. The desired result follows by using Equation (2.24).


Remark

In all the calculations that follows we will first assume that Φ is continuously differentiable

with bounded derivatives and then pass to the limit by a density argument. We will not

explicitly state this.

Proposition 3.2.3 Let Φ : R → R be a function of polynomial growth. Then Γ is given by

Γ =e−rT

σx2TE[Φ(ST )

W 2T

σT− 1

σ−WT

]. (3.42)

49

Proof

Assuming Φ is continuously differentiable, we have

Γ =∂

∂x∆ =

∂

∂x

(e−rT

σxTE[Φ(ST )WT ]

)= − e−rT

σx2TE[Φ(ST )WT ] +

e−rT

σxTE[Φ′(ST )

∂ST∂x

WT ]

= − e−rT


e−rT

σx2TE[Φ′(ST )STWT ]. (3.43)

Substituting for ST in the second term on the right hand side of Equation (3.43) and assuming

that Φ′ is continuously differentiable we proceed as in the proof of ∆ (Proposition 3.38) and

obtain

Γ = − e−rT


e−rT

σx2TE[Φ′(ST )

1

σT

∫ T

0

DtSTdtWT ]

= − e−rT


e−rT

σx2TE[

∫ T

0

Dt(Φ(ST ))1

σTWTdt] by (2.16)

= − e−rT


e−rT

σx2TE[Φ(ST )

1

σTδ(WT )] by (2.22). (3.44)

Using Equation (2.26) with F = WT and u = 1 we calculate δ(WT ) as follows:

δ(WT ) = WT

∫ T

0

dWt −∫ T

0

DtWTdt = W 2T − T. (3.45)

Therefore, we have

Γ = − e−rT


e−rT

σx2TE[Φ(ST )

1

σTW 2

T − T]

=e−rT

σx2TE[Φ(ST )W

2T

σT− 1

σ−WT]. 2

Proposition 3.2.4 Let Φ : R → R be a function of polynomial growth. Then V is given by

V = e−rTE[Φ(ST )

W 2T

σT− 1

σ−WT

]. (3.46)

Proof

We have, from Equation (3.34),

∂ST∂σ

= ST (WT − σT ). (3.47)

By definition of V and Equation (3.47) we have

V =∂

∂σE[e−rTΦ(ST )] = E[e−rTΦ′(ST )

∂ST∂σ

] = e−rTE[Φ′(ST )ST (WT − σT )]. (3.48)

50

Substituting for ST (Equation (3.41)) in Equation (3.48) we have

V = E−rTE[Φ′(ST )1

σT

∫ T

0

DtSTdt(WT − σT )]

=e−rT

σTE[

∫ T

0

Φ′(ST )DtST (WT − σT )dt]

=e−rT

σTE[

∫ T

0

Dt(Φ(ST ))(WT − σT )dt] by (2.16)

=e−rT

σTE[Φ(ST )δ(WT − σT )] by (2.22).

We can write

δ (WT − σT ) = δ (WT )− σTδ(1)

because the Skorohod integral is linear. From Equation (3.45) we have

δ (WT ) = W 2T − T

and we also have

δ(1) =

∫ T

0

1dWt = WT .

Hence

δ (WT − σT ) = W 2T − T − σTWT .

Therefore, we have

V = e−rTE[Φ(ST )

W 2T

σT− 1

σ−WT

]. 2

Comparing Γ and V we note that

Γ =V

σx2T.

In the Black-Scholes context we have the relation

ρ = xT∆− TE[e−rTΦ(ST )]

from which

ρ = E[e−rTΦ(ST )

(WT

σ− T

)].

The formulae given above for ∆, Γ, V and ρ can be computed numerically using Monte Carlo

simulation procedures.

51

3.3 Greeks for Exotic options

Here, we consider a path dependent option. Let the underlying asset St, 0 ≤ t ≤ T be given

by Equation (3.33). An example of a path dependent option is an Asian option. An Asian

option is an option whose payoff function depends on the average price of the underlying

asset over a certain period of time as opposed to at maturity. It is also known as average

option. Thus, the payoff is a function of the average of the stock price 1T

∫ T0Stdt, that is,

Payoff = Φ

(1

T

∫ T

0

Stdt

)for some deterministic function Φ of one variable.

Note that there is no closed formula for the density function of the random variable 1T

∫ T0Stdt

in contrast to the European call option case. The option price at time t = 0 is given by

u(x) = E[e−rTΦ

(1

T

∫ T

0

Stdt

)]. (3.49)


∆ =2e−rT

σ2xE[Φ

(1

T

∫ T

0

Stdt

)ST − S0∫ T

0Stdt

− µ

] (3.50)

where µ = r − σ2

2.

Proof

Assuming first that Φ is continuously differentiable with bounded derivatives, we compute

∆ as follows:

∆ =∂

∂xE[e−rTΦ

(1

T

∫ T

0

Stdt

)] = e−rTE[Φ′

(1

T

∫ T

0

Stdt

)1

T

∫ T

0

∂St∂x

dt]

=e−rT

xE[Φ′

(1

T

∫ T

0

Stdt

)1

T

∫ T

0

Stdt]. (3.51)

Using Proposition 2.5.1 with F = 1T

∫ T0Sudu, G = 1

T

∫ T0Sudu, u = St and f = Φ we obtain

∆ =e−rT

xE[Φ

(1

T

∫ T

0

Stdt

)δ

(1T

∫ T0Sudu · St∫ T

0Du(

1T

∫ T0Sτdτ)Sudu

)]

We note that

Du

(1

T

∫ T

0

Sτdτ

)=

1

T

∫ T

0

DuSτdτ =1

T

∫ T

0

σSτDuWτdτ =σ

T

∫ T

0

Sτ1u<τdτ =σ

T

∫ T

u

Sτdτ,

(3.52)

52

Therefore, we have

∆ =e−rT

xE[Φ

(1

T

∫ T

0

Stdt

)δ

(1T

∫ T0Sudu · St∫ T

0σT

∫ TuSτdτSudu

)].

We also note that∫ T

0

σ

T

(∫ T

u

Sτdτ

)Sudu =

σ

T

∫ T

0

Sτ

(∫ τ

0

Sudu

)dτ

=σ

T

∫ T

0

1

2d

(∫ τ

0

Sudu

)2

dτ

=σ

T

1

2

(∫ T

0

Sudu

)2

. (3.53)

Therefore, we have

∆ =e−rT

xE[Φ

(1

T

∫ T

0

Stdt

)δ

(∫ T0Sudu · St

σ2(∫ T

0Sudu)2

)]

=e−rT

xE[Φ

(1

T

∫ T

0

Stdt

)2

σδ

(St∫ T

0Sudu

)]. (3.54)

Using Proposition 2.3.4 with F = 1∫ T0 Sτdτ

so that DuF = − σ∫ T

u Sτdτ

(∫ T0 Sτdτ)

2 and u = St, direct

computation shows that

δ

(St∫ T

0Stdt

)=

∫ T0StdWt∫ T

0Stdt

+σ

2. (3.55)

Therefore, we have

∆ =e−rT

xE[Φ

(1

T

∫ T

0

Stdt

)2∫ T

0StdWt

σ∫ T

0Stdt

+ 1

]. (3.56)

Recall that

St = S0 +

∫ t

0

rSτdτ +

∫ t

0

σSτdWτ , S0 = x

Rearranging this equation and putting S0 = x we have∫ T0StdWt∫ T

0Stdt

=ST − x

σ∫ T

0Stdt

− r

σ.

Therefore, Equation (3.56) becomes

∆ =e−rT

xE[Φ

(1

T

∫ T

0

Stdt

)2(ST − x)

σ2∫ T

0Stdt

− 2r

σ2+ 1

]

=2e−rT

σ2xE[Φ

(1

T

∫ T

0

Stdt

)ST − x∫ T0Stdt

− r +σ2

2

]

=2e−rT

σ2xE[Φ

(1

T

∫ T

0

Stdt

)ST − x∫ T0Stdt

− µ

].

53


The following two propositions give different formulae for ∆ from the one given in Proposition

3.3.1 (see [67]).


∆ =e−rT

xE[Φ

(1

T

∫ T

0

Stdt

) ∫ T0Stdt∫ T

0tStdt

WT

σ− 1

]. (3.57)

Proof

We have seen from the proof of Proposition 3.3.1 (Equation (3.51) that

∆ =e−rT

xE[Φ′

(1

T

∫ T

0

Stdt

)1

T

∫ T

0

Stdt].

Using Proposition 2.5.1 with F = 1T

∫ T0Stdt, G = 1

T

∫ T0Stdt, and u = 1 we obtain

∆ =e−rT

xE[Φ

(1

T

∫ T

0

Stdt

)δ

(1T

∫ T0Stdt∫ T

0Dt(

1T

∫ T0Sτdτ)dt

)] (3.58)

We note that∫ T

0

Dt

(∫ T

0

Sτdτ

)dt = σ

∫ T

0

dt

∫ T

t

Sτdτ = σ

∫ T

0

Sτdτ

∫ τ

0

dt = σ

∫ T

0

τSτdτ. (3.59)

Therefore, we have

∆ =e−rT

xE[Φ

(1

T

∫ T

0

Stdt

)δ

( ∫ T0Stdt

σ∫ T

0tStdt

)]. (3.60)

Let F =∫ T

0Stdt and u = 1

σ∫ T0 tStdt

. By an application of Equation (2.26) we obtain

δ

( ∫ T0Stdt

σ∫ T

0tStdt

)=

∫ T0Stdt ·WT

σ∫ T

0tStdt

−∫ T

0

Dt

(∫ T

0

Sτdτ

)1

σ∫ T

0tStdt

dt

=

∫ T0Stdt ·WT

σ∫ T

0tStdt

−∫ T

0σtStdt∫ T

0σtStdt

=

∫ T0Stdt ·WT

σ∫ T

0tStdt

− 1.

Therefore,

∆ =e−rT

xE[Φ

(1

T

∫ T

0

Stdt

) ∫ T0Stdt∫ T

0tStdt

WT

σ− 1

]. 2

54


∆ =e−rT

xE[Φ

(1

T

∫ T

0

Stdt

)1

〈S〉

(WT

σ+〈S2〉〈S〉

)− 1

]

where 〈S〉 =∫ T0 tStdt∫ T0 Stdt

and 〈S2〉 =∫ T0 t2Stdt∫ T0 Stdt

.

Proof

The proof follows by letting F =∫ T0 Stdt∫ T0 tStdt

and u = 1σ

in Equation (3.51) and proceed as in the

proof of Proposition 3.3.1. 2

Proposition 3.3.4 Let Φ : R → R be a function of polynomial growth. Then V is given by

V = e−rTE[Φ

(1

T

∫ T

0

Stdt

)∫ T0

∫ T0StWtdtdWτ

σ∫ T

0tStdt

+

∫ T0t2Stdt

∫ T0StWtdt

(∫ T

0tStdt)2

−WT

]. (3.61)

Proof

Assume first that Φ is continuously differentiable with bounded derivatives. By definition of

V we have

V =∂

∂σE[e−rTΦ

(1

T

∫ T

0

Stdt

)]

= e−rTE[Φ′(

1

T

∫ T

0

Stdt

)1

T

∫ T

0

∂St∂σ

dt]

= e−rTE[Φ′(

1

T

∫ T

0

Stdt

)1

T

∫ T

0

St(Wt − σt)dt].

Applying Proposition 2.5.1 with F = 1T

∫ T0Stdt, G = 1

T

∫ T0St(Wt − σt)dt and u = 1 we

obtain

V = e−rTE[Φ

(1

T

∫ T

0

Stdt

)δ

(1T

∫ T0St(Wt − σt)dt∫ T

0Dt(

1T

∫ T0Sτdτ)dt

)]

= e−rTE[Φ

(1

T

∫ T

0

Stdt

)δ

(∫ T0StWtdt− σ

∫ T0tStdt

σ∫ T

0tStdt

)] by (3.59)

= e−rTE[Φ

(1

T

∫ T

0

Stdt

)δ

(∫ T0StWtdt

σ∫ T

0tStdt

− 1

)].

We can write

δ

(∫ T0StWtdt

σ∫ T

0tStdt

− 1

)= δ

(∫ T0StWtdt

σ∫ T

0tStdt

)− δ(1) = δ

(∫ T0StWtdt

σ∫ T

0tStdt

)−WT .

55

Choosing F = 1

σ∫ T0 tStdt

, u =∫ T

0StWtdt and using Proposition 2.3.4 we obtain

δ

(∫ T0StWtdt

σ∫ T

0tStdt

)=

∫ T0

∫ T0StWtdtdWτ

σ∫ T

0tStdt

−∫ T

0

Dt

(1

σ∫ T

0τSτdτ

)∫ T

0

StWtdtdt

=

∫ T0

∫ T0StWtdtdWτ

σ∫ T

0tStdt

+

∫ T0Dt(σ

∫ T0τSτdτ)

∫ T0StWtdtdt

(σ2∫ T

0tStdt)2

=

∫ T0

∫ T0StWtdtdWτ

σ∫ T

0tStdt

+σ2∫ T

0t2Stdt

∫ T0StWtdt

σ(∫ T

0tStdt)2

=

∫ T0

∫ T0StWtdtdWτ

σ∫ T

0tStdt

+

∫ T0t2Stdt

∫ T0StWtdt

(∫ T

0tStdt)2

.

Therefore

V = e−rTE[Φ

(1

T

∫ T

0

Stdt

)∫ T0

∫ T0StWtdtdWτ

σ∫ T

0tStdt

+

∫ T0t2Stdt

∫ T0StWtdt

(∫ T

0tStdt)2

−WT

].


Remark

While there are other choices of ut in the integration by parts formula, the choice of a

constant minimizes the variance of the estimate (see [11]).

3.4 Greeks for Barriers and Look-back options

The ideas presented here are taken from [15], [43] and [58]. Here we consider a one-

dimensional risky asset St, 0 ≤ t ≤ T whose dynamics is given by

St = S0 +

∫ t

0

rSsds+

∫ t

0

σ(Ss)SsdWs, S0 = x (3.62)

where r is the interest rate, σ(·) is the volatility and Wt, 0 ≤ t ≤ T is the standard

Brownian motion. The Barrier and Look-back European options have the payoff function of

the form

Payoff = Φ(maxs∈I

Ss,mins∈I

Ss, ST ) for some deterministic function Φ of three variables

where I ⊂ [0, T ] is the set of all times when the extrema are monitored. The price of the

option at time 0, under the risk neutral probability measure, is given by

u(x) = E[e−rTΦ(maxs∈I

Ss,mins∈I

Ss, ST )].

56

We want to compute

∆ =∂

∂xE[e−rTΦ(max

s∈ISs,min

s∈ISs, ST )] and Γ =

∂2

∂x2E[e−rTΦ(max

s∈ISs,min

s∈ISs, ST )].

The difficulty comes from the lack of differentiability of the minimum and maximum pro-

cesses. The minimum and maximum processes are in general not differentiable and we cannot

use the integration by parts formula directly. The law of the minimum and maximum pro-

cesses is known, hence we can use the likelihood ratio to compute the Greeks. However, the

calculations are cumbersome.

We show how Malliavin calculus techniques can be used to compute ∆ and Γ. We use a

localization procedure. This procedure was first introduced in [70]. We consider a trans-

formation of St which avoids some possible degeneracy problems of the matrix σ namely,

Xt = A(St) where A is a strictly increasing function

A(y) =

∫ y

1

du

uσ(u), σ(y) 6= 0

and A−1(·) exists so that

dXt = d

(∫ St

1

du

uσ(u)

)=

1

Stσ(St)dSt +

1

2

d

du

(1

uσ(u)

)u=St

σ(St)2(St)

2dt

=1

Stσ(St)(rStdt+ σ(St)StdWt)−

(uσ(u))′

2|u=St dt

= [r

σ(St)− (uσ(u))′

2|u=St ]dt+ dWt. (3.63)

Writing Equation (3.63) in integral form, we have

Xt = x+

∫ t

0

h(Xs)ds+Wt (3.64)

where x = A(S0) and h(u) = [ rσ(y)

− (yσ(y))′

2]y=A−1(u). This corresponds to the usual logarithm

in the Black-Scholes model with σ = 1. We assume that h is bounded and continuously

differentiable. Let Mt = maxs≤t, s∈I Xs and mt = mins≤t, s∈I Xs be the maximum and the

minimum of the asset respectively. Thus, we have

Payoff = Φ(MT ,mT , XT ).

We need the following condition (see [43]).

57

Assumption 1 There exists a > 0 such that the function Φ(M,m, z) does not depend on

the variables (M,m) for any (M,m, z) such that 0 ≤M − x < a or 0 ≤ x−m < a.

As it is discussed in [43], the assumption appears to be a necessary condition to enable a

representation of Greeks to take the form E[e−rTΦ(·)π] for an appropriate random variable

π. There are many payoff functions that satisfy Assumption 1.

Example

We consider the following single barrier options:

• Up and out Barrier: Φ(M,m, z) = 1M<UΦ(z) with x < U . The assumption is satisfied

when a = U − x.

• Down and in Barrier option: Φ(M,m, z) = 1m≤DΦ(z) with D < x. The assumption is

satisfied when a = x−D.

Definition 3.4.1 An increasing adapted right-continuous process Yt, 0 ≤ t ≤ T is called

a dominating process for Xt if, for any t,

| Xt − x |≤ Yt. (3.65)

We need the following proposition

Proposition 3.4.2 1. For any t, we have | Xt − x |≤ Yt.

2. Let p, q > 1 such that There exists a positive function α : N → [0, T ] with limq→∞ α(q) =

∞, such that, for any q ≥ 1, we have

E[Y qt ] ≤ Cqt

α(q) for all t ∈ [0, T ].

In particular one has Y0 = 0.

3. For any p ≥ 1, choose an infinitely continuously differentiable and bounded function

ψ : [0,∞) → [0, 1] with

ψ(x) =

1 if x ≤ a

2

0 if x ≥ a.

where a is the real positive number appearing in Assumption 1. The random variable

ψ(Yt) belongs to D1,2 for each t. In addition, we have

| E[Dtψ(Yt)] |≤ Cp for all p ≥ 1 (3.66)

with 1p

+ 1q

= 1.

58

Proof

The proof follows the same arguments in [15]. We omit the details. 2

Remark

We call ψ(Yt) a localizing process.

The computation of Greeks consists of calculating the derivative of the minimum and the

maximum of (St)t∈I . It has been shown in [68] that if the random variable Ut ∈ D1,2 for any

t ∈ [0, T ] then, under some additional conditions, the random variables

mins≤T, s∈I

Us and maxs≤T, s∈I

Us

also belong to D1,2. We have the next lemma.

Lemma 3.4.3 Assume that the random variables M , m and ST belong to D1,2. Then their

derivatives are given, for t < T , by

Dtm = m1t≤τm and DtM = M1t≤τM

where τm = inft : mt = mins≤t, s∈I Xs and τM = inft : Mt = maxs≤t, s∈I Xs.

Proof

The proof is given in [68]. We omit the details. 2

The following theorem gives the integration by parts formula for ∆ (see [43]).

Theorem 3.4.4 Let Assumption 1 hold. Let Y be a dominating process satisfying Equation

(3.66). Then, for any Φ : R → R of polynomial growth, we have

∆ =∂

∂xE[e−rTΦ(M,m,XT )] = E[e−rTΦ(M,m,XT )π] (3.67)

with

π = δ

(ψ(Y )∫ T

0ψ(Yt)dt

)where ψ is as given in Proposition 3.4.2

59

Proof

We first assume that the payoff function Φ is continuously differentiable with bounded deriva-

tives. Therefore, we have

∆ =∂

∂xE[e−rTΦ(M,m,XT )] = E[e−rT (Φ′

1 ·M + Φ′2 ·m+ Φ′

3 ·XT )] (3.68)

where Φ′1 denote the derivative of Φ with respect to the first argument, Φ′

2 denote the

derivative of Φ with respect to the second argument and Φ′3 denote the derivative of Φ with

respect to the third argument. We have omitted the arguments of Φ for simplicity and the

dot denotes the product.

We find a representation for

Φ′1 ·M + Φ′

2 ·m+ Φ′3 ·XT .

We do this as follows. Using Proposition 2.2.8 and Lemma 3.4.3 we calculate the Malliavin

derivative of Φ as

DtΦ =(MΦ′

11t<τM +mΦ′21t<τm +XTΦ′

3

)(3.69)

Multiplying both sides of Equation (3.69) by ψ(Yt) we obtain

DtΦψ(Yt) =(MΦ′

11t<τM +mΦ′21t<τm +XTΦ′

3

)ψ(Yt). (3.70)

To apply an integration by parts formula on the representation in (3.70) we have to remove

terms of the types 1t<τM and 1t<τm and this can be done using the localizing process

ψ(Yt). To this end, for t ∈ [0, T ], we have

Φ′11t<τMψ(Yt) = Φ′

1ψ(Yt) (3.71)

and

Φ′21t<τmψ(Yt) = Φ′

2ψ(Yt) (3.72)

Here we consider Equation (3.71). If M < a+x then Φ does not depend on M (by Assump-

tion 1) and hence Equation (3.71) reduces to 0 = 0. On the other hand, if M ≥ a+ x and if

t is such that ψ(Yt) 6= 0, we have Yt < a; therefore using (3.65) we have maxs≤t(Xs − x) <

a < maxs≤T (Xs− x) that is, t ≤ τM and the proof of Equation (3.71) is complete. A similar

argument can be used to prove Equation (3.72) (see [15]).

Substituting Equations (3.71) and (3.72) into Equation (3.70) we obtain, for t ∈ [0, T ],

DtΦψ(Yt) = (MΦ′1 +mΦ′

2 +XTΦ′3)ψ(Yt). (3.73)

60

We can write ∫ T

0

DtΦψ(Yt)dt =

∫ T

0

(MΦ′1 +mΦ′

2 +XTΦ′3)ψ(Yt)dt

which is equivalent to∫ T

0

DtΦ · (ψ(Yt))dt = (MΦ′1 +mΦ′

2 +XTΦ′3)

∫ T

0

ψ(Yt)dt. (3.74)

Therefore, we have∫ T

0

DtΦ ·

(ψ(Yt)∫ T

0ψ(Yt)dt

)dt = Φ′

1 ·M + Φ′2 ·m+ Φ′

3 ·XT (3.75)

Substituting Equation (3.75) into Equation (3.68) we get

∆ = E[e−rT∫ T

0

DtΦ ·

(ψ(Yt)∫ T

0ψ(Yt)dt

)dt]. (3.76)

The desired result follows by an application of Equation (2.22). The general case follows by

a density argument. 2

Following the same procedure as above we deduce that

Γ =∂2

∂x2E[e−rTΦ(M,m,XT )] = E[e−rTΦ(M,m,XT )π] (3.77)

with

π = δ

(δ

(ψ(Y )∫ T

0ψ(Yt)dt

)ψ(Y )∫ T

0ψ(Yt)dt

)− δ

(ψ(Y )∫ T

0ψ(Yt)dt

).

Remark

The computation of Greeks for Barrier options and Look-back options gives an idea of how

to handle discontinuous payoff functions.

3.5 Greeks for the Heston model

We mention that the calculations of Greeks using Malliavin calculus introduced in [35] fo-

cussed on models with deterministic volatility. Here, we use Malliavin calculus to compute

Greeks for models with stochastic volatility where the underlying asset is driven by a Brow-

nian motion. In particular, we consider the Heston model.

61

Heston (1993) assumed that the stock price process is described by a stochastic differential

equation:

dSt = St(bdt+√vtdW

(1)t ) (3.78)

where W (1)t , 0 ≤ t ≤ T denotes a Brownian motion, St is the stock price at time t, b is the

drift coefficient and√vt is the volatility. In addition, Heston proposed that the variance be

driven by a mean-reverting stochastic process of the form

dvt = κ(θ − vt)dt+ ν√vtdW

(2)t (3.79)

where W (2)t , 0 ≤ t ≤ T denotes a Brownian motion, θ is the long-run mean, κ is the rate of

mean reversion and ν is called volatility of variance (often called volatility of volatility). We

assume that the dynamics described by Equations (3.78) and (3.79) are, under a risk neutral

measure, chosen by the market and that the risk neutral measure is given by P . W(1)t and

W(2)t are two correlated standard Brownian motions. We have

dW(1)t dW

(2)t = ρdt, ρ ∈ (−1, 1) (3.80)

where ρ here represents the correlation coefficient between the two standard Brownian mo-

tions W(1)t and W

(2)t . The Heston model is identical to the Black-Scholes model except that

the volatility is allowed to be stochastic. Hence, this model is a generalization of the Black-

Scholes model to the case of stochastic volatility. The Heston model is described in detail in

[39] and is popular in industry because of its quasi-closed form solution for European options

in terms of Fourier-transforms. We will not use the quasi-closed form solution; instead we

investigate how the Malliavin calculus can be used to compute Greeks for the Heston model.

We observe that the Heston model does not satisfy the standard assumptions of a stochastic

differential equation. The square root√vt is neither a differentiable function of vt at vt = 0

nor even Lipschitz continuous. Hence, we cannot directly apply Malliavin calculus approach

to compute Greeks for the Heston model.

To apply Malliavin calculus in the context of the Heston model, we need vt to satisfy cer-

tain conditions as we will show below. To ensure existence and uniqueness of a solution of

Equations (3.78) and (3.79), b, κ, θ and ν are assumed to be strictly positive constants, and

in addition we require that 2κθ ≥ ν2 to ensure that the variance process is always positive,

that is,

P (vt > 0, ∀ t > 0) = 1.

The condition 2κθ ≥ ν2 is called the Novikov condition. Assuming now that v0 > 0 and that

the Novikov condition holds, we consider the square root process

σt :=√vt. (3.81)

62

By an application of the Ito formula, we have

dσt =∂

∂t(√vt)dt+

∂

∂v(√vt)dvt +

1

2ν2vt

∂2

∂v2(√vt)dt

= (κθ

2− ν2

8

)1

σt− 1

2κσtdt+

ν

2dW

(2)t . (3.82)

The Novikov condition implies in particular that(κθ

2− ν2

8

)≥ 0. (3.83)

It is not apparent that σt, the solution of the stochastic differential equation in (3.82), admits

a unique strong solution, but the Yamada-Wanatabe lemma (see [56] Chapter 5 Proposition

2.18) implies, under the Novikov condition, that vt is a unique solution of the stochastic

differential equation (3.79). If σt is a solution of Equation (3.82) then we can use Ito formula

to show that σ2t is a solution of Equation (3.79) satisfying the condition v0 > 0. As σ2

t is

a unique solution of Equation (3.79) we conclude uniqueness of the solution of stochastic

differential equation (3.82) up to a sign. However, if σt satisfies Equation (3.82) it is obvious

that−σt does not and therefore, for v0 > 0, we have uniqueness of the solution σt of stochastic

differential equation (3.82).

Assuming that the volatility σt is Malliavin differentiable, that is, σt ∈ D1,2 we calculate ∆

as follows. We first construct the Heston stochastic volatility model with correlation ρ from

two independent Brownian motions, which consists of a stock St and a variance process vt

satisfying Equations (3.78) and (3.79) (see [33]). Given an arbitrary W (2) there exists Z

which is independent of W (1) and W (2). Then we can express W (1) as follows

W(1)t = ρW

(2)t +

√1− ρ2Zt. (3.84)

It is convenient in the sequel to think of the dynamics described by Equations (3.78) and

(3.79) as driven by Zt and W(2)t respectively rather than W

(1)t and W

(2)t ) respectively. Sub-

stituting Equation (3.84) into Equation (3.78) we obtain

dSt = St

(bdt+

√vt(ρdW

(2)t +

√1− ρ2dZt)

)(3.85)

with respect to some initial condition S0 = x. In the following we will work with the

logarithmic price logSt rather than the actual price to ensure that the stock price is always

positive. Let Xt = logSt. By an application of Ito formula, we obtain

dXt = (b− vt2

)dt+ ρ√vtdW

(2)t +

√vt√

1− ρ2dZt. (3.86)

63

We can write the Equations (3.82) and (3.86) in integral forms as

Xt = log x+

∫ t

0

(b− 1

2vs

)ds+ ρ

∫ t

0

√vsdW

(2)s +

√1− ρ2

∫ t

0

√vsdZs. (3.87)

σt = σ0 +

∫ t

0

((κθ

2− ν2

8

)1

σs− 1

2κσs

)ds+

∫ t

0

ν

2dW (2)

s . (3.88)

It can be proved (see [22] page 17) that for any parameter choice with 2κθ > ν2 and for any

T > 0 we have

sup0≤t≤T

E[σ−2t ] <∞. (3.89)

The two Equations (3.87) and (3.88) can be thought of as a single two dimensional stochastic

differential equation of the form(Xt

σt

)=

(log x

σ0

)+

∫ t

0

(b− 12σ2s)

(κθ2− ν2

8

)1σs− 1

2κσs

ds

+

∫ t

0

( √1− ρ2σs ρσs

0 ν2

)(dZs

dW(2)s

).

We mention that the exact value of the joint density of Xt and σt is unknown.

The inverse matrix of ( √1− ρ2σs ρσs

0 ν2

)is calculated as follows

2

ν√

1− ρ2σs

(ν2

− ρσs

0√

1− ρ2σs

)=

1√1−ρ2σs

− 2ρ

ν√

1−ρ2

0 2ν

.

We calculate the first variation process Yt of

(Xt

σt

)as follows

Yt :=∂

∂x

(Xt

σt

)=

(1x

0

).

An application of Proposition 3.0.4 with a ∈ Υn gives

∆ = E[e−rTΦ(XT )

∫ T

0

a(s)(σ(Xt)−1Yt)

TdWt]

= E[e−rTΦ(XT )

∫ T

0

a(s)

1√1−ρ2σs

− 2ρ

ν√

1−ρ2

0 2ν

( 1x

0

)T (dZs

dW(2)s

)].

64

Choosing a(s) = 1T

and making use of the matrix property that (AB)T = BTAT , for A and

B matrices, we have


∫ T

0

1

T

(1x

0) 1√

1−ρ2σs

0

− 2ρ

ν√

1−ρ22ν

( dZs

dW(2)s

)]

= E[e−rTΦ(XT )

∫ T

0

1

T

(1

x√

1−ρ2σs

0)( dZs

dW(2)s

)]

= E[e−rTΦ(XT )

∫ T

0

1

xT√

1− ρ2σsdZs].

Thus we have proved the following result.

Theorem 3.5.1 Let the stock price and the variance be given by Equations (3.78) and (3.79)

respectively. Then, for any Φ : R → R of polynomial growth, we have


∫ T

0

1

xT√

1− ρ2σsdZs]. (3.90)

Remark

This result has also been obtained in [22] using an approach different from Proposition 3.0.4.

We now show, following the construction in [2], that σt is Malliavin differentiable. First we

define an approximating sequence ΦN(y). Let ΦN(y), N > 0 be a continuously differentiable

function satisfying

ΦN(y) =

1 if y ≥ 2N

0 if y < N.

In addition, ΦN(y) ≤ 1 for all y ∈ R. Then

Φ′N(y) =

0 if y > 2N

0 if y < N.

Furthermore, we define the function

ΛN(y) := ΦN(y)1

ywith ΛN(0) := 0.

We assume that the function ΛN(y) is bounded and continuously differentiable and it satisfies

Λ′N(y) = Φ′

N(y)1

y− ΦN(y)

1

y2. (3.91)

65

In particular, we have

Λ′N(y) =

−1y2

if y ≥ 2N

0 if y < N.

Then, we define the approximating sequence σNt of σt as the solution of the stochastic

differential equation

dσNt = [

(κθ

2− ν2

8

)ΛN(σNt )− κ

2σNt ]dt+

ν

2dWt (3.92)

with σN0 = σ0 for all N > 0.

Proposition 3.5.2 For each t ∈ [0, T ] the sequence σNt converges to σt in L2(Ω).

Proof

The proof can be found in [2]. We omit the details. 2

Proposition 3.5.3 Assuming that 2κθ ≥ ν2 we have σt ∈ D1,2 and for r < t we have

Drσt =ν

2exp

(∫ t

r

[−κ2−(κθ

2− ν2

8

)1

σ2s

]ds

). (3.93)

Proof

We give an outline of the proof. By Proposition 3.5.2 the sequence σNt converges to σt in

L2(Ω) for each t ∈ [0, T ]. This convergence is pointwise and hence we conclude by using the

properties of the function ΛN(x) that

DrσNt =

ν

2exp

(∫ t

r

[−κ2−(κθ

2− ν2

8

)Λ′N(σNs )]ds

)converges pointwise to

G := Drσt =ν

2exp

(∫ t

r

[−κ2−(κθ

2− ν2

8

)1

σ2s

]ds

).

Using the Novikov condition, we see that the exponent in DrσNt is negative for all choices

of N and therefore | DrσNt |≤ ν

2for all N . By means of the bounded convergence theorem

we conclude that DrσNt converges to G in L2(Ω). Using Lemma 1.2.3 in [70] we have that

σt ∈ D1,2 and the result follows. 2

Remark

All the representations of Greeks we have obtained in this chapter can be evaluated by Monte

Carlo methods.

66

Chapter 4

Application of white noise calculus for

Gaussian Processes to the Calculation

of Greeks

The theory of white noise analysis was first introduced in [45] and was originally applied in

quantum physics. Subsequently new applications have been found in stochastic differential

equations (see [47]). More recently, the white noise analysis has been applied to finance (see

[1], [17], [27], [31], [45], [46], [59], [61], [74] and the references therein).

The Malliavin calculus has been presented in the context of analysis on the Wiener space

Ω = C0([0, T ]), the space of all real continuous functions on [0, T ]. In order to calculate the

Malliavin derivative of a random variable we need to show first that the random variable

belongs to the domain of the Malliavin derivative. This, however, is restrictive as some in-

teresting options do not satisfy this condition, for example, the digital option. Nonetheless,

it is possible to obtain some extension of the domain of the Malliavin derivative to the whole

L2. This extension appears in the literature in the framework of white noise analysis (see

[1]) and Wiener setting (see [85] and [86]). The major application in the above papers is to

calculate the replicating portfolio of a given contingent claim.

The goal of this chapter is to derive explicit formulae for Greeks using Malliavin calculus

approach. We mention that similar results were obtained in [28] in the jump diffusion case.

The authors in [28] obtain a functional representation formula for functionals of jump diffu-

sions in terms of the Fourier transform from which they compute Greeks using the method

related to the likelihood method. The key idea in our case is to express the payoff function

in terms of the Donsker delta function (to be defined later on) and then use the Wick chain

67

rule to compute Greeks. The results of this chapter give a generalization of the computa-

tion of Greeks. The approach is advantageous because we can handle discontinuous and

path-dependent payoffs. In addition, we do not require the knowledge of the density of the

underlying asset.

In the following section we review white noise concepts for Brownian motion as developed

in [1], [17], [27], [47] and [74]. We explore a possible generalization of the computation of

Greeks using both Hida-Malliavin derivative and Hermite transforms. For general back-

ground information about white noise we refer to [61].

4.1 Basic concepts of Gaussian white noise analysis

Let L2(R) denotes the set of measurable functions satisfying

‖ f ‖2L2(R):=

∫Rf 2(t)dt <∞. (4.1)

We will work with the probability space Ω = S ′(R), which is the space of tempered distri-

butions, equipped with its Borel σ-algebra F = B(Ω). The space S ′(R) is the dual of the

Schwartz space S(R) of test functions, that is, the rapidly decreasing smooth functions on

R. By the Bochner-Minlos theorem (see [47] page 14) there exists a probability measure µ

on Ω such that ∫Ω

ei〈ω,f〉dµ(ω) = e− 1

2‖f‖2

L2(R) , f ∈ S(R) (4.2)

where i =√−1 and 〈ω, f〉 = ω(f) denotes the action of ω ∈ Ω = S ′(R) applied to f ∈ S(R).

The measure µ is called the White noise probability measure. The triple (Ω,B(Ω), µ) is called

the white noise probability space.

Lemma 4.1.1 Let f ∈ S(R). Then

E[〈·, f〉] = 0, f ∈ S(R). (4.3)

Moreover, we have the Ito isometry

E[〈·, f〉2] =‖ f ‖2L2(R) for all f ∈ S(R). (4.4)

Proof

Letting f(y) = tg(y), t ∈ R in Equation (4.2) we get∫Ω

ei〈ω,tg〉dµ(ω) = e− 1

2t2‖g‖2

L2(R) .

68

Performing Taylor expansions on both sides we have∫Ω

(1 + i〈ω, tg〉 − 1

2〈ω, tg〉2 + · · · )dµ(ω) = 1− 1

2t2 ‖ g ‖2

L2(R) +1

4t4 ‖ g ‖4

L2(R) + · · ·

which is equivalent to∫Ω

dµ(ω) + it

∫Ω

〈ω, g〉dµ(ω)− 1

2t2∫

Ω

〈ω, g〉2dµ(ω) + · · · = 1− 1

2t2 ‖ g ‖2

L2(R)

+1

4t4 ‖ g ‖4

L2(R) + · · ·

Now comparing terms with the same powers of t we have

it

∫Ω

〈ω, g〉dµ(ω) = 0

which implies that

E[〈ω, f〉] = 0.

This proves Equation (4.3). Comparing terms in t2 we have

−1

2t2∫

Ω

〈ω, g〉2dµ = −1

2t2 ‖ g ‖2

L2(R)

which implies

E[〈ω, f〉2] =‖ f ‖2L2(R) .

This proves Equation (4.4). 2

Using Lemma 4.1.1 we can extend the definition of 〈ω, f〉 from f ∈ S(R) to any f ∈ L2(R)

as follows. We take an approximating sequence fn∞n=1 ∈ S(R) such that

limn→∞

‖ fn − f ‖2L2(R)= 0. (4.5)

By using Equation (4.4) we see that the sequence 〈ω, fn〉∞n=1 is Cauchy in L2(µ), since

E[〈ω, fn〉 − 〈ω, fm〉]2 = E[〈ω, fn − fm〉2]

= ‖ fn − fm ‖2L2(R),

and hence the sequence converges in L2(µ). We denote the limit by 〈ω, f〉. This limit is

independent of the choice of the approximating sequence.

In particular, this makes

W (t) := W (t, ω) = 〈ω, χ[0,t](·)〉 (4.6)

well-defined since χ[0,t] is in L2(R) for all t ∈ R. By means of Kolmogorov’s continuity

theorem, the process W (t) can be shown to have a continuous version which we will denote

69

by W (t), t ∈ R, that is, W (t, ω) is continuous in t for all t, P (Wt = Wt) = 1.

From now on we work with the Brownian motion W (t), t ∈ R on the white noise probability

space (S ′(R),B(S ′(R)), µ). It can now be shown that W (t), t ∈ R is a Gaussian process

with mean

E[W (t)] = W (0) = 0 (4.7)

and covariance

E[W (t1)W (t2)] = E[

∫Rχ[0,t1]dWs

∫Rχ[0,t2](s)dWs]

= min(t1, t2).

(See [17] page 49). W (t) is a Brownian motion with respect to the probability law µ. A

function of the type f(t) =∑

k akχ[tk,tk+1)(t) where t0 < t1 < t2 < . . . < tN and ak ∈ Ris called a simple function and belongs to L2(R). Using Equation (4.6) and the linearity

property of f we obtain

〈ω, f〉 =N−1∑k=0

ak〈ω, χ[tk,tk+1)(·)〉 =N−1∑k=0

ak(W (tk+1)−W (tk)) =N−1∑k=0

ak∆W (tk). (4.8)

Riemann sums suggest that we define

〈ω, f〉 =

∫Rf(t)dWt for all f ∈ L2(R). (4.9)

We introduce Hermite polynomials and Hermite functions (see [1]). Hermite polynomials

play an important role in probability theory. In what follows we let

hn(x) := (−1)ne12x2 dn

dxn

(e−

12x2), x ∈ R, n = 0, 1, 2, ... (4.10)

denote the Hermite polynomials. This gives, for example,

h0(x) = 1, h1(x) = x, h2(x) = x2 − 1, h3(x) = x3 − 3x, (4.11)

h4(x) = x4 − 6x2 + 3, h5(x) = x5 − 10x3 + 15x, · · · (4.12)

The generating function of Hermite polynomial is

exp(tx− t2

2) =

∞∑n=0

tn

n!hn(x) for all t, x ∈ R. (4.13)

We have the following properties of the Hermite polynomials:

h′n(x) = nhn−1(x) and hn+1(x) = xhn(x)− nhn−1(x), n ≥ 1. (4.14)

70

Let ξn∞n=1 be the Hermite functions defined by

ξn(x) := π−14 ((n− 1)!)−

12hn−1(

√2x)e−

12x2

, n = 1, 2, ... (4.15)

The Hermite functions ξn form an orthonormal basis for L2(Rn) (see [82]) and

| ξn(x) |≤

Cn−

112 if | x |≤ 2

√n

Ce−γx2

if | x |>√n

where C and γ are positive constants independent of n (see [31]). Most of the theory can be

developed without the explicit use of a particular basis. However, the choice of an explicit

basis is crucial when we want to deduce that the white noise (to be defined later on) belongs

to the space (S)∗ of Hida distribution.

Let J denote the set of all multi-indices α = (α1, α2, · · · ) of finite length l(α) = maxi, αi 6=0 with non-negative integers αi ∈ N ∪ 0 = 0, 1, 2, . . . for all i. Then, for α =

(α1, . . . , αn) ∈ J we put α! = α1!α2! · · ·αn! and | α |= α1 + α2 + · · · + αn. We can

construct an orthogonal L2(µ) basis Hα(ω)α∈J given by

Hα(ω) := hα1(〈ω, ξ1〉)hα2(〈ω, ξ2〉) · · ·hαn(〈ω, ξn〉), ω ∈ Ω (4.16)

where 〈ω, ·〉 and ξj (resp. hj, j = 1, 2, . . . , n) are Hermite functions (resp. Hermite polyno-

mials). Thus, for example,

H(4,0,3,2)(ω) = h4(〈ω, ξ1〉)h0(〈ω, ξ2〉)h3(〈ω, ξ3〉)h2(〈ω, ξ4〉)

= (〈ω, ξ1〉4 − 6〈ω, ξ1〉2 + 3)(〈ω, ξ3〉3 − 3〈ω, ξ3〉)(〈ω, ξ4〉2 − 1)

where we have applied Equations (4.11) and (4.12). The family Hαα∈J is an orthogonal

sequence that constitutes a basis for the Hilbert space L2(µ). The unit vectors

ε(k) = (0, 0, . . . 0, 1, 0, . . . , 0) (4.17)

with 1 on the kth entry, 0 otherwise, k = 1, 2, . . . are important special cases of multi-indices

(see [27]). We note that

Hε(k)(ω) = h1(〈ω, ξk〉) = 〈ω, ξk〉 =

∫Rξk(t)dWt. (4.18)

More generally, by a fundamental result of Ito [52], we have

In(ξ⊗α) = Hα(ω) (4.19)

71

with H0 := 1. ⊗ and ⊗ denote the tensor product and the symmetrized tensor product

respectively. For example, if f and g are real functions on R then

(f ⊗ g)(x1, x2) = f(x1)g(x2)

and

(f⊗g)(x1, x2) =1

2[f(x1)g(x2) + f(x2)g(x1)].

Lemma 4.1.2 Hαα∈J constitutes an orthogonal basis for L2(µ). Moreover, if

α = (α1, α2, . . .) ∈ J we have the norm expression

‖ Hα ‖2= α! := α1!α2! · · · (4.20)

Proof

The proof is given in [47] on page 24. We omit the details. 2

We can show that E[HαHβ] = α!δαβ. So if F =∑

α∈J aαHα(ω) we have

E[F ·Hβ] = E[

(∑α∈J

aαHα(ω)

)·Hβ] = aαβ!, (4.21)

which gives

aα =1

β!E[F ·Hβ], β! 6= 0. (4.22)

We now state the chaos decomposition for the elements of L2(µ) (see [27], Theorem 5.2).

Theorem 4.1.3 Let F ∈ L2(µ), be an FT -measurable random variable. Then there exists a

unique family aαα∈J of constants aα ∈ R such that

F (ω) =∑α∈J

aαHα. (4.23)

Moreover, the Ito isometry is valid:

‖ F ‖2L2(µ)=

∑α∈J

a2α ‖ Hα ‖2

L2(µ)=∑α∈J

a2αα! (4.24)

72

Example

For each t ∈ R, the random variable W (t) ∈ L2(µ) has the expansion

W (t) = 〈ω, χ[0,t](·)〉 = 〈ω,∞∑k=1

(χ[0,t], ξk)L2(R)ξk(·)〉

=

∫R

∞∑k=1

(χ[0,t], ξk)L2(R)ξk(s)dWs =∞∑k=1

(χ[0,t], ξk)L2(R)

∫Rξk(s)dWs

=∞∑k=1

(∫ t

0

ξk(s)ds

)∫Rξk(s)dWs =

∞∑k=1

(∫ t

0

ξk(s)ds

)Hε(k)(ω) (4.25)

where, in general, (f, g)L2(R) =∫

R f(t)g(t)dt.

Lemma 4.1.4 ∫Rf(s)dWs =

∞∑k=1

(ξk, f)L2(R)Hε(k)(ω) for f ∈ L2(R). (4.26)

Following the construction of the Wiener-Ito chaos expansion discussed in Chapter 2, we can

formulate an equivalent theorem to Theorem 4.1.3 in terms of the multiple Ito integral as

follows (see [27], Theorem 5.3).

Theorem 4.1.5 Let F ∈ L2(µ), be an FT -measurable random variable. Then there exists a

unique sequence fn∞n=1 of functions fn ∈ L2(Rn) such that

F (ω) =∞∑n=0

∫Rn

fn(t)dW⊗n(ω) =

∞∑n=0

In(fn). (4.27)

Moreover, the following Ito isometry is valid:

‖ F ‖2L2(µ)=

∞∑n=0

n! ‖ fn ‖2L2(Rn) . (4.28)

Remark

The connection between the two expansions in Theorem 4.1.3 and Theorem 4.1.5 is given by

fn =∑

α∈J :|α|=n

aαξ⊗α n = 0, 1, 2, . . . (4.29)

where aα are the coefficients in the expansion in Hermite functions given in Theorem 4.1.3.

73

4.2 Stochastic test functions and stochastic distribu-

tion functions

From Theorem 4.1.3, the condition∑α∈J

a2α ‖ Hα ‖2

L2(µ)<∞ (4.30)

ensures that

F =∑α∈J

aαHα ∈ L2(µ). (4.31)

The condition (4.30) can be replaced by various other conditions.

Definition 4.2.1 For 0 ≤ ρ ≤ 1 the Kondratiev test function space (S)ρ consists of all

f =∑

α∈J aαHα ∈ L2(µ), aα ∈ R such that

‖ f ‖2ρ,k:=

∑α∈J

(α!)1+ρa2α(2N)kα <∞ for all k ∈ N (4.32)

where (2N)kα := (2 · 1)kα1(2 · 2)kα2 · · · (2 · j)kαj if kα = (kα1, . . . , kαj) ∈ J .

Definition 4.2.2 For 0 ≤ ρ ≤ 1 the Kondratiev distribution space (S)−ρ consists of all

formal series F =∑

α∈J bαHα ∈ L2(µ), bα ∈ R such that

‖ F ‖2−ρ,−q:=

∑α∈J

(α!)1−ρb2α(2N)−qα <∞ for some q ∈ N (4.33)

(S)ρ is endowed with the projective limit topology and (S)−ρ is endowed with limit topology

induced by the above seminorms. We note that for any f =∑

α∈J aαHα ∈ (S)ρ and

F =∑

α∈J bαHα ∈ (S)−ρ the action

〈F, f〉 :=∑α∈J

aαbαα! (4.34)

is well defined and thus the space (S)−ρ is the dual of (S)ρ

For general 0 ≤ ρ ≤ 1 we have the following inclusions

(S)1 ⊂ (S)ρ ⊂ (S)0 ⊂ L2(µ) ⊂ (S)−0 ⊂ (S)−ρ ⊂ (S)−1. (4.35)

Remark

The spaces (S)0 and (S)−0 coincide with the Hida spaces (S) and (S)∗, respectively, which

74

we introduce below.

The stochastic (Hida) test functions (S) and stochastic (Hida) distribution space (S)∗ relates

to L2(µ) in a natural way (see [17], [27], [47] and [74]). We use Theorem 4.1.3 and Theorem

4.1.5 to define the Hida test function space (S) which is a subspace of L2(µ) and the Hida

distribution space (S)∗ which is a superset of L2(µ) as follows.

Definition 4.2.3 We define the Hida space (S) of stochastic test functions to be all

f ∈ L2(µ) whose expansion

f(ω) =∑α∈J

aαHα ∈ L2(µ) (4.36)

satisfies

‖ f ‖2k:=

∑α∈J

(α!)a2α(2N)kα <∞ for all k ∈ N. (4.37)

Let (S)k be the completion of (S) in the norm ‖ · ‖k. Then (S)k is a Hilbert space and the

following inclusions are continuous

(S) ⊂ · · · ⊂ (S)k+1 ⊂ (S)k ⊂ · · · ⊂ (S)1.

It can be shown that (S) is complete if and only if

(S) =∞⋂k=1

(S)k.

((S)k, ‖ · ‖k); k ≥ 1 is a sequence of Hilbert space with respect to the norm ‖ · ‖k such

that (S)k+1 is continuously embedded in (S)k for each k. Let (S) =⋂∞k=1(S)k and endow

(S) with projective limit topology, that is, the coarsest topology such that for each k the

inclusion from (S) into (S)k is continuous. This topological space (S) is called the projective

limit of (S)k, k ≥ 1. A base of neighborhoods of zero in this projective limit topology is

given by the choice of ε > 0, k ≥ 1 and the set f ∈ (S)k; ‖ f ‖k< ε. The projective limit

topology of (S) is induced by a decreasing sequence of Hilbert spaces and therefore (S) is

a countably Hilbert space (see [61]). A sequence fk, k ≥ 1 converges to f in (S) with

respect to the norm ‖ · ‖k if and only if it converges to f in every Hilbert space (S)k, that

is,

‖ fn − f ‖k→ 0 as n→∞ for all k.

75

Definition 4.2.4 We define the Hida space (S)∗ of stochastic distributions to be the set of

formal expansions G(ω) =∑

α∈J bαHα such that

‖ G ‖2−q:=

∑α∈J

(α!)b2α(2N)−qα <∞ (4.38)

for some q ∈ N.

Let (S)∗−q be the completion of L2 with respect to the norm ‖ · ‖−q. Then (S)∗−q is a Hilbert

space. The dual space (S)∗ of (S) is given by

(S)∗ =∞⋃q=1

(S)∗−q

where (S)∗−q, q ≥ 1 are Hilbert spaces and we have the inclusions

(S)∗−1 ⊂ · · · ⊂ (S)∗−q ⊂ (S)∗−q−1 ⊂ · · · ⊂ (S)∗.

((S)∗−q, ‖ · ‖−q); q ≥ 1 is a sequence of Hilbert spaces with respect to the norm ‖ · ‖−qsuch that (S)∗−q is continuously imbedded in (S)∗−q−1 for each q. Let (S)∗ =

⋃∞q=1(S)∗−q

and endow (S)∗ with the finest topology such that for each q the inclusion from (S)∗−q into

(S)∗ is continuous. This topological space (S)∗ is called the inductive limit of the sequence

(S)∗−q; q ≥ 1 (see [61]). A base of neighborhoods of zero in this inductive limit topology

is given by the choice of ε > 0, q ≥ 1 and the set f ∈ (S)∗−q; ‖ f ‖−q< ε. A sequence

fk, k ≥ 1 converges to f in (S)∗ if and only if there exists some q such that fk ∈ (S)∗−q

and

‖ fk − f ‖−q→ 0 as k →∞, for some q,

that is, fk converges to f in (S)∗−q as k →∞.

Definition 4.2.5 If F =∑

α∈J aαHα ∈ (S)∗ we define the generalized expectation E[F ] of

F by

E[F ] := a0. (4.39)

Definition 4.2.6 The action of G(ω) =∑

α∈J bαHα ∈ (S)∗ on f(ω) =∑

α∈J aαHα ∈ (S)

is defined by

〈G, f〉 =∑α∈J

α!aαbα. (4.40)

76

Equation (4.40) is well-defined since∑α∈J

aαbαα! =∑α∈J

aαbα(α!)1/2(α!)1/2(2N)−qα/2(2N)qα/2

≤

(∑α∈J

a2α(α!)(2N)−qα

) 12(∑α∈J

b2α(α!)(2N)qα

) 12

< ∞

for q large enough. We have the inclusions:

(S) ⊂ L2(µ) ⊂ (S)∗. (4.41)

The quantity (2N)α plays an important role in white noise theory. We note that if α = ε(k)

we obtain

(2N)ε(k)

= 2k. (4.42)

We can, in a natural way, define (S)∗-valued integrals as follows (see [17], [46] and [47]).

Definition 4.2.7 Suppose Z : R → (S)∗ has the property that

〈Z(t), f〉 ∈ L1(R) for all f ∈ (S). (4.43)

Then∫

R Z(t)dt is defined to be the unique element of (S)∗ such that

〈∫

RZ(t)dt, f〉 =

∫R〈Z(t), f〉dt for all f ∈ (S). (4.44)

We can show that Equation (4.44) defines∫

R Z(t)dt as an element of (S)∗ (see [46] Proposi-

tion 8.1). If expression (4.43) holds, we say that Z(t) is integrable in (S)∗.

One of the important features of the Hida space (S)∗ is that it contains the singular white

noise Wt for all t (see [74]). By formally differentiating (4.25) we arrive at the following

definition.

Definition 4.2.8 The white noise process W (t) is defined by the following formal expansion

W (t) =∞∑k=1

ξk(t)Hε(k)(ω), t ∈ R (4.45)

where ξk(t) is the Hermite function and ε(k) is given in Equation (4.17).

77

The following lemma says that the white noise W (t) belongs to (S)∗.

Lemma 4.2.9 For each t ∈ R, W (t) is a generalized function, that is, W (t) ∈ (S)∗.

Proof

We need to show that the expansion (4.45) satisfies the condition (4.38). To do this we recall

that

| ξk(t) |≤

Ck−

112 if | t |≤ 2

√k

Ce−γt2

if | t |>√k

for some constants C and γ independent of k and the well-known estimate

supt∈R

| ξk(t) |= O(k−112 ). (4.46)

For each t, we have

‖ W (t) ‖2−q :=

∞∑k=1

ξ2k(t)ε

(k)!((2N)ε(k)

)−q

=∞∑k=1

ξ2k(t)(2k)

−q by (4.42)

≤ C∑k=1

k−16 (2k)−q

= C∞∑k=1

k−16−q2−q

for some constant C. Hence, for any q ≥ 2, we have

‖ W (t) ‖2−q<∞ (4.47)

and so W (t) ∈ (S)∗ for all t. Thus, W (t) is a generalized function. 2

A process B : R → (S)∗ is differentiable in (S)∗ if the limit

limh→0

B(t+ h)−B(t)

h(4.48)

exists in (S)∗ for all t. We denote this limit by ddtB(t).

ddtB(t) = B′

t if and only if there exists a q ∈ N such that

limh→0

‖ B(t+ h)−B(t)

h−B′

t ‖−q= 0.

78

Lemma 4.2.10d

dtW (t) exists in (S)∗ for all t ∈ R. (4.49)

Proof

By Equation (4.25) we have

W (t) =∞∑k=1

∫ t

0

ξk(s)dsHε(k)(ω) and W (t+ h) =∞∑k=1

∫ t+h

0

ξk(s)dsHε(k)(ω). (4.50)

Then

W (t+ h)−W (t) =∞∑k=1

∫ t+h

0

ξk(s)dsHε(k)(ω)−∞∑k=1

∫ t

0

ξk(s)dsHε(k)(ω)

=∞∑k=1

∫ t+h

t

ξk(s)dsHε(k)(ω)

and so, for h 6= 0, we have

1

hW (t+ h)−W (t) =

∞∑k=1

1

h

∫ t+h

t

ξk(s)dsHε(k)(ω). (4.51)

Set

Wh(t) :=∞∑k=1

1

h

∫ t+h

t

ξk(s)dsHε(k)(ω)

and so

| Wh(t) |≤ K, for fixed t

for K constant. We need to show that

Wh(t) →∞∑k=1

ξk(t)Hε(k)(ω) (4.52)

as h→ 0, that is, for some q ∈ N,

‖∞∑k=1

1

h

∫ t+h

t


ξk(t)Hε(k)(ω) ‖2−q→ 0 as h→ 0. (4.53)

We have

∞∑k=1

1

h

∫ t+h

t


ξk(t)Hε(k)(ω) =∞∑k=1

1

h

∫ t+h

t

[ξk(s)−ξk(t)]dsHε(k)(ω). (4.54)

Put

ak(h) :=1

h

∫ t+h

t

[ξk(s)− ξk(t)]ds.

79

Therefore we have

‖∞∑k=1

1

h

∫ t+h

t


ξk(t)Hε(k)(ω) ‖2−q =

∞∑k=1

| ak(h) |2 ε(k)!(2N)−qε(k)

by (4.38)

=∞∑k=1

| ak(h) |2 (2k)−q by (4.42).

Since supt∈R | εk(t) |= O(k−12 ), we see that

suphak(h), h ∈ [0, 1] k = 1, 2, . . . <∞.

So

| ak(h) |2 (2k)−q ≤ C(2k)−q

for some constant C and the right hand side is the term of a convergent series. In addition,

since

ak(h) → 0 as h→ 0, k = 1, 2, . . .

we have

‖∞∑k=1

1

h

∫ t+h

t


ξk(t)Hε(k)(ω) ‖2−q→ 0 as h→ 0

for all q ≥ 1 by dominated convergence and the result then follows by Lemma 4.2.9. 2

Remark

(S)∗ is too small for the purpose of solving stochastic ordinary and partial differential equa-

tions. However, we can find a unique solution in (S)−1.

4.3 The Wick product

In this section review the definition of a Wick product on the space (S)−1 (see [27]).

The Wick product was first introduced by G. Wick in the early 1950’s as a renormalization

technique in the theory of quantum physics. New applications have been developed in

stochastic analysis (see [27], [47] and [74]).

Definition 4.3.1 The Wick product FG of F =∑

α∈J aαHα ∈ (S)−1 and G =∑

β∈J bβHβ ∈(S)−1 with aα, bβ ∈ R is defined by

(F G)(ω) =∑α,β∈J

aαbβHα+β =∑γ∈J

( ∑α+β=γ

aαbβ

)Hγ(ω). (4.55)

80

The Wick product is a commutative, associative and distributive binary operation on each

of the spaces (S)1, (S), (S)∗ and (S)−1 (see [47] page 47).

As an example, we have

(Wt Wt)(ω) = W 2t (ω)− t (4.56)

which follows from the next lemma. We need the following property:

Hε(j)+ε(k) =

Hε(j)Hε(k) if j 6= k

H2ε(j)− 1 if j = k

Lemma 4.3.2 We have(∫Rf(s)dWs

)(∫

Rg(s)dWs

)=

(∫Rf(s)dWs

)·(∫

Rg(s)dWs

)−∫

Rf(s)g(s)ds (4.57)

for all f, g ∈ L2(R).

Proof

Let f, g ∈ L2(R). Then(∫Rf(s)dWs

)(∫

Rg(s)dWs

)=

(∞∑j=1

(ξj, f)L2(R)Hε(j)

)

(∞∑k=1

(ξk, g)L2(R)Hε(k)

)by (4.26)

=∞∑

j,k=1

(ξj, f)L2(R)(ξk, g)L2(R)Hε(j)+ε(k) by (4.55)

=

(∞∑j=1

(ξj, f)L2(R)Hε(j)

)·

(∞∑k=1

(ξk, g)L2(R)Hε(k)

)

−∞∑j=1

(ξj, f)L2(R)(ξj, g)L2(R) (by the property above)

=

(∫Rf(s)dWs

)·(∫

Rg(s)dWs

)−

∞∑j=1

(ξj, f)L2(R)(ξj, g)L2(R)

=

(∫Rf(s)dWs

)·(∫

Rg(s)dWs

)− (f, g)L2(R). 2

The Wick powers Xn, n = 0, 1, 2, . . . of X ∈ (S)−1 are defined as follows:

X0 := 1, and Xn := X X · · · X (n factors). (4.58)

The Wick exponential expX of X ∈ (S)−1 is defined by

expX :=∞∑n=0

1

n!Xn (4.59)

81

provided the series converges in (S)−1. We also have the following useful rules.

(X + Y )2 = X2 + 2X Y + Y 2 X, Y ∈ (S)−1 (4.60)

and

exp(X + Y ) = exp(X) exp(Y ) X,Y ∈ (S)−1. (4.61)

We note that

Hnε(k) = Hε(k) Hε(k) · · · Hε(k) (n times) by (4.58)

= Hε(k)+ε(k)+···+ε(k) by (4.55)

= Hnε(k) (4.62)

Lemma 4.3.3 We have

exp(〈ω, f〉) = exp〈ω, f〉 − 1

2‖ f ‖2 (4.63)

for f ∈ L2(R).

Proof

By basis independence (see [47] page 73) we assume that f = cε1 where c is a constant. We

have

exp〈ω, f〉 = exp(〈ω, cε1〉) =∞∑n=0

1

n!〈ω, cε1〉n by (4.59)

=∞∑n=0

cn

n!Hnε(1)(ω) =

∑n=0

cn

n!Hnε(1)(ω) by (4.62)

=∞∑n=0

cn

n!hn(〈ω, ε1〉) by (4.18)

= expc〈ω, ε1〉 −1

2c2

= exp〈ω, f〉 − 1

2‖ f ‖2

where the second last equality follows by the generating property of the Hermite polynomi-

als. 2

We now use the white noise and the Wick product to define an integration of a general class

of processes with respect to W (t) (see [47]).

82

Definition 4.3.4 If X is S∗-integrable, then so is X1[a,b] for all a, b ∈ R and we put∫ b

a

X(t)dt :=

∫RX(t)1[a,b](t)dt. (4.64)

Proposition 4.3.5 Suppose X : R → (S)∗ is such that X(t)Wt is integrable in (S)∗. Then∫RX(t)δWt =

∫RX(t) Wtdt. (4.65)

Proof

The proof follows by the arguments in [47] Theorem 2.5.9 on page 57. 2

Remark

The integral on the left hand side of Equation (4.65) denotes the Skorohod integral of the

stochastic process X(t) = X(t, ω). The integral on the right hand side of Equation (4.65)

is interpreted as an (S)∗-valued integral. We note that the integral on the right may exist

even if X is not Skorohod integrable. Thus, the right hand side of Equation (4.65) may be

regarded as an extension of the Skorohod integral.

Definition 4.3.6 Suppose X is an (S)∗-valued process such that∫RXt Wtdt ∈ (S)∗

then we call this integral the generalized Skorohod integral of X.

Lemma 4.3.7 Let f ∈ (S) and Gt ∈ (S)−q for all t ∈ R, for some q ∈ N. Put q = q + 1log 2

.

Then ∫R| 〈Gt Wt, f〉 | dt ≤‖ f ‖q

(∫R‖ Gt ‖2

−q dt

) 12

. (4.66)

The following theorem gives conditions for the generalized Skorohod integral to exist (see

[27] page 77).

Theorem 4.3.8 1. Suppose G : R → (S)−q satisfies∫

R ‖ Gt ‖2−q dt <∞ for some q ∈ N.

Then ∫RGt Wtdt exists in (S)∗.

2. Suppose F (t) and Fn(t), n = 1, 2, . . . . are elements of (S)∗ for all t ∈ R and∫R ‖ Fn(t)− F (t) ‖2

−q dt→ 0, n→∞. Then∫RFn(t) Wtdt→

∫RF (t) Wtdt, n→∞

in the weak∗-topology on (S)∗

83

Proof

1. The proof follows from Lemma 4.3.7 and Definition 4.2.7 .

2. We have

| 〈∫

R(Fn(t)− F (t)) Wtdt, f〉 | ≤

∫R| 〈(Fn(t)− F (t)) Wt, f〉 | dt by (4.44)

≤ ‖ f ‖q∫

R‖ Fn(t)− F (t) ‖2

−q dt by (4.66)

→ 0, n→∞. 2

4.4 The Hermite Transform

The Wick product faces challenges when limit operations are involved in computations. To

handle such situations we use a transformation called Hermite transform or H-transform

(see [47] Section 2.6) which transforms an element F ∈ (S)−1 into deterministic functions

HF (z1, z2, . . .) of complex variables zj ∈ C, j = 1, 2, . . . with values in C.

Definition 4.4.1 Let F (ω) =∑

α∈J cαHα(ω) ∈ (S)−1. Then the Hermite transform of F ,

denoted by HF or F , is defined by

HF (z) = F (z) =∑α∈J

cαzα ∈ C (4.67)

where z = (z1, z2, . . . , ) ∈ CN (the set of all sequences of complex numbers) and

zα = zα11 zα2

2 · · · zαnn · · · if α = (α1, α2, . . .) ∈ J where z0

j = 1.

One can show that the sum in Equation (4.67) converges on the infinite dimensional neigh-

borhood

Kq(R) = (z1, z2, . . .) ∈ CN;∑α∈J

(2N)qα | zα |2< R2 (4.68)

for some 0 < q,R <∞ (see [47] Proposition 2.6.5).

Example

Let W (x, ω) =∑∞

j=0

∫ x0ξj(s)dsHε(j)(ω). Then we have

HW (z) = W (x)(z) =∞∑j=0

∫ x

0

ξj(s)dszj, z = (z1, z2, . . .) ∈ CNc .

The following proposition is an immediate consequence of Definitions 4.3.1 and 4.4.1.

84

Proposition 4.4.2 If F,G ∈ (S)−1 then

H(F G)(z) = HF (z) · HG(z). (4.69)

for all z such that HF (z) and HG(z) exist. In general

H(f (F ))(z) = f(HF (z)) (when convergent) (4.70)

if f : C → C is entire, f(R) ⊂ R and f (F ) :=∑

α∈J cαFn ∈ (S)−1.

One can use the Hermite transform to characterize distributions in (S)−1. This is given in

the next theorem. Its proof is found in [47] on page 68.

Theorem 4.4.3 1. If F =∑

α∈J aαHα ∈ (S)−1 then there exists q,Mq <∞ such that

| HF (z) |≤∑α∈J

| aα || zα |≤Mq

(∑α∈J

(2N)qα | zα |2) 1

2

for all z ∈ (CN)c. In particular, HF is a bounded analytic function on Kq(R) for all

R <∞.

2. Conversely, assume that g(z) :=∑

α∈J bαzα is a power series of z ∈ (CN)c such that

there exist q < ∞ and δ > 0 with g(z) absolutely convergent and bounded on Kq(δ).

Then there exists a unique G ∈ (S)∗ such that HG = g, namely

G =∑α∈J

bαHα.

The Hermite transform also serves as a useful tool to describe the topology of (S)−1. In

particular, the convergence of sequences of Hida distributions can be characterized as follows.

Theorem 4.4.4 A sequence Xn, n ≥ 1 converges to X in (S)−1 if and only if there exists

q,M <∞, R > 0 such that

supz∈Kq(R)

| HXn(z) |≤M (4.71)

for all n ≥ 1 and

HXn(z) → HX(z) (4.72)

as n→∞ for all z ∈ Kq(R).

85

Proof

Details of the proof is given in [47] on page 81. 2

We have the following important lemma (see [47] pages 85.

Lemma 4.4.5 Suppose X(t, ω) and F (t, ω) are (S)−1 processes such that

dX(t, z)

dt= F (t, z) for each t ∈ (a, b), z ∈ Kq(δ) (4.73)

and that F (t, z) is a bounded function of (t, z) ∈ (a, b) × Kq(δ), continuous in t ∈ (a, b) for

each z ∈ Kq(δ). Then X(t, ω) is a differentiable (S)−1 process and

dX(t, ω)

dt= F (t, ω) (4.74)

for all t ∈ (a, b).

Definition 4.4.6 An (S)−1-process X is strongly integrable over an interval [a, b] if

∫ b

a

X(t, ω)dt := lim4tk→0

n−1∑k=0

X(t∗k, ω)4tk (4.75)

exists in (S)−1 for all partitions a = t0 < t1 < . . . < tn = b of [a, b], 4tk = tk+1−tk and

t∗k ∈ [tk, tk+1] for k = 1, . . . , n− 1.

Taking the Hermite transform in Equation (4.75) and using Lemma 4.4.5, we have the

following result (see [47] page 86).

Lemma 4.4.7 Let X(t) be an (S)−1 process. Suppose there exists q <∞, δ > 0 such that

supX(t, z) : t ∈ [a, b], z ∈ Kq(δ) <∞ (4.76)

and X(t, z) is a continuous function of t ∈ [a, b] for each z ∈ Kq(δ). Then X(t) is strongly

integrable and

H(∫ b

a

X(t)dt

)=

∫ b

a

X(t)dt (4.77)

where the integral to the right is the Lebesgue integral.

We also have the following useful lemma.

86

Lemma 4.4.8 Let G be a bounded open subset of R+×R. Assume a process F : G→ (S)−1

with H(F ) = F such that F and its derivatives are bounded on G×Kq(R), continuous with

respect to t for all z ∈ Kq(R) and analytic in z ∈ Kq(R) for all t, q <∞, R > 0. Then

H(d

dtF

)=

d

dt(H(F )) =

d

dtF . (4.78)

on Kq(R).

Proof

The proof is based on Lemma 4.4.5. The mean value theorem implies that

F (t+ h)(z)− F (t)(z)

h=

d

dtF (t+ εh)(z)

for some ε ∈ [0, 1], for all z ∈ Kq(R). By the assumptions on F we conclude that

F (t+ h)(z)− F (t)(z)

h→ d

dtF (t)(z) as h→ 0 (4.79)

pointwise boundedly for z ∈ Kq(R). Since (S)1 is a nuclear space (see Lemma 2.8.2 in [47])

we can show using Theorem 4.4.4 that statement (4.79) is equivalent to convergence in (S)−1.

That is

F (t+ h)− F (t)

h→ d

dtF (t) in (S)−1

for all t. The result then follows since the Hermite transform is a continuous linear functional

in (S)−1 2

We have the following chain rule in (S)−1.

Proposition 4.4.9 Suppose that t→ Xt : R → (S)−1 is continuously differentiable and let

f : C → C be entire (analytic on C) such that f(R) ⊂ R and f (Xt) ∈ (S)−1 for all t, then

d

dtf (Xt) = (f ′)(Xt)

d

dtXt in (S)−1. (4.80)

87

Proof

H(f ′(Xt)

d

dtXt

)(z) = H(f ′(Xt))(z) · H

(d

dtXt

)(z) by (4.69)

= f ′(H(Xt)(z)) ·d

dtH(Xt)(z) by (4.70) and (4.78)

=d

dtf(H(Xt)(z))

=d

dtH(f (Xt))(z) by (4.70)

= H(d

dtf (Xt)

)(z) by (4.78)

The result follows by the uniqueness of the Hermite transform (see Theorem 4.4.3). 2

Example

d

dtexp(Wt) = exp(Wt)

dWt

dt= exp(Wt) Wt.

Note

The Hermite transform is closely related to the so called S-transform (see [46] and [47]

page 80). The S-transform maps random variables into non-random functionals. The use of

Hermite transform has some advantages, for instance it enables the application of methods

of complex analysis.

4.5 Hida-Malliavin derivative

Now that the basic white noise theory have been given, we can proceed to define the Hida-

Malliavin derivative. We follow the construction in [27].

Definition 4.5.1 Let F : S ′(R) → R be a given function and let γ(t) =∫ t

0g(s)ds be deter-

ministic, g ∈ L2([0, T ]). We say that F has a directional derivative in (S)∗ in the direction

γ if

DγF (ω) := limε→0

F (ω + εγ)− F (ω)

ε(4.81)

exists in (S)∗.

88

Example

Let F (ω) = 〈ω, f〉 =∫

R f(t)dWt(ω) for some f ∈ S(R). Then

DγF (ω) = limε→0

1

ε[〈ω + εγ, f〉 − 〈ω, f〉]

= limε→0

1

ε〈εγ, f〉 = 〈γ, f〉

=

∫Rf(t)γ(t)dt

for all γ ∈ L2([0, T ]).

Definition 4.5.2 We say that F : S ′(R) → R is differentiable if there exists a map

ψ : R → (S)∗ such that ψ(t)g(t) = ψ(t, ω)g(t) is (S)∗-integrable and

DγF =

∫Rψ(t, ω)g(t)dt for all g ∈ L2([0, T ]). (4.82)

Then DtF (ω) is defined to be ψ(t, ω). This is the Hida-Malliavin derivative of F at t in

(S)∗.

Example

Let F (ω) = 〈ω, f〉 =∫

R f(t)dWt(ω) for some f ∈ S ′(R). Then by the above example, F is

differentiable and its Hida-Malliavin derivative is

DtF (ω) = f(t) for all (t, ω).

Assume that the Hida-Malliavin derivative of F ∈ L2(µ) exists and suppose that ϕ is con-

tinuously differentiable, DtF ∈ L2(R) for all t ∈ R and ϕ′(F )DtF ∈ L2(λ × µ) then the

Hida-Malliavin derivative of ϕ(F ) exists and

Dtϕ(F ) = ϕ′(F )DtF. (4.83)

More generally, we have the following extension (see [27] page 82).

Theorem 4.5.3 Assume that the Hida-Malliavin derivative of F1, . . . , Fn ∈ L2(µ) exists and

that ϕ is continuously differentiable in Rn, DtFi ∈ L2(µ) for all t ∈ R and∑n

i=1∂ϕ(F )∂xi

DFi ∈L2(µ × λ) for i = 1, . . . , n where F = (F1, . . . , Fn). Then the Hida-Malliavin derivative of

ϕ(F ) exists and

Dtϕ(F ) =n∑i=1

∂ϕ(F )

∂xiDtFi. (4.84)

89

By using Equation (4.83) and the example above we have

Dt(hn(〈ω, f〉)) = h′n(〈ω, f〉)f(t) = nhn−1(〈ω, f〉)f(t) (4.85)

where hn(t)n≥0, t ∈ R are Hermite polynomials of order n and f ∈ L2(R). In particular,

choosing n = 1 we have

Dt(h1(〈ω, f〉) = f(t).

The following theorem gives some versions of the Wick chain rule for Hida-Malliavin deriva-

tives (see [27] page 87).

Theorem 4.5.4 1. Let F,G ∈ D1,2. Then F G ∈ D1,2 and

Dt(F G) = F DtG+DtF G, t ∈ R.

2. Let F ∈ D1,2. Then F n ∈ D1,2 and

Dt(Fn) = nF (n−1) DtF, n = 1, 2, . . .

3. Let F ∈ D1,2 be Hida-Malliavin differentiable and assume that

exp F =∞∑n=0

1

n!F n ∈ D1,2.

Then

Dt exp F = exp F DtF.

Proof

The proofs for parts 1 and 3 are given in [27] page 87. Here we give the proof to part 2 which

is not given. By the closability of the Hida-Malliavin derivative (see Theorem 6.12 in [27]

page 86) it suffice to prove 2 in the case when F = exp〈ω, f〉 = exp(〈ω, f〉 − 12‖ f ‖2

L2(R))

where f ∈ L2(R) is a deterministic function and 〈ω, f〉 =∫

R f(s)dWs. Then

DtFn = Dt(F F · · · F )

= Dt(exp〈ω, f〉 exp〈ω, f〉 · · · exp〈ω, f〉)

= Dt(exp〈ω, f + f + · · ·+ f〉) by (4.61)

= Dt(exp(〈ω, f + f + · · ·+ f〉 − 1

2‖ f + f + · · ·+ f ‖2

L2(R))) by (4.63)

= exp(〈ω, f + f + · · ·+ f〉 − 1

2‖ f + f + · · ·+ f ‖2

L2(R))

(f(t) + f(t) + · · ·+ f(t)) by (4.84)

= exp(〈ω, nf〉)nf(t) by (4.63). (4.86)

90

On the other hand,

nF (n−1) DtF = n(F F · · · F ) Dt(exp〈ω, f〉)

= n(exp〈ω, f〉 exp〈ω, f〉 · · · exp〈ω, f〉)

Dt(exp(〈ω, f〉 − 1

2‖ f ‖2

L2(R))) by (4.63)

= n exp(〈ω, f + f + · · ·+ f〉) exp(〈ω, f〉 − 1

2‖ f ‖2

L2(R))f(t) by (4.84)

= n exp(〈ω, (n− 1)f〉) exp(〈ω, f〉)f(t)

= n exp(〈ω, nf〉)f(t) by (4.63). (4.87)

The result then follows by comparing (4.86) and (4.87). 2

Example

For f ∈ L2(R) we have

Dt(〈ω, f〉n) = n〈ω, f〉n−1f(t) and Dt(〈ω, f〉n) = n〈ω, f〉(n−1) f(t) for a.a t. (4.88)

These examples illustrate that the Hida-Malliavin derivative satisfies the chain rule.

We recall that, for α = (α1, . . . , αn) ∈ J , we have

Hα(ω) = hα1(〈ω, ξ1〉) · · ·hαn(〈ω, ξn〉) = 〈ω, ξ1〉α1 · · · 〈ω, ξn〉αn . (4.89)

Motivated by the examples above, we now extend Definition 4.5.2 to elements in (S)∗ (see

[27] page 83).


α∈J cαHα ∈ (S)∗. Then we define the Hida-Malliavin

derivative DtF of F at t by

DtF (ω) :=∑α∈J

cα

∞∑k=1

αkHα−ε(k)(ω)ξk(t) =∑β∈J

(∞∑k=1

cβ+ε(k)(βk + 1)ξk(t)

)Hβ(ω) (4.90)

whenever the sum converges in (S)∗.

The following result shows the Hida-Malliavin derivative operator is closable.

Lemma 4.5.6 Let F ∈ Dom(Dt) ⊂ (S)∗ for a.a t and q ∈ N. Then

1.∫

R ‖ DtF ‖2−q dt ≤‖ F ‖2

−q, q ≥ 2q + 1log 2

.

91

2. DtF ∈ (S)∗ for a.a. t ∈ R.

3. Suppose Fn ∈ Dom(Dt) ⊂ (S)∗ for a.a t and for all n = 1, 2, . . . and Fn → F, n→∞in (S)∗. Then there exists a subsequence Fnk

, k = 1, 2, . . . such that

DtFnk→ DtF, k →∞, in (S)∗ (4.91)

for a.a. t ∈ R.

Proof

The proof is similar to the one given in [27] on page 96. We give it here for completeness.

1. Suppose F =∑

α∈J cαHα. Then, by Equation (4.90), we have

DtF =∑α∈J

∞∑k=1

cααkξk(t)Hα−ε(k) =∑β∈J

(∞∑k=1


)Hβ =

∑β∈J

gβ(t)Hβ.

where gβ(t) =∑∞

k=1 cβ+ε(k)(βk + 1)ξk(t). Since F ∈ (S)∗ there exists a q ∈ N such that

‖ F ‖2−q:=

∑α∈J

c2αα!(2N)−qα <∞.

We note that∫Rg2β(t)dt =

∫R

(∞∑k=1


)2

dt =∞∑k=1

c2β+ε(k)(βk + 1)2.

Therefore, using (x+ 1)e−x ≤ 1 for all x ≥ 0, we obtain∫R‖ DtF ‖2

−q dt =

∫R

∑β∈J

g2β(t)β!(2N)−qβd =

∑β∈J

∞∑k=1

c2β+ε(k)(βk + 1)2β!(2N)−qβ

=∑β∈J

∞∑k=1

c2β+ε(k)(βk + ε(k))!(2k)−βk

log 2 (2N)−2qβk

≤∑β∈J

∞∑k=1

c2β+ε(k)(βk + ε(k))!(2N)−q(β+ε(k))

≤∑α∈J

c2αα!(2N)−qα =‖ F ‖2−q

which proves 1 and hence 2.

3. For a given F and Fn, n = 1, 2, . . . we have∫R‖ Dt(Fn − F ) ‖2

−q dt ≤‖ Fn − F ‖2−q→ 0, n→∞

by 1, if q ∈ N is large enough. Hence, there exists a subsequence Fnk, k = 1, 2, . . . such that

‖ Dt(Fnk− F ) ‖2

−q→ 0, k →∞ for a.a t ∈ R. 2

92

4.6 Conditional expectation on (S)∗

Definition 4.6.1 Let F =∑∞

n=0

∫Rn fndW

⊗n ∈ (S)∗. Then the conditional expectation of

F with respect to Ft is defined by

E[F | Ft] =∞∑n=0

∫Rn

fn · χ[0,t]ndW⊗n (4.92)

when convergent in (S)∗

Lemma 4.6.2 Let F,G,E[F | Ft],E[G | Ft] ∈ (S)∗. Then

E[F G | Ft] = E[F | Ft] E[G | Ft]. (4.93)

Proof

Assume without loss of generality that F = In(fn) =∫

Rn fndW⊗n and

G = Im(gm) =∫

Rm gmdW⊗m for some fn ∈ L2(Rn) and gm ∈ L2(Rm). Then we have

E[F G | Ft] = E[In(fn) Im(gm) | Ft] = E[In+m(fn⊗gm) | Ft]

= E[

∫Rn+m

fn⊗gm · χ[0,t]n+mdW⊗(n+m) | Ft] by (4.27)

=

∫Rn+m

fn · χ[o,t]n⊗gm · χ[0,t]mdW⊗(n+m) by (4.92)

= E[F | Ft] E[G | Ft]. 2

Lemma 4.6.3 Suppose F ∈ (S)∗ and exp F ∈ (S)∗. Then

E[exp F | Ft] = exp E[F | Ft]. (4.94)

In particular, if F ∈ L1(µ), we have

E[exp F ] = expE[F ]. (4.95)

Proof

By Lemma 4.6.2 we have

E[exp F | Ft] = E[∞∑n=0

1

n!F n | Ft] =

n∑n=0

1

n!E[F | Ft]n = expE[F | Ft]

where we have used Equations (4.59) and (4.92). 2

93

Let P (x) =∑

α∈J cαxα be a polynomial where xα = xα1

1 xα22 , . . . x ∈ RN , cα ∈ R and x0

j = 1.

Then we can define its Wick version at X = (X1, . . . , Xn) ∈ ((S)∗)n by

P (X) =∑α∈J

cαXα.

Let X(t) = (X(t)1 , . . . , X

(t)n ) be of the form

X(t)i (ω) =

∫ t

0

ξi(s)dWs =

∫Rξi(s)1[0,t](s)dWs, i = 1, 2, . . .

Then t→ X(t)i is differentiable in (S)∗ and

d

dt

(∫ t

0

ξi(s)dWs

)=

d

dt

(∫ t

0

ξi(s)Wsds

)= ξi(t)Wt ∈ (S)∗. (4.96)

The following Wick chain rule for polynomials follows by induction (see [27] page 94).

Lemma 4.6.4 Let P (x) =∑

α∈J cαxα be a polynomial in x = (x1, . . . , xn) ∈ Rn. Let

X(t)j =

∫ t0ξj(s)dWs, j = 1, . . . , n. Then

d

dtP (X(t)) =

n∑j=1

(∂P

∂xj

)(X(t)) ξj(t)Wt. (4.97)

Proof

Since X(t)j =

∫R ξj(s)dWs, j = 1, . . . , n, we can write this, using Equation (4.65), as

X(t)j =

∫ t

0

ξj(s) Wsds.

Now

d

dtP (X(t)) =

n∑j=1

(∂P

∂xj

)(X(t))

d

dt

(∫ t

0

ξj(s) Wsds

)

=n∑j=1

(∂P

∂xj

)(X(t)) ξj(t)Wt. 2

4.7 The Donsker delta function

The Donsker delta function is a generalized white noise functional which has been studied

in several monographs within white noise analysis (see [27], [46], [59], [61] and the references

therein). Here we give its definition within the white noise framework as discussed in the

previous sections. We follow the presentation in [27].

94

Definition 4.7.1 Let X : Ω → R be a random variable which also belongs to (S)−1. Then

a continuous function

δX(·) : R → (S)−1

is called a Donsker delta function of X if it satisfies∫Rϕ(x)δX(x)dx = ϕ(X) a.e (4.98)

for all (measurable) ϕ : R → R such that the integral on the left hand side converges in

(S)−1.

Proposition 4.7.2 Suppose X is a normally distributed random variable with variance v >

0. Then δX is unique and is given by the expression

δX(x) =1√2πv

exp[−(x−X)2

2v] ∈ (S)−1 (4.99)

Proof

We follow the proof in [27] Proposition 7.2. Let

f(x) =1√2πv

exp[−(x−X)2

2v].

The characterization theorem for (S)−1 (see Theorem 4.4.3) ensures that f(x) ∈ (S)−1 for

all x and that x → f(x) is continuous for x ∈ R. We show that f(x) satisfies Equation

(4.98), that is, ∫Rϕ(x)f(x)dx = ϕ(X) a.s. (4.100)

We first assume that ϕ has the form

ϕ(x) = eλx for some λ ∈ C. (4.101)

Taking the Hermite transform of the left hand side of Equation (4.100) gives

H(∫

Rϕ(x)f(x)dx

)=

∫ReλxH(f(x))dx =

∫Reλx

1√2πv

exp

(−(x− X)2

2v

)dx (4.102)

where X = X(z) is the Hermite transform of X at z = (z1, z2, . . .) ∈ CN.

Put

E[eλZ ] =

∫Reλx

1√2πv

exp

(−(x− X)2

2v

)dx

95

where Z is a normally distributed random variable with mean X and variance v. Denote

the mean of X by m. Now Z := X −m+ X is such a random variable. Hence,

E[eλ(X−m+X)] =

∫Reλx

1√2πv

exp

(−(x− X)2

2v

)dx = exp(λX +

1

2λ2v)

where we have used a well known formula for the characteristic function of a normal random

variable in the last equality. We, therefore, have

H(∫

Rϕ(x)f(x)dx

)= exp

(λX +

1

2λ2v

)= H

(exp

(λX +

1

2λ2v

))by (4.70)

= H(exp(λX)) by (4.63)

= H(ϕ(X)) by (4.101).

This proves that Equation (4.100) holds for functions ϕ given by Equation (4.101). Therefore

Equation (4.100) also holds for linear combinations of such functions. By a density argument,

Equation (4.100) holds for all ϕ such that the integral on the left hand side of Equation

(4.98) converges. We then prove uniqueness in the following way: if %1 : R → (S)−1 and

%2 : R → (S)−1 are two continuous functions such that∫Rϕ(x)%i(x)dx = ϕ(X), i = 1, 2 (4.103)

for all ϕ such that the integral on the left hand side converges, then in particular Equation

(4.103) must hold for all continuous functions with compact support. But then clearly we

must have

%1(x) = %2(x) for a.a x ∈ R

and hence for all x by continuity. 2

Lemma 4.7.3 Let ψ : [0, T ] → R, φ : [0, T ] → R be deterministic functions such that∫ T0| ψs | ds <∞ and ‖ φ ‖2

L2([0,T ]):=∫ T

0φ2sds <∞. Define

Xt =

∫ t

0

ψsds+

∫ t

0

φsdWs, 0 ≤ t ≤ T. (4.104)

Then

exp

(− (x−XT )2

2 ‖ φ ‖2L2([0,T ])

)= exp

(− x2

2 ‖ φ ‖2L2([0,T ])]

)+

∫ T

0

exp

(− (x−Xt)

2

2 ‖ φ ‖2L2([0,T ])

) x−Xt

‖ φ ‖2L2([0,T ])

(ψt + φtWt)dt (4.105)

where W denotes the white noise of W .

96

Proof

The proof is a result of an application of the fundamental theorem of calculus and Proposition

4.4.9. Define % : [0, T ] → (S)−1 by

%t = exp

(− (x−Xt)

2

2 ‖ φ ‖2L2([0,T ])

), 0 ≤ t ≤ T. (4.106)

Then

%T = %0 +

∫ T

0

d%

dtdt

= exp

(− x2

2 ‖ φ ‖2L2([0,T ])

)+

∫ T

0

exp

(− (x−Xt)

2

2 ‖ φ ‖2L2([0,T ])

) d

dt

(− (x−Xt)

2

2 ‖ φ ‖2L2([0,T ])

)dt

= exp

(− x2

2 ‖ φ ‖2L2([0,T ])

)+

∫ T

0

exp

(− (x−Xt)

2

2 ‖ φ ‖2L2([0,T ])

) −2

x−Xt

2 ‖ φ ‖2L2([0,T ])

d

dt(x−Xt)

= exp

(− x2

2 ‖ φ ‖2L2([0,T ])

)+

∫ T

0

exp

(− (x−Xt)

2

2 ‖ φ ‖2L2([0,T ])

) x−Xt

‖ φ ‖2L2([0,T ])

(ψt + φtWt)dt 2

Next we state the main theorem for the Donsker delta function as presented in [27] (see

Theorem 7.4).

Theorem 4.7.4 Let φ : [0, T ] → R, α : [0, T ] → R, ψ = φα be deterministic functions such

that 0 <‖ φ ‖2L2([0,T ])=

∫ T0φ2sds <∞ and 0 ≤

∫ T0ψ2sds <∞. Define

Xt =

∫ t

0

φsαsds+

∫ t

0

φsdWs, 0 ≤ t ≤ T.

Let f : R → R be bounded. Then

f(XT ) = V0 +

∫ T

0

ut (ψ + Wt)dt (4.107)

where

V0 =

∫R

f(x)√2π ‖ φ ‖L2([0,T ])

exp

(− x2

2 ‖ φ ‖2L2([0,T ])

)dx

and

u(t) = φ(t)

∫R

f(x)√2π ‖ φ ‖L2([0,T ])

exp

(− (x−XT )2

2 ‖ φ ‖2L2([0,T ])

) x−XT

‖ φ ‖2L2([0,T ])

dx. (4.108)

97

Proof

The proof is a consequence of Proposition 4.7.2 and Lemma 4.7.3 for ψs = φsαs. 2

Remark

If φ is continuous at t = T it can be shown that Equation (4.108) implies

limt→T

ut = φT

∫R

g(x)√2π ‖ φ ‖L2([0,T ])

exp

(− (x−XT )2

2 ‖ φ ‖2L2([0,T ])

) x−XT

‖ φ ‖2L2([0,T ])

dx. (4.109)

This limit clearly exists in (S)−1. The following corollary provides a more explicit represen-

tation than the one in Theorem 4.7.4.

Corollary 4.7.5 Let φ and Xt be as in Theorem 4.7.4. In addition, assume ‖ φ ‖2L2([0,T ])> 0

for all t < T . Let f : R → R be bounded. Then

f(XT ) = V0 +

∫ T

0

ut(ψdt+ dWt) (4.110)

where

V0 =

∫R

f(x)√2π ‖ φ ‖L2([0,T ])

exp

(− x2

2 ‖ φ ‖2L2([0,T ])

)dx

and

u(t) = φ(t)

∫R

f(x)√2π ‖ φ ‖L2([0,T ])

exp

(− (x−XT )2

2 ‖ φ ‖2L2([0,T ])

)x−XT

‖ φ ‖2L2([0,T ])

dx. (4.111)

Proof

The proof follows the same arguments in Theorem 4.7.4 and using the fact that∫ T0ut Wtdt =

∫ T0utdWt in the L2-case. We omit the details. 2

4.8 Financial Application: Calculating Greeks

We consider the following model with two securities (see [1] and [74]):

1. A risk-free asset (for example a bank account) where the price At at time t is given by

dAt = rtAtdt, A0 = 1 (4.112)

2. A risky asset (for example a stock) where the price St at time t is given by

dSt = µtStdt+ σtStdWt, S0 = x > 0. (4.113)

98

where rt, µt and σt are deterministic functions satisfying the property∫ T

0

(| rt | + | µt | +σ2

t

)ds <∞.

We assume that σ is bounded away from zero. The exact solutions of the differential Equa-

tions (4.113) is given by

St = x exp

(∫ t

0

(µs −

1

2σ2s

)ds+

∫ t

0

σsdWs

). (4.114)

Then, we consider ∆ of a digital option. The digital option payoff then takes the form

χ[K,∞)(ST ) (4.115)

with strike price K and ST is the value of a stock price at final time T . Following the

constructions in the preceding sections, we apply the concept of white noise analysis together

with the Donsker delta function to compute ∆ for the digital option. Here we only illustrate

the computation of ∆.

We define ν by:

d(logS) = (µt −1

2σ2t )dt+ σtdW := νtdt+ σtdW. (4.116)

Let

vT =

∫ T

0

σ2udu. (4.117)

Then

logST ∼ N

(∫ T

0

νudu, vT

).

So we may apply (4.98) to get a.s.

f(ST ) = f(elogST ) =1√

2πvT

∫Rf(ey) exp

(−(y − logST )2

2vT

)dy. (4.118)

We note that for f ∈ L1(R) and with compact support the integral belongs to the distribution

space (S)−1 (by Lemma 4.4.7). The option price with the payoff function of the form (4.115)

is given by

u(x) = E[e−rTf(ST )] = E[e−rT√2πvT

∫Rf(ey) exp

(−(y − logST )2

2vT

)dy].

Theorem 4.8.1 Let f : R → R be a function of polynomial growth. Then

∂

∂xE[

e−rT√2πvT

∫Rf(ey) exp

(−(y − logST )2

2vT

)dy]

= E[e−rT√2πvT

∫Rf(ey) exp

(−(y − logST )2

2vT

) (y − logST )

vT

1

xdy]. (4.119)

99

Proof

Let

u(x) = E[e−rTf(ST )] = E[e−rT√2πvT

∫Rf(ey) exp

(−(y − logST )2

2vT

)dy].

First we assume that f ∈ L2(R) and has compact support. Then by Lemma 4.4.7 the strong

integral exists in (S)−1. Taking the Hermite transform on both sides and using Lemma 4.4.7

since condition (4.76) holds we obtain the following deterministic equation

u(x)(z) = E[e−rT√2πvT

∫Rf(ey)H

(exp

(−(y − logST (x))2

2vT

))dy]

where u denote the Hermite transforms of u and the expectation is taken in the generalized

sense.

We note that

H(

exp(−(y − logST (x))2

2vT

))= exp

(−(y − log ST (x)(z))2

2vT

)where we have used Proposition 4.4.2. Therefore we have

u(x)(z) = E[e−rT√2πvT

∫Rf(ey) exp

(−(y − log ST (x)(z))2

2vT

)dy].

where ST is the Hermite transform of ST .

Then

u(x+ ε)(z)− u(x)(z)

= E[e−rT√2πvT

∫Rf(ey)[exp

(−(y − log ST (x+ ε)(z))2

2vT

)− exp

(−(y − log ST (x)(z))2

2vT

)]dy]

and so, for ε 6= 0, we have

limε→0

u(x+ ε)(z)− u(x)(z)

ε

= limε→0

E[e−rT√2πvT

∫Rf(ey)

1

ε[exp

(−(y − log ST (x+ ε)(z))2

2vT

)− exp

(−(y − log ST (x)(z))2

2vT

)]dy].

Put

Zε(y)(z) :=1

ε[exp

(−(y − log ST (x+ ε)(z))2

2vT

)− exp

(−(y − log ST (x)(z))2

2vT

)]

= exp

(−(y − log ST (x)(z))2

2vT

)

· 1

εexp

(ε(y log ST (x)(z)− log S2

T (x)(z))

vT− log ε2S2

T (x)(z)

2vT

)− 1.

100

Using Taylor expansions and letting ε→ 0, we have

Zε(y)(z) → exp

(−(y − log ST (x)(z))2

2vT

)(y log ST (x)(z)− log S2

T (x)(z)

vT

)in L2(µ) as ε→ 0

since

| ST (x+ ε) | = | (x+ ε) | exp

(∫ T

0

(µ− 1

2σ2

)dt+

∫ T

0

σdWt

)≤ (x+ 1) exp

(∫ T

0

(µ− 1

2σ2

)ds+

∫ T

0

σdWt

)∈ L1(du).

Thus, we have the following estimate

| Kε(y)(z) | := | e−rT√2πvT

f(ey)1

ε[exp

(−(y − log ST (x+ ε)(z))2

2vT

)− exp

(−(y − log ST (x)(z))2

2vT

)] |

= | e−rT√2πvT

f(ey) exp

(−(y − log ST (x)(z))2

2vT

)(y log ST (x)(z)− log S2

T (x)(z)

vT

)|

≤ A1· | f(ey) | e−A2(y2−|y|)(A3 | y | +A4) ∈ L1(R) (4.120)

for some positive constants A1, A2, A3, A4 for fixed z. Using a similar estimate as in (4.120)

we obtain

∫R| Kε(y)(z) | dy ≤ A1

∫R| f(ey) | e−A2(y2−|y|)(A3 | y | +A4)dy <∞

for fixed z with constants independent of ε. Since f(ey) grows polynomially, we can use the

dominated convergence theorem to interchange the order of taking the limit and expectation

and obtain

limε→0

u(x+ ε)(z)− u(x)(z)

ε

= E[e−rT√2πvT

limε→0

∫Rf(ey)

1

ε[exp

(−(y − log ST (x+ ε)(z))2

2vT

)− exp

(−(y − log ST (x)(z))2

2vT

)]dy].

Using the estimate (4.120) for some positive constants A1, A2, A3, A4 which are independent

of ε and for fixed z we can use the dominated convergence theorem to interchange the order

101

of taking the limit and the integral and obtain

limε→0

u(x+ ε)(z)− u(x)(z)

ε

= E[e−rT√2πvT

∫Rf(ey) lim

ε→0

1

εexp

(−(y − log ST (x+ ε)(z))2

2vT

)− exp

(−(y − log ST (x)(z))2

2vT

)dy]

= E[e−rT√2πvT

∫Rf(ey)

d

dx

(exp

(−(y − log ST (x)(z))2

2vT

))dy]

= E[e−rT√2πvT

∫Rf(ey) exp

(−(y − log ST (x)(z))2

2vT

)(y − log ST (x)(z))

vT

1

xdy]

where we have used the chain rule in the last equality. Thus we have

d

dxu(x)(z) = E[

e−rT√2πvT

∫Rf(ey) exp

(−(y − log ST (x)(z))2

2vT

)(y − log ST (x)(z))

vT

1

xdy].

We can write the above equation as

d

dxu(x)(z) = E[

e−rT√2πvT

∫Rf(ey)H

(exp

(−(y − logST (x))2

2vT

))·H(

(y − logST (x))

vT

1

x

)dy].

Since ST (x) ∈ (S)−1 the application of Proposition 4.4.2 yields

d

dxu(x)(z) = E[

e−rT√2πvT

∫Rf(ey)H

(exp

(−(y − logST (x))2

2vT

) (y − logST (x))

vT

1

x

)dy].

By using Lemma 4.4.7 twice (recall that f has compact support) on the right hand side and

Lemma 4.78 on the left hand side of the above equation we obtain

H(d

dxu(x)

)(z) = H

(E[

e−rT√2πvT

∫Rf(ey) exp

(−(y − logST (x))2

2vT

) (y − logST (x))

vT

1

xdy]

).

The result then follows by the uniqueness of the Hermite transform.

For the general case we consider, for f of polynomial growth, the sequence

fn = fχ[−n,n]

whose functions have compact support so that | fn |≤| f |. Then fn → f in L2 as n → ∞.

Let

un(x) = E[e−rT∫

Rfn(e

y)1√

2πvTexp

(−(y − logST (x))2

2vT

)dy].

Then it is clear that

un(x) → u(x) in (S)−1. (4.121)

102

By Theorem 4.4.4 this is equivalent to

H(un(x))(z) → H(u(x))(z).

Put

g(x) = E[e−rT∫

Rf(ey)

1√2πvT

exp(−(y − logST )2

2vT

) (y − logST )

vT 1

xdy].

Using the Cauchy-Schwartz inequality we have

| ddxun(x)− g(x) |

= | E[e−rT∫

R(fn(e

y)− f(ey))1√

2πvTexp

(−(y − logST (x))2

2vT

) (y − logST (x))

vT 1

xdy] |

≤ E[e−rT∫

R| fn(ey)− f(ey) || 1√

2πvTexp

(−(y − logST (x))2

2vT

) (y − logST (x))

vT 1

x| dy]

≤ A1E[

∫R| fn(ey)− f(ey) | e−A2(y2−|y|)(A3 | y | +A4)dy].

Since f(ey) grows polynomially the dominated convergence theorem implies that

| fn(ey) − f(ey) | converges to 0 as n → ∞. Therefore using (4.120) the above inequality

proves thatd

dxun(x) → g(x) pointwise. (4.122)

From (4.121) and (4.122) we can deduce that u(x) is continuously differentiable and that

d

dxu(x) = g(x). 2

103

Chapter 5

Malliavin calculus for Pure Jump

Levy SDEs

In this chapter we aim at deriving a Malliavin derivative representation for a pure jump Levy

stochastic differential equation Xt in terms of its first variation process. We are inspired

by the ideas discussed Chapter 2 Section 2.4 where we have reviewed a representation for

the Malliavin derivative of Xt in the Brownian motion case. The result we obtain can be

considered as a generalization of the ideas developed in Chapter 2 Section 2.4 to pure jump

Levy case.

We begin this chapter by introducing the Levy process with both continuous part and jump

part. We then restrict ourselves to a pure jump case. As in the second half of Chapter 2,

we define a chaos expansion valid in a pure jump setting. We review the definition of the

Skorohod integral and the stochastic derivative for the pure jump case (see [27]). We then

give some known formulae for the Skorohod integral and the stochastic derivative in the case

of pure jump Levy process. We mention that these formulae generalize the known results for

the Malliavin calculus in the case of Brownian motion. We define the first variation process

of the pure jump Levy stochastic differential equation Xt and then derive the Malliavin

derivative representation (see Section 5.5).

In Section 5.6 we derive the necessary and sufficient conditions for a function to serve as

a weighting function in the pure jump Levy case. Here we are motivated by the ideas in

[9] and [10] where the author gives the necessary and sufficient conditions for a function to

serve as a weighting function in the Brownian motion case.

104

5.1 Basic definitions and results for Levy processes

This section presents the basic concepts and results for Levy processes. For a detailed

account of Levy processes we refer to [3], [16], [79] and [80].

Definition 5.1.1 Let (Ω,F , Ftt≥0, P ) be a filtered probability space. An Ft-adapted pro-

cess (Xt)t≥0 with X0 = 0 a.s is called a Levy process if Xt is continuous in probability and

has stationary and independent increments.

A Levy process has a cadlag modification (see [16]) and we will always assume that we are

using the cadlag version. By cadlag we mean right continuous and having left limits. Let

Ft, t ≥ 0 be the natural filtration of Xt completed with the P -null sets of F . For the Levy

process Xt we denote by

Xt− = lims→t, s<t

Xs, t ≥ 0 (5.1)

the left limit process and by

4Xt = Xt −Xt− (5.2)

the jump size at time t. Put R0 := R\0 and let B(R0) be the σ-algebra generated by the

family of all Borel subsets Λ ⊂ R whose closure Λ does not contain 0. The jump measure of

Xt is defined by

N(t,Λ) := N(t,Λ, ω) =∑

0≤s≤t

χΛ(4Xs), Λ ∈ B(R0). (5.3)

N(t,Λ) describes the number of jumps whose size 4Xs belongs to Λ and which occur before

or at time t. The derivative form is denoted by N(dt, dz), t ≥ 0, z ∈ R0. Moreover, N(t,Λ)

defines, in a natural way, a Poisson random measure N on B(0,∞)× B(R0) given by

(a, b]× Λ → N(b,Λ)−N(a,Λ), Λ ∈ B(R0), 0 < a ≤ b (5.4)

and its standard extension. The Levy measure ν of Xt is defined by

ν(Λ) := E[N(1,Λ)], Λ ∈ B(R0). (5.5)

The Levy measure ν always satisfies∫R0

min(1, z2)ν(dz) <∞, (5.6)

but it is possible that ∫R0

min(1, | z |)ν(dz) = ∞. (5.7)

105

This places a bound on the sizes of small jumps. The situation is of interest in financial

modelling (see [71]).

The Law of Xt is infinitely divisible (see [80]) with characteristic function of the form

E[exp(iuXt)] = (φ(u))t = etψ(u) (5.8)

where ψ(u) = log φ(u) and φ(u) is the characteristic function of X1. The function ψ(u) is

called the characteristic exponent and it satisfies the following Levy-Khintchine formula (see

[16]):

ψ(u) = iau− 1

2σ2u2 +

∫ ∞

−∞(eiuz − 1− iuz1|z|<1)ν(dz) (5.9)

where a ∈ R0, σ2 > 0 and ν is a measure on R0.

We assume further that the Levy measure ν satisfies the following condition: for each ε > 0

there exists λ > 0 such that ∫(−ε,ε)c

exp(λ | z |)ν(dz) <∞ (5.10)

where (−ε, ε)c denotes the complement of the interval (−ε, ε). This condition implies that∫ ∞

−∞| z |i ν(dz) <∞, i ≥ 2 (5.11)

and that the characteristic function E[exp(iuXt)] is analytic in a neighborhood of 0. As a

consequence, Xt has moments of all orders and the polynomials are dense in L2(R0, P X−1t )

for all t ≥ 0 (see [71]). The Levy process Xt admits a decomposition

Xt = a1t+ σWt +

∫ t

0

∫|z|<1

zN(ds, dz) +

∫ t

0

∫|z|≥1

zN(ds, dz) (5.12)

for some constants a1, σ ∈ R0 where W = Wt, t ≥ 0 is a standard Brownian mo-

tion, N(dt, dz) := N(dt, dz)− ν(dz)dt is a compensated Poisson random measure of Xt and

N(dt, dz) is a Poisson random measure. The Brownian motion Wt is independent of the

compensated Poisson random measure N(dt, dz).

We can write the representation in Equation (5.12) as

Xt = at+ σWt +

∫ t

0

∫R0

zN(dt, dz) (5.13)

where a = a1 +∫|z|≥1

zν(dz). Levy processes may be regarded as natural generalizations of

the Brownian motion to discontinuous processes. Following the representation in Equation

106

(5.13) we consider the more general stochastic differential equation of the form

dXt = α(t)dt+ σ(t)dWt +

∫R0

γ(t, z)N(dt, dz) (5.14)

where α, σ and γ are predictable processes satisfying∫ t

0

| α(s) | +σ2(s) +

∫R0

γ2(s, z)ν(dz)ds <∞ P. a.s. for all t ≥ 0, z ∈ R0. (5.15)

This is called an Ito-Levy process.

Theorem 5.1.2 (1-dimensional Ito formula [75])

Let X = Xt, t ≥ 0 be the Ito-Levy process given by Equation (5.14) and let f ∈ C2(R2) and

define

Yt := f(t,Xt).

Then Y = Yt, t ≥ 0 is also an Ito-Levy process and

dYt =∂f

∂t(t,Xt)dt+

∂f

∂x(t,Xt)[α(t)dt+ σ(t)dWt] +

1

2

∂2f

∂x2(t,Xt)σ

2(t)dt

+

∫R0

(f(t,Xt) + γ(t, z))− f(t,Xt)−∂f

∂x(t,Xt)γ(t, z)ν(dz)dt

+

∫R0

f(t,Xt + γ(t, z))− f(t,Xt)N(dt, dz). (5.16)

Assumption

From now onwards we will assume that

Xt =

∫ t

0

∫R0

γ(t, z)N(ds, dz), 0 ≤ t ≤ T, (5.17)

that is, α = σ = 0 in Equation (5.14).

Lemma 5.1.3 Fix T > 0. The set of random variables

f(Xt1 , · · · , Xtn); ti ∈ [0, T ] i = 1, 2, . . . , n, (5.18)

where f : Rn → R is an infinitely many times continuously differentiable function with

bounded derivatives, is dense in L2(FT , P ).

Proof

The proof follows the same arguments as in [73]. We omit the details. 2

107

Let ψ = ψ(t, z, ω) be predictable. Then, the following Ito isometry is valid:

E[

(∫ T

0

∫R0

ψ(t, z)N(dt, dz)

)2

] = E[

∫ T

0

∫R0

ψ(t, z)2ν(dz)dt]. (5.19)

We introduce a continuous function γ : R0 → (−1, 0) ∪ (0, 1) given by

γ(z) =

ez − 1 if z < 0

1− e−z if z > 0

(see [64]) which is totally bounded and has an inverse γ−1 : (−1, 0)∪(0, 1) → R0. In addition,

limz→0γ(z)z

= 1, so γ approaches zero just as fast as z. ρ((−∞, 1] ∪ [1,∞)) < ∞ and since

γ is totally bounded it follows that γ ∈ L2(ρ) and eλγ − 1 ∈ L2(ρ) for all λ ∈ R. Hence, if

h ∈ C([0, T ]) then

ehγ − 1 ∈ L2(ν), hγ ∈ L2(ν) and ehλ − 1− hγ ∈ L1(ν). (5.20)

The function γ(z) ensures that the exponential function eλXt belongs to L2(µ).

Lemma 5.1.4 The linear span of random variables of the type

exp∫ T

0

∫R0

h(t)γ(z)N(ds, dz)−∫ T

0

∫R0

(eh(t)γ(z) − 1− h(t)γ(z))ν(dz)dt, (5.21)

where h ∈ C([0, T ]), is dense in L2(µ).

Proof


5.2 Chaos expansion

Here, we extend the Wiener-Ito chaos expansion for Brownian motion developed in the

second half of Chapter 2 to pure jump Levy processes (see [27]). However, in this case the

corresponding iterated integrals are with respect to the compensated Poisson measure rather

than the Levy process itself. Let λ(dt) = dt denotes the Lebesgue measure on [0, T ] and let

L2((λ×ν)n) = L2([0, T ]×R0)n be the space of all deterministic functions f : ([0, T ]×R0)

n →R0 such that

‖ f ‖2L2((λ×ν)n):=

∫([0,T ]×R0)n

f 2(t1, z1, . . . , tn, zn)dt1ν(dz1) · · · dtnν(dzn) <∞. (5.22)

108

If f is a real function on ([0, T ] × R0)n we define its symmetrization f with respect to the

variables (t1, z1, . . . , tn, zn) by

f(t1, z1, . . . , tn, zn) =1

n!

∑σ

f(tσ(1), zσ(1), . . . , tσ(n), zσ(n)) (5.23)

where the sum is taken over all permutations σ of (1, . . . , n). A function f ∈ L2((λ× ν)n) is

called symmetric if f = f and we denote by L2((λ× ν)n) the space of symmetric functions

in L2((λ × ν)n). We assume that the symmetric function f ∈ L2(λ × ν)n vanishes on the

diagonal, that is,

f(t1, z1, . . . , tn, zn) = 0 if ti = tj and zi = zj for some i 6= j. (5.24)

Define

Gn := (t1, z1, . . . , tn, zn) : 0 ≤ t1 ≤ . . . ≤ T, zi ∈ R0 and i = 1, n (5.25)

and let L2(Gn) be the set of functions g : Gn → R0 such that

‖ g ‖2L2(Gn):=

∫Gn

g2(t1, z1, . . . , tn, zn)dt1ν(dz1) · · · dtnν(dzn) <∞. (5.26)

Definition 5.2.1 For any g ∈ L2(Gn) the n-fold iterated Ito integral Jn(g) is the random

variable with respect to N(·, ·) in L2(µ) defined as

Jn(g) :=

∫ T

0

∫R0

· · ·∫ t−2

0

∫R0

g(t1, z1, . . . , tn, zn)N(dt1, dz1) · · · N(dtn, dzn). (5.27)

We set J0(g) = g for any g ∈ R.

Note that in each step the corresponding integrand is adapted because of the limits of the

preceding integrals. For f ∈ L2((λ× ν)n) we have

‖ f ‖2L2((λ×ν)n)= n!

∫ T

0

∫R0

· · ·∫ t−2

0

∫R0

f 2(t1, z1, . . . , tn, zn)dt1ν(dz1) · · · dtnν(dzn). (5.28)

If f ∈ L2((λ× ν)n) we also define

In(f) := n!Jn(f). (5.29)

For f ∈ L2((λ× ν)n) and g ∈ L2((λ× ν)m) we obtain the following orthogonality relation

E[In(f)Im(g)] =

0 if n 6= m

n!(f, g)L2((λ×ν)n) if n = m m,n = 1, 2, . . .

109

where (·, ·)L2((λ×ν)n) is the inner product in L2((λ × ν)n). The relation follows by applying

Ito isometry iteratively and due to the fact that the expected value of an Ito integral is zero.

With the notation above the following chaos expansion in terms of iterated integrals with

respect to N(dt, dz) holds.

Theorem 5.2.2 Let F ∈ L2(µ) be an FT -measurable random variable. Then there exists a

unique sequence of functions fn∞n=0 where fn ∈ L2((λ× ν)n), n ≥ 1 such that

F =∞∑n=0

In(fn). (5.30)

Moreover, the following Ito isometry is valid:

‖ F ‖2L2(µ)=

∞∑n=0

n! ‖ fn ‖2L2((λ×ν)n) . (5.31)

Proof

The proof is similar the corresponding Brownian motion case given in [27] on page 11. We

omit the details. 2

Example

Choose h ∈ L2(λ × ν) and let F (ω) =∫∞

0

∫R h(s, z)N(ds, dz). Then we have the following

chaos expansion

F (ω) = I1(h).

Corollary 5.2.3 Let F = YT where

Yt = exp

(∫ t

0

∫R0

h(s)γ(z)N(ds, dz)−∫ t

0

∫R0

(eh(s)γ(z) − 1− h(s)γ(z))ν(dz)ds

), t ∈ [0, T ]

(5.32)

with h(s) ∈ L2([0, T ]). Then

F = 1 +∞∑n=1

In(1

n!(ehγ − 1)⊗n). (5.33)

Proof

Put

Xt =

∫ t

0

∫R0

h(s)γ(z)N(ds, dz)−∫ t

0

∫R0

(eh(s)γ(z) − 1− h(s)γ(z))ν(dz)ds

so that

Yt = eXt . (5.34)

110

Applying the Ito formula (5.16) with

α(t) = −∫

R0

(eh(t)γ(z) − 1− h(t)γ(z))ν(dz)dt, β(t) = 0 and γ(t,Xt, z) = h(t)γ(z)

we obtain

dYt = eXt [−∫

R0

(eh(t)γ(z) − 1− h(t)γ(z))ν(dz)dt] +

∫R0

(eXt+h(t)γ(z) − eXt − eXth(t)γ(z))ν(dz)dt

+

∫R0

(eXt+h(t)γ(z) − eXt)N(dt, dz)

=

∫R0

eXt(eh(t)γ(z) − 1)N(dt, dz)

=

∫R0

Y −t (eh(t)γ(z) − 1)N(dt, dz).

Hence

YT = 1 +

∫ T

0

∫R0

Y −s (eh(s)γ(z) − 1)N(ds, dz).

Applying the Ito formula to Equation (5.34) for a second time and then integrate from 0 to

T we obtain

YT = 1 +

∫ T

0

∫R0

(1 +

∫ t−1

0

∫R0

Yt2(eh(t2)γ(z2) − 1)N(dt2, dz2))(e

h(t1)γ(z1) − 1)N(dt1, dz1)

= 1 +

∫ T

0

∫R0

(eh(t1)γ(z1) − 1)N(dt1, dz1)

+

∫ T

0

∫R0

∫ t−1

0

∫R0

Yt−2 (eh(t2)γ(z2) − 1)(eh(t1)γ(z1) − 1)N(dt2, dz2)N(dt1, dz1).

Applying the Ito formula to Equation (5.34) repeatedly and then integrate from 0 to T we

obtain

YT =N−1∑n=0

In(fn) +

∫ T

0

∫R0

· · ·∫ t−2

0

∫R0

Yt−1

N∏i=1

(eh(ti)γ(zi) − 1)N(dt1, dz1) · · · N(dtN , dzN)

where

fn(t1, z1, . . . , tn, zn) =1

n!

n∏i=1

(eh(ti)γ(zi) − 1) =1

n!(eh(t)γ(z) − 1)⊗n. (5.35)

This leads to a chaos expansion

YT =∞∑n=0

In(fn).

Next we show that YT converges in L2(µ). By the Ito isometry (5.19) we have

E[Y 2T ] = 1 +

∫ T

0

∫R0

· · ·∫ t−2

0

∫R0

E[Y 2t−1

] | (eh(t)z − 1)⊗N |2 dt1ν(dz1) · · · dtNν(dzN)

111

which has a unique solution given by

E[Y 2T ] = exp

(∫ T

0

∫R0

(eh(t1)z1 − 1)2dt1ν(dz1)

)2

.

Therefore

‖ YT ‖2L2(µ)= exp(‖ ehγ − 1 ‖2

L2(ν)).

Since ehγ − 1 ∈ L2(ν) by (5.20) we conclude that YT ∈ L2(µ) and the proof is com-

plete. 2


We recall the definition of the Skorohod integral in terms of Wiener-Ito chaos expansion (see

[27]). Let X = X(t, z), t ∈ [0, T ], z ∈ R0 be a stochastic process such that X(t, z) is a

FT -measurable random variable for all (t, z) ∈ [0, T ]× R0 and

E[X(t, z)2] <∞, (t, z) ∈ [0, T ]× R0. (5.36)

Then, by Theorem 5.2.2, the random variable X(t, z) has a chaos expansion of the form

X(t, z) =∞∑n=0

In(fn(t1, z1, . . . , tn, zn; t, z)) for each (t, z) (5.37)

where fn(. . . , t, z) ∈ L2((λ×ν)n), n ≥ 1 and where I0(f0) := E[X(t, z)]. Let fn(t1, z1, . . . , tn+1, zn+1)

be the symmetrization of fn(t1, z1, . . . , tn, zn; t, z) with respect to the n+1 pairs of variables

(t1, z1), . . . , (tn+1, zn+1) with tn+1 = t and zn+1 = z.

Definition 5.3.1 Assume that

∞∑n=0

(n+ 1)! ‖ fn ‖2L2((λ×ν)n+1)<∞. (5.38)

Then the Skorohod integral of X with respect to N , denoted by

δ(X) =

∫ T

0

∫R0

X(t, z)N(δt, dz),

is defined by

δ(X) :=∞∑n=0

In+1(fn(t1, z1, . . . , tn+1, zn+1)). (5.39)

112

Condition (5.38) and Equation (5.39) imply that the Skorohod integral belongs to L2(µ) and

‖∫ T

0

∫R0

X(t, z)N(δt, dz) ‖2L2(µ)=

∞∑n=0

(n+ 1)! ‖ fn ‖2L2((λ×ν)n+1)<∞. (5.40)

The following proposition says that ifX(t, z) is adapted, then the Skorohod integral coincides

with the Ito integral.

Proposition 5.3.2 Suppose X(t, z), t ∈ [0, T ], z ∈ R0 is a stochastic process such that

E[

∫ T

0

∫R0

X2(t, z)ν(dz)dt] <∞. (5.41)

Then the Skorohod integral and the Ito integral coincide in L2(µ), that is,

∫ T

o

∫R0

X(t, z)N(δt, dz) =

∫ T

0

∫R0

X(t, z)N(dt, dz). (5.42)

Proof

The proof follows the same arguments as in [27] on page 23. We omit the details. 2

Corollary 5.3.3 Assume the conditions in Proposition 5.3.2 hold. Then

E[

∫ T

0

∫R0

X(t, z)N(δt, dz)] = 0. (5.43)

Proof

E[

∫ T

0

∫R0

X(t, z)N(δt, dz)] = E[

∫ T

0

∫R0

X(t, z)N(dt, dz)] by (5.42)

= E[

∫ T

0

∫R0

X(t, z)(N(dt, dz)− ν(dz)dt)]

= E[

∫ T

0

∫R0

X(t, z)N(dt, dz)]− E[

∫ T

0

∫R0

X(t, z)ν(dz)dt]

= T

∫R0

X(t, z)ν(dz)− T

∫R0

X(t, z)ν(dz) = 0. 2

113

5.4 Stochastic derivative

The stochastic derivative Dt,z can be defined in several ways. Here we recall the definition

of the stochastic derivative by means of a chaos expansion (see [27] page 176).

We introduce the set D1,2 ⊂ L2(µ) defined by

D1,2 := F =∞∑n=0

In(fn) :∞∑n=1

nn! ‖ fn ‖2L2((λ×ν)n)<∞. (5.44)

Definition 5.4.1 The stochastic derivative Dt,z : D1,2 → L2(λ× ν × µ) is defined by

Dt,zF :=∞∑n=1

nIn−1(fn(·, t, z)) (5.45)

where F is an FT -measurable random variable of the form F =∑∞

n=0 In(fn) with fn(·, t, z) =

fn(t1, z1, . . . , tn−1, zn−1, t, z) and In−1(fn(·, t, z)) means that the (n− 1)-fold iterated integral

of fn is regarded as a function of its (n − 1) first pairs of variables (t1, z1), . . . , (tn−1, zn−1)

while the final pair (t, z) is kept as a parameter.

Dt,zF is well-defined since

‖ Dt,zF ‖2L2(λ×ν×µ) =

∫ T

0

∫R0

‖∞∑n=1

nIn−1(fn(·, t, z)) ‖2L2(µ) ν(dz)dt

=

∫ T

0

∫R0

∞∑n=1

n2(n− 1)! ‖ fn(·, t, z) ‖2L2((λ×ν)n−1) ν(dz)dt

=∞∑n=1

nn! ‖ fn ‖2L2((λ×ν)n)

< ∞. (5.46)

This implies that Dt,zF ∈ L2(λ× ν × µ) if F ∈ D1,2 which shows that D1,2 is the domain of

Dt,z. We can write Equation (5.45) as

Dt,zF =∞∑n=1

nIn−1(fn(·))f(t, z), F ∈ D1,2. (5.47)

This suggests that the operator Dt,z satisfy a chain rule. The form is appealing because it

has some resemblance to the derivative of a monomial:

d

dxxn = nxn−1.

114

Thus, it is natural to call Dt,zF the stochastic derivative of F at (t, z).

Example

For

F =

∫ T

0

∫R0

f(t, z)N(dt, dz), f ∈ L2(λ× ν)

we have seen that F = I1(f) from the previous example. Then

Dt,zF = I0(f(·, t, z)) = f(t, z).

In particular, if F =∫ T

0

∫R0zN(dt, dz) we have

Dt,zF = z.

Example

For η(t) =∫ T

0

∫R0zN(ds, dz), let F = η2(T ).

Define

Yt = η2(t) =

(∫ T

0

∫R0

zN(ds, dz)

)2

.

An application of the Ito formula (5.16) with α(, t, Xt) = β(t,Xt) = 0, γ(t,Xt, z) = z

and f = η2(t) gives

dYt = dη2(t) =

∫R0

[(η(t) + z)2 − η2(t)− 2η(t)z]ν(dz)dt+

∫R0

[(η(t) + z)2 − η2(t)]N(dt, dz)

=

∫R0

z2ν(dz)dt+

∫R0

[2η(t)z + z2]N(dt, dz).

Hence

η2(T ) = T

∫R0

z2ν(dz) +

∫ T

0

∫R0

[2η(t)z + z2]N(dt, dz).

Therefore

Dt,zF = Dt,zη2(T ) = 2η(t)z + z2.

Example

By Corollary 5.2.3 we have

F = YT = 1 +∞∑n=1

In

(1

n!(ehγ − 1)⊗n

).

Then

Dt,zF =∞∑n=1

nIn−1

(1

n!(ehγ − 1)⊗(n−1)(ehγ − 1)

)=

∞∑n=1

n

n!(eh(t)γ(z) − 1)In−1

((ehγ − 1)⊗(n−1)

)= (eh(t)γ(z) − 1)

∞∑n=1

1

(n− 1)!In−1((e

hγ − 1)⊗(n−1)) =(eh(t)γ(z) − 1

)F.

115

The following theorem is taken from [27] on page 177. We include the proof here for easy

reading.

Theorem 5.4.2 Assume F =∑∞

n=0 In(fn) ∈ L2(µ). Let Fk∞k=1 be a sequence with

Fk =∑∞

n=0 In(f(k)n ) ∈ D1,2, k = 1, 2, . . . such that Fk → F, k → ∞ in L2(µ) and that

Dt,zFk, k = 1, 2, . . .converges in L2(λ× ν × µ). Then F ∈ D1,2 and

Dt,zF = limk→∞

Dt,zFk

in L2(λ× ν × µ).

Proof

limk→∞

∞∑n=0

‖ f (k)n − fn ‖L2((λ×ν)n)= 0 in L2((λ× ν)n) for all n = 0, 1, . . .

Since Fk ∈ D1,2 and Dt,kFk converges, then Dt,kFk is a Cauchy sequence in L2 and therefore

from calculations in (5.46) it follows that

E[

∫ T

0

∫R0

(Dt,zFk −Dt,zFj)2ν(dz)dt] =

∞∑n=0

nn! ‖ f (k)n − f (j)

n ‖2L2(λ×ν×µ)→ 0 k, j →∞.

Thus, by the Fatou lemma, we have

limk→∞

∞∑n=0

nn! ‖ f (k)n − fn ‖2

L2((λ×ν)n) = limk→∞

(∞∑n=0

limj→∞

nn! ‖ f (k)n − f (j)

n ‖2L2((λ×ν)n)

)

≤ limk→∞

(limj→∞

inf∞∑n=0

nn! ‖ f (k)n − f (j)

n ‖2L2((λ×ν)n)

)= 0

which implies that F ∈ D1,2 and

Dt,zFk → Dt,zF, k →∞ in L2(λ× ν × µ). 2

The stochastic derivative Dt,z satisfies the following “product rule” (see [29]).

Lemma 5.4.3 Let F,G ∈ D1,2 with G bounded. Then FG ∈ D1,2 and

Dt,z(FG) = FDt,zG+GDt,zF +Dt,zFDt,zG λ× ν a.e. (5.48)

116

Proof

With the help of Theorem 5.4.2 the result is satisfied for F and G of the form f(Xt1 , · · · , Xtk)

and g(Xt1 , · · · , Xtk) respectively where f and g are differentiable functions with compact sup-

port. Then, by using a limit argument the proof follows from the closedness ofDt,z. 2

Let G = F in Equation (5.48) so that we have

Dt,z(F2) = 2FDt,zF +Dt,zFDt,zF = (F +Dt,zF )2 − F 2.

By induction it follows that if F ∈ D1,2 then we have

Dt,z(Fn) = (F +Dt,zF )n − F n. (5.49)

The following result is the “chain rule” which is useful in the evaluation of stochastic deriva-

tives.

Theorem 5.4.4 Let F ∈ D1,2 and let ϕ be a real continuous function on R0. Suppose

ϕ(F ) ∈ L2(µ) and ϕ(F +Dt,zF ) ∈ L2(λ× ν × µ). Then ϕ(F ) ∈ D1,2 and

Dt,zϕ(F ) = ϕ(F +Dt,zF )− ϕ(F ) (5.50)

Proof

We follow the proof in [27] on page 178. We first assume that ϕ has compact support and

F ∈ D1,2. Then ϕ has the inverse Fourier transform of its Fourier transform ϕ:

ϕ(F ) =1√2π

∫R0

eiyF ϕ(y)dy

where

ϕ(y) =1√2π

∫R0

e−ixyϕ(x)dx.

By Equation (5.49) and Theorem 5.4.2 we have

Dt,zϕ(F ) = Dt,z

(1√2π

∫R0

∞∑n=0

1

n!(iy)n(F n)ϕ(y)dy

)

=1√2π

∫R0

∞∑n=0

1

n!(iy)n((F +Dt,zF )n − F n)ϕ(y)dy

=1√2π

∫R0

(eiy(F+Dt,zF ) − eiyF )ϕ(y)dy

= ϕ(F +Dt,zF )− ϕ(F ).

117

Hence the result holds in this case. To prove the general case we proceed by approximation.

We choose Fn ∈ D1,2, n = 1, 2 . . . such that Fn → F, n → ∞ in D1,2. Then ϕ(Fn) → ϕ(F )

in L2(µ) by the dominated convergence theorem and

ϕ(Fn +Dt,zFn)− ϕ(Fn) → ϕ(F +Dt,zF )− ϕ(F )

in D1,2. Hence the result holds for all F ∈ D1,2 in the case of ϕ with compact support. The

extension to the case when ϕ(F ) ∈ L2(µ) and ϕ(F + Dt,zF ) ∈ L2(λ × ν × µ) follows by a

limit argument using the closedness of Dt,z. 2

Examples

1. For ϕ(F ) = ln(F ) where F = 1 + θ(t, z) with θ(t, z) ∈ D1,2 we have

Dt,zϕ(F ) = Dt,z ln(1 + θ(s, z))

= ln(1 + θ(t, z) +Dt,z(1 + θ(s, z)))− ln(1 + θ(t, z))

= ln(1 + θ(t, z) +Dt,zθ(s, z))− ln(1 + θ(t, z))

= ln

(1 +

Dt,zθ(s, z)

1 + θ(t, z)

).

2. Let η(T ) =∫ T

0

∫R zN(ds, dz) and F = (η(T )−K)+. Then, we have

Dt,zF = (η(T ) +Dt,zη(T )−K)+ − (η(T )−K)+ = (η(T ) + z −K)+ − (η(T )−K)+.

The next theorem gives the relationship between the stochastic derivative and the Skorohod

integral (see [27] page 180).

Theorem 5.4.5 Let X(t, z), t ∈ [0, T ], z ∈ R0 be Skorohod integrable and F ∈ D1,2. Then

E[

∫ T

0

∫R0

X(t, z)Dt,zFν(dz)dt] = E[F

∫ T

0

∫R0

X(t, z)N(δt, dz)]. (5.51)

Proof

The proof is similar to the proof of the corresponding Brownian motion given in [27] on page

34. We omit the details. 2

The following corollary gives the closability of the Skorohod integral (see [27] page 181).

Corollary 5.4.6 Suppose that Xn(t, z), t ∈ [0, T ], z ∈ R0 is a sequence of Skorohod integrable

stochastic processes and that

δ(Xn) :=

∫ T

0

∫R0

Xn(t, z)N(δt, dz), n = 1, 2, .. (5.52)

118

converges in L2(µ). In addition if

limn→∞

Xn(t, z) = 0 in L2(λ× ν × µ) (5.53)

then we have

limn→∞

δ(Xn) = 0 in L2(µ). (5.54)

Proof

By Theorem 5.4.5 and assumption in Equation (5.53) we have that

E[

∫ T

0

∫R0

Xn(t, z)Dt,zFν(dz)dt] = E[F

∫ T

0

∫R0

Xn(t, z)N(dt, dz)] → 0 as n→∞

for all F ∈ D1,2. Then we have limn→∞ δ(Xn) = 0 weakly in L2(µ). The result follows because

the sequence δ(Xn), n = 1, 2, . . . is convergent in L2(µ). 2

If Xn, n = 1, 2, . . . is a sequence of Skorohod integrable random variable such that

X(t, z) = limn→∞

Xn(t, z) in L2(λ× ν × µ)

then the Skorohod integrable of X(t, z) can be defined as

δ(X) :=

∫ T

0

∫R0

X(t, z)N(δt, dz) = limn→∞

∫R0

Xn(t, z)N(δt, dz) := limn→∞

δ(Xn) (5.55)

provided the limit exists in L2(µ). The following theorem is the fundamental theorem of

calculus (see [27] page 181).

Theorem 5.4.7 Let X = X(s, y), (s, y) ∈ [0, T ]× R0 be a stochastic process such that

E[

∫ T

0

∫R0

X2(s, y)ν(dy)ds] <∞. (5.56)

Assume that X(s, y) ∈ D1,2 for all (s, y) ∈ [0, T ] × R0 and that Dt,zX(·, ·) is Skorohod

integrable with

E[

∫ T

0

∫R0

(∫ T

0

∫R0

Dt,zX(s, y)N(δs, dy)

)2

ν(dz)dt] <∞. (5.57)

Then∫ T

0

∫R0X(s, y)N(δs, dy) ∈ D1,2 and

Dt,z

∫ T

0

∫R0

X(s, y)N(δs, dy) =

∫ T

0

∫R0

Dt,zX(s, y)N(δs, dy) +X(t, z). (5.58)

119

Proof

The proof can be found in [27] on page 182. We omit the details. 2

Example

Let

F =

∫ T

0

∫R0

ln(1 + θ(s, z))N(δs, dz).

Then

Dt,zF = Dt,z

(∫ T

0

∫R0

ln(1 + θ(s, z))N(δs, dz)

)=

∫ T

0

∫R0

Dt,z(ln(1 + θ(s, z)))N(δs, dz) + ln(1 + θ(t, z))

=

∫ T

0

∫R0

(ln(1 + θ(s, z) +Dt,z(1 + θ(s, z)))− ln(1 + θ(s, z)))N(δs, dz) + ln(1 + θ(t, z))

=

∫ T

0

∫R0

ln

(1 +

Dt,zθ(s, z)

1 + θ(s, z)

)N(δs, dz) + ln(1 + θ(t, z)).

Theorem 5.4.8 Let X ∈ L2(λ× ν × µ) and Dt,zX ∈ L2(λ× ν × µ). Then the Ito isometry

holds

E[

(∫ T

0

∫R0

X(t, z)N(δt, dz)

)2

] = E[

∫ T

0

∫R0

X2(t, z)ν(dz)dt]

+ E[

∫ T

0

∫R0

∫ T

0

∫R0

Dt,zX(s, y)Ds,yX(t, z)ν(dy)dsν(dz)dt].

Proof

The proof is given in [29]. We omit the details. 2

5.5 Differentiability of pure jump Levy stochastic dif-

ferential equation

We consider the following pure jump Levy stochastic differential equation

Xt = x+

∫ t

0

∫R0

γ(s,Xs− , z)N(ds, dz), X0 = x ∈ R (5.59)

where N(dt, dz) = N(dt, dz) − ν(dz)dt is a compensated Poisson random measure. The

function γ : [0, T ]×R0 ×R → R is assumed to be continuously differentiable with bounded

120

derivatives and satisfies the linear growth condition:∫R0

| γ(t, x, z) |2 ν(dz) ≤ C(1+ | x |2), 0 ≤ t ≤ T, x ∈ R (5.60)

for a constant C < ∞. In addition, we assume that the function γ satisfies the Lipschitz

condition:∫R0

| γ(t, x, z)− γ(t, y, z) |2 ν(dz) ≤ K | x− y |2, 0 ≤ t ≤ T, x, y ∈ R, (5.61)

for a constant K < ∞. The conditions (5.60) and (5.61) ensure the existence of a unique

solution Xt, 0 ≤ t ≤ T to the stochastic differential equation (5.59). That is, there exists

a unique cadlag adapted solution Xt, 0 ≤ t ≤ T, such that

E[ sup0≤t≤T

| Xt |2] <∞.

The following theorem says that the solution to the stochastic differential equation (5.59) is

Malliavin differentiable, that is, the random variable Xt belongs to D1,2 (see [27] page 310).

Theorem 5.5.1 Suppose the conditions (5.60) and (5.61) hold. Then there exists a unique

strong solution Xt, 0 ≤ t ≤ T to the stochastic differential equation (5.59) such that

Xt ∈ D1,2 for all 0 ≤ t ≤ T ,

sup0≤r≤t

E[ supr≤s≤T

| Dr,zXs |2] <∞ (5.62)

and the stochastic derivative Dr,zXt follows a linear equation

Dr,zXt =

∫ t

r

∫R0

(γ(s,Xs +Dr,zXs, z)− γ(s,Xs, z))N(ds, dz) + γ(r,Xr, z) (5.63)

for r ≤ t a.e and Dr,zXt = 0 for r > t a.e.

Proof

We consider the Picard approximation Xn(t), n ≥ 0 to Xt given by

Xn+1(t) = x+

∫ t

0

∫R0

γ(s,Xn(s−), z)N(ds, dz), X0(t) = x. (5.64)

We first prove, by induction on n, that

Xn(t) ∈ D1,2 for all 0 ≤ t ≤ T (5.65)

and that

ψn+1(t) ≤ k1 + k2

∫ t

0

ψn(s)ds (5.66)

121

for all 0 ≤ t ≤ T, n ≥ 0 where k1 and k2 are constants and

ψn(t) := sup0≤r≤t

E[

∫R0

supr≤s≤t

| Dr,zXn(s) |2 ν(dz)] <∞.

It can be shown that conditions (5.65) and (5.66) hold for n = 0 since

Dt,z

∫ T

0

∫R0

γ(s, x, ζ)N(ds, dζ) = γ(t, x, z).

Suppose that it holds for all n. Then, the closability of the stochastic derivative Dt,z and

Theorem 5.4.4 imply that

Dr,zγ(t,Xn(t−), z) = γ(t,Xn(t

−) +Dr,zXn(t−), z)− γ(t,Xn(t

−), z)

for r ≤ t a.e and ν- a.e. Using Theorem 5.4.7 we deduce that the Ito integral∫ t0

∫R0γ(s,Xn(s

−), z)N(ds, dz) belongs to D1,2 and that for r ≤ t we have

Dr,zXn+1(t) =

∫ t

0

∫R0

Dr,zγ(s,Xn(s−), ζ)N(ds, dζ) + γ(r,Xn(r

−), z)

=

∫ t

0

∫R0

(γ(s,Xn(s−) +Dr,zXn(s

−), ζ)− γ(s,Xn(s−), ζ))N(ds, dζ) + γ(r,Xn(r

−), z)

=

∫ t

r

∫R0

(γ(s,Xn(s−) +Dr,zXn(s

−), ζ)− γ(s,Xn(s−), ζ))N(ds, dζ) + γ(r,Xn(r

−), z)

for r ≤ t a.e and ν- a.e. Then, by Doob maximal inequality, Fubini Theorem, Ito isometry,

(5.60) and (5.61) we get

E[

∫R0

supr≤s≤t

(Dr,zXn+1(s))2ν(dz)] ≤ 8K

∫ t

r

E[

∫R0

| Dr,zXn(u−) |2 ν(dz)]du

+ 2C(1 + E[| Xn(r−) |2])

≤ 8K

∫ t

r

E[

∫R0

| Dr,zXn(u−) |2 ν(dz)]du

+ 2C(1 + λ) (5.67)

for all 0 ≤ r ≤ t where C is a constant and

λ := supn≥0

E[ sup0≤s≤T

| Xn(s) |2] <∞.

Applying a discrete version of Gronwall’s inequality to (5.67) we get

supn≥0

E[

∫ T

0

∫R0

| Ds,zXn(t) |2 ν(dz)ds] <∞

122

for all 0 ≤ t ≤ T . Thus, the inequality (5.67) shows that the conditions (5.65) and (5.66)

hold for n+ 1. Moreover, we note that

E[ sup0≤s≤T

| Xn(s)−X(s) |2] → 0

as n goes to infinity by Picard approximation. Hence, by Theorem 5.4.2, we conclude that

Xt belongs to D1,2.

Finally, applying the operator Dt,z to Equation (5.59) and using Theorem (5.4.4) and

Theorem (5.4.7) we obtain Equation (5.63). 2

We will use the following assumption (see [48]).

Assumption 2 Let the function γ(t,Xt, z) be of the form

γ(t,Xt, z) = g(t, z)Xt (5.68)

for some deterministic function g.

Assumption 2 implies, for any adapted random variable Y , the following

1. γ(t,X + Y, z) = γ(t,X, z) + γ(t, Y, z).

2. γ(t,XY, z) = Y γ(t,X, z).

Under Assumption 2 we have

Dr,zXt =

∫ t

r

∫R0

(g(s, z)(Xs +Dr,zXs)− g(s, z)Xs)N(ds, dz) + g(r, z)X(r)

=

∫ t

r

∫R0

g(s, z)Dr,zXsN(ds, dz) + g(r, z)X(r)

The following proposition gives a representation for Dt,zXt under Assumption 2.

Proposition 5.5.2 Let γ be as in Assumption 2 and let the first variation process Yt of Xt

satisfies the following equation

Yt = Y0 +

∫ t

0

∫R0

g(s, z)YsN(ds, dz) (5.69)

with Y0 = 1. Then the stochastic derivative of the process Xt in Equation (5.59) is given by

Dr,zXt = YtY−1r γ(r,Xr, z)1r≤t. (5.70)

123

Proof

Using Assumption 2 we have

Xt = x+

∫ t

0

∫R0

γ(s,Xs, z)N(ds, dz) = x+

∫ t

0

∫R0

g(s, z)XsN(ds, dz).

Taking the partial derivative with respect to x on both sides we have

∂

∂xXt = I +

∫ t

0

∫R0

g(s, z)∂Xs

∂xN(ds, dz). (5.71)

Setting Yt = ∂∂xXt we get

Yt = I +

∫ t

0

∫R0

g(s, z)YsN(ds, dz).

That is,

dYt =

∫R0

g(t, z)YtN(dt, dz) Y0 = I. (5.72)

Using the Ito formula the solution to Equation (5.72) is given by

Yt = exp∫ t

0

∫R0

[log(1+g(s, z))−g(s, z)]ν(dz)ds+

∫ t

0

∫R0

log(1+g(s, z))N(ds, dz) (5.73)

On the other hand, an application of Theorem 5.4.7 to the stochastic differential equation

(5.59) yields

Dr,zXt =

∫ t

r

∫R0

(γ(s,Xs +Dr,zXs, z)− γ(s,Xs, z))N(ds, dz) + γ(r,Xr, z)

Under Assumption 2 we have

Dr,zXt =

∫ t

r

∫R0

(g(s, z)(Xs +Dr,zXs)− g(s, z)Xs)N(ds, dz) + γ(r,Xr, z)

=

∫ t

r

∫R0

g(s, z)Dr,zXsN(ds, dz) + γ(r,Xr, z).

Fix r, z and set Zt := Dr,zXt. We have

Zt =

∫ t

r

∫R0

g(s, z)ZsN(ds, dz) + γ(r,Xr, z).

That is,

dZt =

∫R0

g(t, z)ZtN(dt, dz) (5.74)

124

with initial condition Zr = γ(r,Xr, z). Using the Ito formula the solution to Equation (5.74)

is given by

Zt = γ(r,Xr, z) exp∫ t

r

∫R0

[log(1 + g(s, z))− g(s, z)]ν(dz)ds

+

∫ t

r

∫R0

log(1 + g(s, z))N(ds, dz) (5.75)

Matching Equation (5.73) with Equation (5.75) yields

Dr,zXt = YtY−1r γ(r,Xr, z)1r≤t. 2

Proposition 5.5.3 The first variation process Yt is invertible a.s. Further the inverse Y −1t

satisfies a.s

Y −1t = I −

∫ t

0

∫R0

Y −1s (I + γ′(s,Xs− , z))

−1γ′(s,Xs− , z)2ν(dz)ds

−∫ t

0

∫R0

Y −1s (I + γ′(s,Xs− , z))

−1γ′(s,Xs− , z)N(ds, dz) (5.76)

where γ′(·, ·, ·) denotes the derivative of γ(·, ·, ·) with respect to the second argument.

Proof

Since Xt satisfies the stochastic differential equation (5.59) its first variation process satisfies

Yt = I +

∫ t

0

∫R0

γ′(s,Xs− , z)YsN(ds, dz).

Now we consider the linear stochastic differential equation for unknown matrix valued Zt:

Zt = I −∫ t

0

∫R0

Zs(I + γ′(s,Xs− , z))−1γ′(s,Xs− , z)

2ν(dz)ds

−∫ t

0

∫R0

Zs(I + γ′(s,Xs− , z))−1γ′(s,Xs− , z)N(ds, dz).

It has a unique solution Zt. It can be shown that the product ZtYt satisfies

d(ZtYt) = 0.

Therefore

ZtYt = I

holds a.s, proving that Yt is invertible, that is, Zt = Y −1t and the inverse Y −1

t satisfies

(5.76). 2

125

5.6 The necessary and sufficient condition for a func-

tion to serve as a weighting function

In this section we use the results discussed in the preceding sections. Greeks can be expressed

as (see [35])

Greeks = E[e−rTΦ(XT )π | X0 = x]

where π is a weight function and Φ(XT ) is the payoff function which is square integrable.

The weight function could be expressed in the form of a Skorohod integral. Here we examine

the set of functions expressed as a Skorohod integral and determine which conditions these

functions should satisfy to serve as a weighting function. We mention that similar condi-

tions were given in [7] in the Brownian motion case. We consider the market model whose

dynamics is given by Equation (5.59).

The following result shows the necessary and sufficient conditions to be satisfied by a function

to serve as a weight function of ∆ in the pure jump case. This is an extension of the work

of [7] to pure jump cases.

Theorem 5.6.1 Let Xt be a stochastic process of the form (5.59). Let Φ : R → R be of

polynomial growth and let u ∈ L2([0, T ]×Ω). Then a necessary and sufficient conditions for

a function u(·, ·) to serve as a weight function of ∆ is that it satisfy the following

1. u(·, ·) is Skorohod integrable.

2.

E[e−rTΦ′(XT )YT | XT ]

= E[e−rT∫ T

0

∫R0

u(t, z)(Φ(XT +Dt,zXT )− Φ(XT ))ν(dz)dt | XT ]. (5.77)

where Dt,zXT = YTY−1t γ(t,Xt, z)1t<T(t) and YT = ∂

∂xXT .

Proof

Necessary condition: First, assume that the payoff function Φ is continuously differentiable

126

with bounded derivatives. Then

∆ =∂

∂xE[e−rTΦ(XT ) | X0 = x]

= E[e−rTΦ′(XT )∂

∂xXT | X0 = x]

= E[e−rTΦ′(XT )YT | X0 = x] (5.78)

where the interchange of the derivative and the expectation operator is justified by the

dominated convergence theorem.

We want to write E[e−rTΦ′(XT )YT | X0 = x] as E[e−rTΦ(XT )δ(u) | X0 = x] where δ(u) is

the Skorohod integral of a certain u ∈ L2([0, T ]× Ω), that is,

∆ = E[e−rTΦ(XT )δ(u) | X0 = x].

Using the notation in Definition 5.3.1 we have


∫ T

0

∫R0

u(t, z)N(δs, dz) | X0 = x]

= E[e−rT∫ T

0

∫R0

u(t, z)Dt,z(Φ(XT ))ν(dz)dt | X0 = x] by (5.51)

= E[e−rT∫ T

0

∫R0

u(t, z)[Φ(XT +Dt,zXT )− Φ(XT )]ν(dz)dt | X0 = x] (5.79)

where we have used Equation (5.50) in the last equality.

So u(t, z) should satisfy the following equation

E[e−rTΦ′(XT )YT | X0 = x]

= E[e−rT∫ T

0

∫R0

u(t, z)[Φ(XT +Dt,zXT )− Φ(XT )]ν(dz)dt | X0 = x]. (5.80)

Using the fact that Equation (5.80) should hold for any continuously differentiable function

Φ, we get that the following equality holds on any function measurable, leading to conditions

expressed with conditional expectations:

E[e−rTΦ′(XT )YT | XT ]

= E[e−rT∫ T

0

∫R0

u(t, z)(Φ(XT +Dt,zXT )− Φ(XT ))ν(dz)dt | XT ] (5.81)

and this is Equation (5.77).

127

Sufficient condition: We assume that the function u ∈ L2([0, T ]×Ω) satisfies Equation (5.81)

and its Skorohod integral. Then the proof can be conducted backwards. 2

Next we give the necessary and sufficient conditions for a function to serve as a weight

function for V . We interpret V as the Vega for pure jump Levy stochastic differential

equation (5.59). We first define a jump-perturbed process Xεt , 0 ≤ t ≤ T, for small ε > 0,

as the solution of a perturbed stochastic differential equation in the direction of γ:

dXεt =

∫R0

(γ(s,Xεs , z) + εγ(s,Xε

s , z))N(ds, dz) (5.82)

where γ is a continuously differentiable function with bounded derivatives. Writing Equation

(5.82) in integral form we have

Xεt = x+

∫ t

0

∫R0

(γ(s,Xεs , z) + εγ(s,Xε

s , z))N(ds, dz).

The corresponding option price (under the risk-neutral probability measure) is given by

uε(x) = E[e−rTΦ(XεT ) | Xε

0 = x]. (5.83)

We define V as follows.

Definition 5.6.2

V :=∂

∂εuε(x) |ε=0 . (5.84)

The variation process Zεt of Xε

t satisfies the following equation

Zεt =

∫ t

0

∫R0

(γ′(s,Xεs , z) + εγ′(s,Xε

s , z))ZεsN(ds, dz). (5.85)

Note that Zεt :=

∂Xεt

∂ε.

The following theorem gives the necessary and sufficient conditions for a function to serve

as a weight function for V .

Theorem 5.6.3 Let Xt be a stochastic process of the form (5.59). Let Φ : Rn → R be of

polynomial growth and let u ∈ L2([0, T ]×Ω). Then a necessary and sufficient conditions for

a function u(·, ·) to serve as a weight function of V is that it satisfy the following

1. u(·, ·) is Skorohod integrable.

128

2.

E[e−rTΦ′(XεT )Zε

T | XεT ]

= E[e−rT∫ T

0

∫R0

u(t, z)(Φ(XεT +Dt,zX

εT )− Φ(Xε

T ))ν(dz)dt | XεT ] (5.86)

where Zεt = ∂

∂εXεt and Dt,zX

εT = Y ε

T (Y εt )−1γ(t,Xε

t , z)1t<T(t).

Proof

Necessary condition: As in the proof of Theorem 5.6.1, we first assume that the payoff

function Φ is continuously differentiable with bounded derivatives. Then we have

∂

∂εuε(x) =

∂

∂εE[e−rTΦ(Xε

T ) | Xε0 = x]

= E[e−rTΦ′(XεT )∂

∂εXεT | Xε

0 = x]

= E[e−rTΦ′(XεT )Zε

T | Xε0 = x] (5.87)

where the interchange of the derivative operator and the expectation operator is justified by

the dominated convergence theorem.

We want to write E[e−rTΦ′(XεT )Zε

T | Xε0 = x] as E[e−rTΦ(Xε

T )δ(u) | Xε0 = x] where δ(u) is

the Skorohod integral of a certain u ∈ L2([0, T ]× Ω), that is,

∂

∂εuε(x) = E[e−rTΦ(Xε

T )δ(u) | Xε0 = x].

Using the notation in Definition 5.3.1 we have

∂

∂εuε(x) = E[e−rTΦ(Xε

T )

∫ T

0

∫R0

u(t, z)N(δs, dz) | Xε0 = x]

= E[e−rTΦ(XεT )

∫ T

0

∫R0

u(t, z)N(ds, dz) | Xε0 = x] by (5.42)

= E[e−rT∫ T

0

∫R0

u(t, z)Dt,z(Φ(XεT ))ν(dz)dt | Xε

0 = x] by (5.51)

= E[e−rT∫ T

0

∫R0

u(t, z)[Φ(XεT +Dt,zX

εT )− Φ(Xε

T )]ν(dz)dt | Xε0 = x] (5.88)

where we have used Equation (5.50) in the last equality.

So u(t, z) should satisfy the following equation


T | Xε0 = x]

= E[e−rT∫ T

0

∫R0

u(t, z)[Φ(XεT +Dt,zX

εT )− Φ(Xε

T )]ν(dz)dt | Xε0 = x]. (5.89)

129

Using the fact that Equation (5.89) should hold for any continuously differentiable function

Φ, we get that the following equality holds on any function measurable, leading to conditions

expressed with conditional expectations:


T | XεT ]

= E[e−rT∫ T

0

∫R0

u(t, z)(Φ(XεT +Dt,zX

εT )− Φ(Xε

T ))ν(dz)dt | XεT ] (5.90)

and this is Equation (5.86).

Sufficient condition: We assume that the function u ∈ L2([0, T ]×Ω) satisfies Equation (5.90)

and its Skorohod integral. Then the proof can be conducted backwards. 2

Remark

As in the pure Brownian motion case (see [7]) the necessary and sufficient conditions for a

function to serve as a weight function are different for each Greek.

130

Chapter 6

Calculations of Greeks for Jump

Diffusion Processes

In this chapter we compute Greeks of processes driven by both continuous process and jump

process. We mention that there are several papers that have considered the computation

of Greeks of processes driven by both continuous process and jump process (see [25],[26],

[32], [34], [76] and the references therein). In particular, Davis and Johansson [25] calculate

Greeks of models driven by a Brownian motion and a Poisson process with deterministic

jump sizes. El-Khatib and Privault [32] consider a market driven by jumps alone. The

authors defined a Malliavin calculus on the Poisson space and were able to calculate Greeks

of processes having Poisson jump times with random jump sizes but imposing a regularity

condition on the payoff function. We note that the papers mentioned above have advantages

for specific applications. We extend the results in [25] to more general cases.

The main difficulty is in establishing a chain rule that is valid for both the continuous part

and the jump part since the stochastic derivative for the jump part is a difference operator.

We, however, are able to circumvent the difficulty by working on the Wiener-Poisson space

on which a chain rule has been defined (see [62]). Using the chain rule, Davis and Johansson

[25] obtain Greeks for jump diffusion models which satisfy a separability condition. We

review the generalized chain rule following the work in [76] and use it to compute Greeks

for different models. In particular, we compute ∆ for the Heston model with jumps. This is

new.

In the last section we use a slightly different approach. We are inspired by ideas developed in

[4] and [13]. In [4] the authors give approximation of a Levy process by a Brownian motion

while in [13] the authors use the likelihood approach to compute Greeks for approximation

131

of Levy processes. We use the Malliavin calculus approach. We first approximate the

small jumps for the Levy process by a Brownian motion (see [4]). Then we calculate the

corresponding Greeks by using the chain rule and the integration by parts formula and then

use limit arguments to obtain the Greeks for the original Levy process. The approach is

applicable to more cases as it can be applied to random variables whose density function is

not explicitly known.

6.1 Basic elements of a Levy chaotic calculus

Let X = Xt, 0 ≤ t ≤ T be a real-valued Levy process defined on a complete probability

space (Ω,F , P ). We assume, as before, that we are using a cadlag version. As in Chapter 5,

we assume that the Levy measure ν of Xt satisfies the condition (5.10), that is,∫(−ε,ε)

eλ|z|ν(dz) <∞. (6.1)

We will consider a Levy process Xt of the form

Xt = σW (t) +

∫ t

0

∫R0

zN(ds, dz), 0 ≤ t ≤ T (6.2)

where Wt, 0 ≤ t ≤ T is a standard Brownian motion, N(ds, dz) is the compensated Poisson

random measure associated with Xt and∫R0

z2ν(dz)dt <∞. (6.3)

We perform the following transformation to the Levy process Xt and this will play an im-

portant role in the analysis (see [30] and [71]). We set

X(i)t =

∑0<s≤t

(∆Xs)(i), i = 2, 3, . . . (6.4)

For convenience we put

X(1)t = Xt, 0 ≤ t ≤ T. (6.5)

The processes X(i) = X(i)t , 0 ≤ t ≤ T, i = 1, 2, . . . are again Levy processes and are called

power jump processes which jump at the same points as the original Levy process Xt.

We have

E[Xt] = E[X(1)t ] = tm1 <∞ (6.6)

132

where m1 = E[X1] and

E[X(i)t ] = E[

∑0<s≤t

(∆Xs)(i)] = t

∫R0

ziν(dz) := mit <∞, i ≥ 2 (6.7)

where mi =∫

R0ziν(dz) (see [79] page 29). We define the compensated power jump processes

Y (i)t , 0 ≤ t ≤ T of order i as follows

Y(i)t := X

(i)t − E[X

(i)t ] = X

(i)t −mit, i = 1, 2, . . . (6.8)

Example

Y(1)t = σWt +

∫ t

0

∫R0

zN(ds, dz) and Y(i)t =

∫ t

0

∫R0

ziN(ds, dz), i ≥ 2.

It turns out that the processes Y (i) = Y (i)t , 0 ≤ t ≤ T are martingales. An important

question is the orthogonalization of the set Y (i), i = 1, 2, . . . of martingales. We denote by

M2 the space of square integrable martingales M such that

supt

E[M2t ] <∞ and M0 = 0 a.s. (6.9)

We note that if M ∈M2 then

limt→∞

E[M2t ] = E[M2

∞] <∞ and Mt = E[M∞ | Ft]. (6.10)

Thus, each M ∈M2 can be identified with the terminal value M∞. Let N ∈M2 be another

martingale. Two martingales are said to be orthogonal if E[N∞M∞] = 0. We give a stronger

notion of orthogonality for martingales in M2 (see [79] page 179).

Definition 6.1.1 Two martingales M and N are said to be strongly orthogonal if the product

L = MN is a (uniformly integrable) martingale.

We note that if N and M are strongly orthogonal then E[N∞M∞] = E[L∞] = E[L0] = 0, so

strong orthogonality implies orthogonality. The converse is not true (see [79] page 179).

Definition 6.1.2 The predictable quadratic covariation process of Y (i) and Y (j) denoted by

〈Y (i), Y (j)〉t is defined by

〈Y (i), Y (j)〉t = E[

(∫ t

0

∫R0

ziN(ds, dz)

)(∫ t

0

∫R0

zjN(ds, dz)

)]

=

∫ t

0

∫R0

zi+jν(dz)ds = mi+jt, i, j ≥ 2

where 〈·, ·〉 denotes the inner product in L2(ν).

133

Definition 6.1.3 Let Y 1, Y 2 ∈ Rn be two real-valued Levy type stochastic integrals. The

quadratic covariation of Y 1 and Y 2 denoted by [Y 1, Y 2]t is defined as

Y 1(t) · Y 2(t) = Y 1(0) · Y 2(0) +

∫ t

0

Y 1(s−)dY 2(s) +

∫ t

0

Y 2(s−)dY 1(s) + [Y 1, Y 2]t. (6.11)

Example

For an Ito-Levy process of the form

Y(i)t = σiWt +

∫ t

0

∫R0

ziN(ds, dz), i ≥ 1, (6.12)

the quadratic covariation process of Y (i) and Y (j) using Definition 6.1.3 is given by

[Y (i), Y (j)]t =

∫ t

0

σiσjds+

∫ t

0

∫R0

zizjν(dz)ds+

∫ t

0

∫R0

zizjN(ds, dz)

= σiσjt+ t

∫R0

zi+jν(dz) +

∫ t

0

∫R0

zi+jN(dz)ds

= 1i=j=1σ2t+mi+jt+X

(i+j)t , i, j ≥ 1

where σ is a parameter corresponding to the Gaussian part of the Levy process Xt.

We seek for a set of pairwise strongly orthogonal martingales H(i), i = 1, 2, . . . , such that

each H(i) is a linear combination of the Y (j), j = 1, 2, . . . , i with leading coefficient equal to

1. We set

H(i) = Y (i) + ai,j−1Y(i−1) + · · ·+ ai,1Y

(1) =i∑

j=1

ai,jY(j)t , i ≥ 1 (6.13)

where the constants ai,j are chosen in such a way that ai,i = 1 and H(i) are martingales.

Moreover, H(i), i = 1, 2, . . . are strongly orthogonal martingales. It is shown in [71] that the

coefficients ai,j correspond to the coefficients of the orthonormalization of the polynomials

zn, n ≥ 0 with respect to the measure µ(dz) = z2ν(dz) + σ2δ0(dz) where δ0 is the Dirac

measure at point 0. In particular, the polynomials pn(z) defined by

pn(z) =n∑j=1

an,jzj−1 (6.14)

are orthogonal with respect to the measure dµ, that is,∫R0

pn(z)pm(z)dµ(z) = 0, n 6= m. (6.15)

134

Furthermore, the processes H(i)t , 0 ≤ t ≤ T are martingales with predictable quadratic

variation process given by

〈H(i), H(i)〉t = qit (6.16)

where

qi = a2i,1σ

2 +∑

j,j′=1,...,i

ai,jai,j′mj+j′ . (6.17)

Using the Definition 6.1.3 we deduce that the quadratic covariation of H(i) and H(j) is given

by

[H(i), H(j)]t = ai,1aj,1σ2t+ ai,kaj,k′mk+k′t+

i∑k=1

j∑k′=1

ai,kaj,k′X(k+k′)(t). (6.18)

The H(i)t are called the orthogonal power jump processes of a Levy process Xt. We can now

state the result in [71] on the chaos expansion of a random variable in terms of iterated

integral with respect to H(i).

Theorem 6.1.4 Every random variable F ∈ L2(P ) can be represented in the form

F = E[F ] +∞∑n=1

∑0≤i1,...,in≤m

∫ ∞

0

∫ tm

0

· · ·∫ t2

0

fi1,...,in(t1, . . . , tn)dH(i1)t1 · · · dH(in)

tn (6.19)

for any deterministic function fi1,...,in ∈ L2([0, T ]n).

An immediate consequence of Theorem 6.1.4 is the following corollary (see [71]).

Corollary 6.1.5 Every F ∈ L2(P ) can be written as

F = E[F ] +∑n≥1

∫ ∞

0

ϕ(n)(t)dH(n)t (6.20)

where ϕ(n), n = 1, 2, . . . are predictable processes.

6.2 Chaos expansion

In order to define a stochastic integral with respect to Levy processes we use iterated integrals

(instead of multiple integrals that is used in the case of Gaussian processes or Poisson

processes) because of the chaotic representation given in [71] which consists of H(i), i =

1, 2, . . . as integrators. We will adopt the following notation

U0 = [0, T ] and U1 = [0, T ]× R0. (6.21)

135

ulk =

t if l = 0

(t, z) if l = 1.

H0(du) = dWt and H1(du) = N(dt, dz).

〈H0〉(du) = dt and 〈H1〉(du) = ν(dz)dt.

For i1, . . . , in = 0, 1, we let

Gi1,...,in = (ui11 , . . . , uinn ) ∈n∏k=1

Uik : 0 < t1 < t2 < . . . < tn < T, k = 1, . . . , n (6.22)

be a positive simplex of Rn.

Definition 6.2.1 We define the (n-fold) iterated integrals L(i1,...,in)n (fi1,...,in) of fi1,...,in ∈

L2(Gi1,...,in) with respect to Hi1 , . . . , Hin by

L(i1,...,in)n (fi1,...,in) :=

∫Gi1,...,in

fi1,...,in(ui11 , . . . , uinn )Hi1(du

i11 ) · · ·Hind(u

inn ). (6.23)

The integrals are well-defined since all the processes Hj, j = 1, . . . , n are martingales with

respect to the filtration Ft0≤t≤T . Using the Ito isometry it can be shown that Li1,...,inn

and Lj1,...,jmm are orthogonal whenever n = m and (i1, . . . , in) 6= (j1, . . . , jm). We now give a

chaotic representation property of the Levy process which can be considered as a reformu-

lation of the result in [71].

Proposition 6.2.2 Let F ∈ L2(Ω). Then there exists a unique sequence fi1,...,in∞n=0,

i1, . . . , in = 0, 1, where fi1,...,in ∈ L2(Gi1,...,in) such that

F = E[F ] +∞∑n=1

∑i1,...,in=0,1

L(i1,...,in)n (fi1,...,in) (6.24)

where fi1,...,in are deterministic functions in L2(Gi1,...,in). We also have the Ito isometry:

‖ F ‖2L2(P )= [ | F |]2 +

∞∑n=1

∑i1,...,in=0,1

‖ Li1,...,inn (fi1,...,in) ‖2L2(Gi1,...,in ) . (6.25)

6.2.1 Directional derivative

In this subsection we present the properties of the directional derivative in the context of

calculus of variations (see [76]).

136

Definition 6.2.3 Let fi1,...,in ∈ L2(Gi1,...,in). Then we define the derivative of Li1,...,inn (fi1,...,in)

in the lth-direction by

D(l)

ul Li1,...,inn (fi1,...,in) :=

n∑k=1

1ik=lLi1,...,ik−1,ik+1,...,inn−1 (fi1,...,in(. . . , ul, . . .)1

G(k)i1,...,in

(t)) (6.26)

where ul appears in the kth position, l = 0, 1 and

G(k)i1,...,in

(t) = (ui11 , . . . , uik−1

k−1 , uik+1

k+1 , . . . , uinn ) ∈ Gi1,...,ik−1,ik+1,...,in :

0 < t1 < . . . < tk−1 < t < tk+1 < . . . < tn < T. (6.27)

We note that, for k 6= m, we have G(k)i1,...,in

(t) ∩ G(m)i1,...,in

(t) = ∅. We define the space of

random variables that are differentiable in the lth-direction and its respective derivative in

the following definitions (see [62]).

Definition 6.2.4 We say that F is differentiable in the lth-direction (l = 0, 1) if F ∈ D(l)1,2

where

D(l)1,2 := F ∈ L2(Ω), F = E[F ] +

∑n=1

∑i1,...,in=0,1

L(i1,...,in)n (fi1,..,in) :

∞∑n=1

∑i1,...,in=0,1

n∑k=1

1ik=lqi1 · · · qik−1qik+1

· · · qin

×∫Gi1,...,in

‖ fi1,...,in(. . . , ul, . . .)1G(k)i1,...,in

(t) ‖2L2(Gk

i1,...,in) 〈Hik〉(dul) <∞

and q0 = 1.

Definition 6.2.5 For F ∈ D(l)1,2 such that

F = E[F ] +∞∑n=1

∑i1,...,in=0,1

L(i1,...,in)n (fi1,...,in) (6.28)

we define the derivative of F in the lth-direction (l = 0, 1) as the element of L2(Ω × [0, T ])

given by

D(l)

ul F =∞∑n=1

∑i1,...,in=0,1

n∑k=1

1ik=lL(i1,...,ik−1,ik+1,...,in)n−1 (fi1,...,in(. . . , ul, . . .)1G(k)

i1,...,in(t)). (6.29)

where the convergence of series is in L2(Ω× [0, T ]) and ul is appearing in the kth position.

137

We note that 1ik=l in the formula (6.29) indicates the direction in which we are taking

our derivative from and 1G(k)i1,...,in

(t) indicates the interval in which the integral is defined.

We observe that the derivative reduces the order of integration by 1. This is also the case

for Malliavin derivative for Brownian motion. Here, we are not multiplying by a factor n

because we are dealing with non-symmetric functions.

We also note that Definition 6.2.4 ensures that the stochastic derivative D(l)

ul F belongs to

L2(Ω× [0, T ]). We observe, as in the standard situation for Gaussian processes, that D(l)1,2 is

dense in L2(Ω) since every F ∈ L2(Ω) of the form (6.24) is in D(l)1,2. It can be shown that if we

remove the Poisson processes, the space D(0)1,2 coincides with the standard Gaussian space D1,2.

We can define the derivative in the directions Wt and Nt − λt through iterated integrals

using Definition 6.2.5. We denote the respective derivatives by D(0)t and D

(1)t,z . D

(0)t is the

derivative with respect to the Wiener direction and D(1)t,z is the derivative with respect to the

Poisson random measure direction. Thus, for the Wiener direction, we have

D(0)t F =

∞∑n=1

∑i1,...,in=0,1

n∑k=1

1ik=0L(i1,...,ik−1,ik+1,...,in)n−1 (fi1,...,in(. . . , t, . . .)1G(k)

i1,...,in(t)) (6.30)

where t is a parameter occurring in the kth position and for the Poisson random measure

direction we have

D(1)t,z F =

∞∑n=1

∑i1,...,in=0,1

n∑k=1

1ik=1L(i1,...,ik−1,ik+1,...,in)n−1 (fi1,...,in(. . . , t, z, . . .)1G(k)

i1,...,in(t)) (6.31)

where (t, z) is kept as a parameter occurring in the kth position.

The integrand ϕn in Corollary 6.1.5 can be given explicitly in terms of the stochastic deriva-

tive.

Theorem 6.2.6 Let F ∈ D1,2. Then

F = E[F ] +∞∑n=1

∫ T

0

E[D(l)

ul F | Ft− ]dH(n)t . (6.32)

In particular, we have the following theorem.

Theorem 6.2.7 Let F ∈ L2(Ω). Then

F = E[F ] +

∫ T

0

E[D(0)t F | Ft− ]dWt +

∫ T

0

∫R0

E[D(1)t,z F | Ft− ]dN(ds, dz). (6.33)

138

This is the Clark-Haussman-Ocone formula.

Proof

We have∫ T

0

E[D(0)t F | Ft]dWt +

∫ T

0

∫R0

E[D(1)t,z F | Ft]dN(ds, dz)

=∑l=0,1

∫ul

E[D(l)

ul F | Ft]dHl(ul)

=∑l=0,1

∫ul

E[∞∑n=1

∑i1,...,in=0,1

n∑k=1


i1,...,in(t)) | Ft]dHl(u

l).

We note that the expectation of the integrals that have tk > t, k = 1, . . . , n − 1 are zero.

Thus, the only non-zero integrals are the ones that are adapted, that is, those with k = n.

As a result the third sum of the chaos expansion vanishes and we have∑l=0,1

∫ul

E[∞∑n=1

∑i1,...,in=0,1

n∑k=1



l)

=∑l=0,1

∫ul

∞∑n=1

∑i1,...,in=0,1

1in=lL(i1,...,in−1,in+1,...,in)n−1 (fi1,...,in(. . . , ul, . . . )1G(n)


l)

=∞∑n=1

∑i1,...,in=0,1

L(i1,...,in)n (fi1,...,in)

= F − E[F ]. 2

6.2.2 Wiener-Poisson space

We recall some basic results on the Wiener-Poisson space (Ω,F , P ) from [51], [62], [64], [76]

and [77]).

Fix T > 0. Let ΩW = C0([0, T ]) be the space of continuous functions defined on [0, T ], null

at the origin, with the topology of the uniform convergence. Let FW be the Borel σ-algebra

of ΩW with respect to which ω(t), t ∈ [0, T ] are measurable and PW be the probability

measure on (ΩW ,FW ) such that

Wt(ω) := ω(t) (6.34)

is a standard Brownian motion.

Let ΩN be the set of all integer-valued measures ω′ on [0, T ] × Rn such that ω′(t, z) ≤ 1

for any point (t, z) ∈ [0, 1]× [0, T ]× Rn and ω′(A× B) <∞ when ν(A× B) <∞. Let PN

be the probability measure on (ΩN ,F) such that

N(ω′, A×B) := ω′(A×B), A×B ∈ [0, T ]× Rn (6.35)

139

is a Poisson random measure with intensity measure ν(dz) where ν is a Levy measure.

Hence, N(A × B) is a Poisson variable with mean ν(A × B) and the variables N(Ai × Bi)

are independent when Ai ×Bj are disjoint.

Let (Ω,F) be the product measure space, that is, Ω := ΩW × ΩN , Ft = FWt × FN

t where

ω ∈ ΩW , ω′ ∈ ΩN and consider the product probability measure P := PW × PN on (Ω,F).

The triple (Ω,F , P ) is called the Wiener-Poisson space with Levy measure ν.

Since in ΩW × ΩN there is a product measure, there exists an isometry

L2(ΩW × ΩN) ' L2(ΩW ;L2(ΩN)) (6.36)

where

L2(ΩW ;L2(ΩN)) = F : ΩW → L2(ΩN) :

∫ΩW

‖ F (ω) ‖2L2(ΩN ) dPW (ω) <∞. (6.37)

Every F ∈ L2(ΩW , L2(ΩN)) can be considered as a functional F : ω → F (ω, ω′). Therefore,

we can define a derivative operator D(0)t on the space L2(ΩW ;L2(ΩN)) in the usual Malliavin

calculus sense. The derivative operator D(0)t is closed from L2(ΩW ;L2(ΩN)) into L2(ΩW ×

[0, T ];L2(ΩN)). Thus, if F ∈ D1,2 then

D(0)t F ∈ L2(ΩW ;L2(ΩN ;L2([0, T ]))) ' L2(ΩW ;L2(ΩN × [0, T ]))

' L2(ΩW × ΩN × [0, T ]).

Similarly, the derivative operatorD(1)t,z is closed from L2(ΩN ;L2(ΩW )) into L2(ΩN×[0, T ];L2(ΩW )).

Thus, if F ∈ D1,2 then

D(1)t,z F ∈ L2(ΩN ;L2(ΩW ;L2([0, T ]))) ' L2(ΩN ;L2(ΩW × [0, T ]))

' L2(ΩN × ΩW × [0, T ])

Now, it turns out that through the identification of these spaces, the directional derivative

D(0)t is equivalent to the standard Malliavin derivative Dt and the directional derivative D

(1)t,z

is equivalent to the standard stochastic derivative Dt,z (see [62]). We formally state this

result in the following proposition.

Proposition 6.2.8 Assume that W is a Brownian motion and N is the compensated Pois-

son process. Then the directional derivative operator D(0)t in the direction W coincides with

the standard Malliavin derivative Dt for Brownian motion and the directional derivative op-

erator D(1)t,z in the direction N coincides with the standard stochastic derivative operator Dt,z

for compensated Poisson processes.

140

Proof

The proof is given in [62] on page 211. We omit the details. 2

The following chain rule holds (see [76]).

Proposition 6.2.9 1. Let F = f(Z,Z1) ∈ L2(Ω) where Z only depends on the Brownian

motion Wt and Z1 only depends on the Poisson process N(1)t . Assume that f(x, y)

is a continuously differentiable function with partial derivatives in the variable x and

that Z ∈ D(0)1,2. Then F ∈ D(0)

1,2 and

D(0)t F = f ′(Z,Z1)D

(0)t Z (6.38)

where D(0)t Z is the usual Gaussian Malliavin derivative.

2. If F ∈ D(1)1,2 then

D(1)t,z F = F ε+

t,z − F (6.39)

where ε+t,z is a transformation on ΩN , that implies that we have a jump of size z at

time t, given by

ε−t,zω′(A×B) = ω′(A×B ∩ t, zc)

and

ε+t,zω

′(A×B) = ε−t,zω′(A×B) + 1A(t)1B(z)

where t, zc stands for the complement of t, z.

The following assumption enables Davis and Johansson [25] to calculate stochastic weights

for jump diffusion models using the chain rule (6.38).

Assumption: (Separability)

Assume that b, σ and α are continuously differentiable functions with bounded derivatives

and consider Markov jump diffusion of the form

dXt = b(Xt)dt+ σ(Xt)dWt +m∑k=1

αk(Xt)(dN(k)t − λkdt), X0 = x (6.40)

for which we have a continuously differentiable function f with bounded derivative in the

first argument such that

Xt = f(Xct , X

dt ), Xc

0 = x (6.41)

141

where Xct denote the continuous part and Xd

t denote the jump part (discontinuous part).

Here Xct satisfies a stochastic differential equation

dXct = bc(X

ct )dt+ σc(X

ct )dWt, Xc

0 = x (6.42)

with continuous coefficients bc and σc while Xdt is adapted to the natural filtration FN

t of the

Poisson processes (N(1)t , . . . , N

(m)t ). In particular, Xd

t does not depend on x. We say that

the jump diffusion process is separable.

Example

Consider the stochastic differential equation

dSt = bStdt+ σStdWt + αSt(dNt − λdt), S0 given. (6.43)

St satisfies the separability assumption. Using the Ito formula the solution of (6.43) is given

by

St = S0 exp(b− λα− 1

2σ2)t+ σWt + αNt ≡ Xt · Yt (6.44)

with

dXt = bXtdt+ σXtdWt, X0 = x (6.45)

dYt = αYt(dNt − λdt), Y0 = y (6.46)

and x, y are such that S0 = x · y. Therefore St is separable.

The chain rule given in Proposition 6.2.9 is applicable only to random variables that satisfy

the separability condition above. We extend the application of the chain rule to include

random variables that do not satisfy the separability condition. This is done in the following.

Lemma 6.2.10 The linear span of random variables of the form

exp∫ T

0

σh(t)dWt +

∫ T

0

∫R0

h(t)γ(z)N(ds, dz)− 1

2

∫ T

0

σ2h2(t)dt

−∫ T

0

∫R0

(eh(t)γ(z) − 1− h(t)γ(z))ν(dz)dt (6.47)

where γ : R0 → (−1, 0) ∪ (0, 1) is the continuous function introduced in Chapter 5 and

h ∈ C([0, T ]), is dense in L2(FT , P ).

Proof

The proof is found in [63]. We omit the details. 2

We need a technical result (see [76]).

142

Lemma 6.2.11 Let F ∈ D(0)1,2 and Fnk

∞k=1 be a sequence such that Fnk∈ D(0)

1,2 and Fnk→ F

in L2(P ). Then there exists a subsequence Fkm∞km=1 and a constant 0 < C <∞ such that

‖ D(0)t Fkm ‖2

L2([0,T ]×Ω)< C and

D(0)t F = lim

m→∞D

(0)t Fkm (6.48)

in L2([0, T ]× Ω).

Proof

The proof follows the same steps as in Theorem 5.4.2 (see Chapter 5). We omit the de-

tails. 2

We now state the chain rule that we will use to compute the Greeks (see [76]).

Theorem 6.2.12 Let F ∈ D(0)1,2 and f be a continuously differentiable function and of poly-

nomial growth. Then f(F ) ∈ D(0)1,2 and

D(0)t f(F ) = f ′(F )D

(0)t F. (6.49)

Proof

Let F ∈ D(0)1,2, then there exists a sequence Fn∞n=0 where Fn ∈ S for all n ∈ N

(Fn = fn(W (h1), . . . ,W (hnk), ω′) for fn continuously differentiable with polynomial growth)

that converges to F in L2(Ω) as n→∞. Every term of Fn is in D(0)1,2. By Lemma 6.2.11 there

exists a subsequence Fnk∞k=0 such that limk→∞D(0)Fnk

= D(0)F in L2([0, T ]×Ω). We note

that the elements of the sequence Fnk∞k=0 are separable, thus we can apply Proposition

6.2.9 to the process f(Fnk) and we have

D(0)t f(Fnk

) = f ′(Fnk)D

(0)t Fnk

.

f is continuously differentiable and of polynomial growth, hence

limk→∞

f(Fnk) = f(F ) in L2(Ω).

By the dominated convergence theorem we have

limk→∞

f ′(Fnk) = f ′(F ) in L2(Ω).

Thus, we have

limk→∞

f(Fnk)D

(0)t Fnk

= f ′(F )D(0)t F in L2([0, T ]× Ω) for all t ∈ [0, T ].

The operator D(0)t is closable, hence limk→∞D

(0)t f(Fnk

) = D(0)t f(F ) in L2([0, T ]× Ω) for all

t ∈ [0, T ]. This completes the proof. 2

143


Let δ(l) : L2(Ω × Ul) → L2(Ω) be the adjoint operator of the directional derivative D(l)

ul ,

l = 0, 1 where Ul is given in (6.21). We denote by Dom(δ(l)) the domain of δ(l).

Definition 6.3.1 Let h ∈ L2(Ω × Ul). Then h belongs to Dom(δ(l)) if for all F ∈ D(l)1,2 we

have

| E[

∫Ul

D(l)

ul Fh(ul)〈Hl〉(dul)] |≤ C ‖ F ‖L2(Ω) (6.50)

where C is some constant depending on h. For every h ∈ Dom(δ(l)) we can define the

Skorohod integral in the lth direction as

E[

∫Ul

D(l)

ul Fh(ul)〈Hl〉(dul)] = E[Fδ(l)(ul)] (6.51)

for any F ∈ D(l)1,2.

Proposition 6.3.2 Let F belongs to L2(Ω) and h ∈ L2(Ul) with chaos expansion

F = E[F ] +∞∑n=1

∑i1,...,in=0,1

Li1,...,inn (fi1,...,in). (6.52)

Then

δ(l)(Fh) =

∫Ul

E[F ]h(ul)Hl(dul)

+∞∑n=1

∑i1,...,in=0,1

n∑k=1

∫Uin

. . .

∫Uik+1

∫Ul

∫Uik

· · ·∫Ui1

fi1,...,in(uI11 , . . . , uinn )h(ul)1Gi1,...,in

· 1tk<t<tk+1Hi1(dui11 ) · · ·Hik(du

ikk )Hl(du

l)Hik+1(du

ik+1

k+1 ) · · ·Hin(duinn ) (6.53)

if the infinite sum converges in L2(Ω).

Proof


As in the Brownian motion case, the Skorohod integral and the Ito integral coincide in the

case of adapted processes.

Proposition 6.3.3 Let h be an adapted process such that E[∫Ulh2(ul)〈Hl〉(dul)] <∞. Then

h ∈ Dom(δ(l) for l = 0, 1 and

δ(l)(h) =

∫Ul

h(ul)Hl(dul). (6.54)

144

The following proposition gives the relationship between the Skorohod integral δ(l) and the

directional derivative D(l)

ul .

Proposition 6.3.4 Let h be an adapted process such that E[∫Ulh2(ul)〈Hl〉(dul)] <∞. Then,

for h ∈ D(0)1,2,

D(l)

ul

∫ T

0

h(s)dWs =

h(t) +

∫ TtD

(0)t h(s)dWs if l = 0∫ T

tD

(1)t,z h(s)dWs if l = 1.

and, for h ∈ D(1)1,2,

D(l)

ul

∫ T

0

∫R0

h(s, θ)N(ds, dθ) =

∫ TtD

(0)t h(s, θ)N(ds, dθ) if l = 0

h(t, z) +∫ Tt

∫R0D

(1)t,z h(s, θ)N(ds, dθ) if l = 1.

Proof

The proof follows from the definition of the directional derivative. 2

6.4 Greeks for jump diffusion models

Let Xt, 0 ≤ t ≤ T be an n-dimensional process satisfying the following stochastic differ-

ential equation

dXt = b(Xt)dt+ σ(Xt)dWt +

∫R0

γ(Xt, z)N(dt, dz), X0 = x, x ∈ Rn (6.55)

where Wt, 0 ≤ t ≤ T is a d-dimensional Brownian motion, N(dt, dz) is a compensated

Poisson random measure. The coefficient b : R×Rn → Rn represents the deterministic drift,

σ : R×Rn → Rn×Rd represents the volatility and γ : R×R×Rn → Rn×R represents the

jump size. We assume that the coefficients b, σ and γ are continuously differentiable with

bounded derivatives. The coefficients also satisfy the linear growth condition

| b(x) |2 +σ(x)2 +

∫R0

| γ(x, z) |2 ν(dz) ≤ C(1+ | x |2) (6.56)

for each x ∈ Rn where C is a positive constant as well as the Lipschitz condition

| b(x)− b(y) | + | σ(x)− σ(y) | + | γ(z, x)− γ(z, y) |≤ K | x− y | (6.57)

for all x, y ∈ Rn and z ∈ R0, for a constant K < ∞. These conditions ensure the existence

of a unique solution Xt, 0 ≤ t ≤ T of Equation (6.55).

145

We assume that the matrix σ satisfies the uniform ellipticity condition. As in the diffusion

case, we introduce the first variation process Yt, 0 ≤ t ≤ T associated with the solution

Xt, 0 ≤ t ≤ T. The process Yt, 0 ≤ t ≤ T satisfies the stochastic differential equation

dYt = b′(Xt)Ytdt+n∑i=1

σ′i(Xt)YtdW(i)t +

∫R0

γ′(Xt, z)YtN(dt, dz), Y0 = I (6.58)

where I is the identity matrix of Rn, the primes denote derivatives and σi is the ith column

vector of σ. We can show that the first variation process Yt, 0 ≤ t ≤ T of Xt, 0 ≤ t ≤ Tis the derivative of Xt with respect to x, that is,

Yt :=∂Xt

∂x. (6.59)

Assuming that Xt ∈ D1,2, we can show, following the same steps as for the Brownian motion

case, that the Malliavin derivative of Xt is given by

D(0)s Xt = YtY

−1s σ(Xs)1s≤t (6.60)

We assume that the payoff Φ depends on a finite set of payments dates: t1, t2, . . . , tn, with

the convention that t0 = 0 and tn = T . Given 0 < t1 < . . . < tn = T , the option price, under

the risk-neutral probability measure, is given by

u(x) = E[e−rTΦ(Xt1 , · · · , Xtn) | X0 = x] (6.61)

for some underlying assets Xt1 , . . . , Xtn . The payoff Φ : Rn → R is assumed to be square

integrable and of polynomial growth. The following proposition gives the explicit formula

for calculating ∆ (see [25]).

Proposition 6.4.1 Let a ∈ L2([0, T ]) be an adapted process such that∫ ti

0

a(t)dt = 1, for all i = 1, .., n. (6.62)

Assume that b, σ and γ (in Equation (6.55)) are continuously differentiable with bounded

partial derivatives and that the matrix σ satisfies the uniform ellipticity condition. Let Φ :

Rn → R be a function of polynomial growth. Then we have

∆ = E[e−rTΦ(Xt1 , · · · , Xtn)

∫ T

0

a(t)(σ−1(Xt)Yt)TdWt | X0 = x]. (6.63)

146

Proof

We first assume that Φ is continuously differentiable. Then we can calculate the derivative

inside the expectation and we get

∆ =∂


n∑i=1

∂

∂xiΦ(Xt1 , · · · , Xtn)

∂Xti

∂x| X0 = x]

= E[e−rTn∑i=1

∂

∂xiΦ(Xt1 , · · · , Xtn)Yti | X0 = x] (6.64)

where Yti is the first variation process of Xti . As a consequence of Lemma 3.0.2 we have

Yti =

∫ T

0

a(t)(D(0)t Xti)σ

−1(Xt)Ytdt (6.65)

for any a ∈ L2([0, T ]). Substituting Equation (6.65) into Equation (6.64) we obtain

∆ = e−rTE[

∫ T

0

n∑i=1

∂

∂xiΦ(Xt1 , · · · , Xtn)D

(0)t Xtia(t)(σ

−1(Xt)Yt)Tdt | X0 = x]. (6.66)

An application of the chain rule gives

∆ = e−rTE[

∫ T

0

D(0)t Φ(Xt1 , · · · , Xtn)a(t)(σ−1(Xt)Yt)

Tdt | X0 = x]. (6.67)

Since the matrix σ is uniformly elliptic by assumption, we can deduce that

a(t)σ−1(Xt)Yt ∈ Dom(δ0). An application of the relation (2.22) to Equation (6.67) gives

∆ = E[e−rTΦ(Xt1 , · · · , Xtn)δ0(a(t)(σ−1(Xt)Yt)

T)| X0 = x]. (6.68)

The proof is completed by using the Equation (2.24).

The general case is obtained by using limit arguments (see Proposition 3.0.4). We omit

details. 2

Remark

The expression for ∆ is given in terms of the Ito integral. This is because the Ito integrals

are in general easy to simulate. We note that the formula for ∆ is essentially the same as the

one obtained in the Brownian motion case. The only difference is in the underlying assets

which in this case consist of jump parts.

We provide an example that illustrates how we apply Proposition 6.4.1 to calculate ∆.

147

Example

We consider the geometric Levy process

dSt = bStdt+ σStdWt +

∫R0

zStN(dt, dz), S0 = x. (6.69)

By Ito formula the solution of Equation (6.69) at final time t = T is given by

ST = x exp(b− 1

2σ2)T +σWT +

∫ T

0

∫R0

ln(1+z)−zν(dz)ds+

∫ T

0

∫R0

ln(1+z)N(ds, dz).

(6.70)

We note that

YT =∂ST∂x

=STx, σ−1(Xt) =

1

σST. (6.71)

Choose

a(t) =1

T.

We therefore have

∆ = E[e−rTΦ(ST )

∫ T

0

1

T

1

σST

STxdWt] =

e−rT

σxTE[e−rTΦ(ST )WT ].

where

WT =1

σln(STx

)−(b− 1

2σ2)T−

∫ T

0

∫R0

ln(1+z)−zν(dz)ds−∫ T

0

∫R0

ln(1+z)N(ds, dz).

Next, we compute the derivative of the option price with respect to the drift coefficient. As

in the Brownian motion case, we introduce the perturbed process

dXεt = (b(Xε

t ) + εb(Xεt ))dt+ σ(Xε

t )dWt +

∫R0

γ(z,Xεt )N(dt, dz), Xε

0 = x (6.72)

where ε is a small parameter and b is a bounded function. The corresponding option price

has the form

uε(x) = E[e−rTΦ(Xεt1, · · · , Xε

tn) | Xε0 = x]. (6.73)

The following result gives the derivative of the option price with respect to the parameter ε

(see [25]).

Proposition 6.4.2 Assume that b, b, σ and γ (in Equation (6.72)) are continuously differ-

entiable with bounded partial derivatives and that the matrix σ satisfies the uniform ellipticity

condition. Let Φ : Rn → R be a function of polynomial growth. Then we have

ρ :=∂

∂εuε(x) |ε=0= E[e−rTΦ(Xt1 , · · · , Xtn)

∫ T

0

(σ−1(Xt)b(Xt))TdWt | X0 = x]. (6.74)

148

Proof

The proof follows the same steps as in Proposition 3.1.3. We omit the details. 2

Next we consider the derivative of the option price with respect to the volatility term. Here,

we define the perturbed process Xεt , 0 ≤ t ≤ T by

dXεt = b(Xε

t )dt+ (σ(Xεt ) + εσ(Xε

t ))dWt +

∫R0

γ(z,Xεt )N(dt, dz), Xε

0 = x. (6.75)

where ε is a small parameter and σ is a continuously differentiable function with bounded

derivative. The variation process Zεt , 0 ≤ t ≤ T of Xε

t , 0 ≤ t ≤ T satisfies the following

stochastic differential equation

dZεt = b′(Xε

t )Zεt dt+(σ′(Xε

t )+εσ′(Xε

t ))Zεt dWt+σ(Xε

t )dWt+

∫R0

γ′(z,Xεt )Z

εt N(dt, dz), Zε

0 = 0

where 0 is the column vector in Rn. Further, we recall

Υn = a(t) ∈ L2([0, T ]) :

∫ ti

ti−1

a(t)dt = 1 for all i = 1, . . . , n. (6.76)

Proposition 6.4.3 Assume that b, σ, σ and γ (in Equation (6.75)) are continuously dif-

ferentiable with bounded partial derivatives and that the matrix σ is uniformly elliptic. Let

βti = ZtiY−1ti , i = 1, . . . , n such that σ−1(Xt)Ytβt belongs to the domain of the Skorohod

integral for all t ∈ [0, T ]. Let Φ : Rn → R be a function of polynomial growth. Then, for any

a(t) ∈ Υn and for any Φ of polynomial growth, we have

V :=∂

∂εuε(x) |ε=0= E[e−rTΦ(Xt1 , · · · , Xtn)δ0

(σ−1(Xt)Y βt

)| X0 = x] (6.77)

where βt =∑n

i=1 a(t)(βti − βti−1)1ti−1≤t≤ti.

Proof

The proof follows the same steps as in Proposition 3.1.5. We omit the details. 2

We denote the derivative of the option price with respect to the amplitude parameter γ by

V1. We define the perturbed process

dXεt = b(Xε

t )dt+ σ(Xεt )dWt +

∫R0

(γ(z,Xεt ) + εγ(z,Xε

t ))N(dt, dz), Xε0 = x. (6.78)

where ε is a small parameter and γ is a continuously differentiable function with bounded

derivative. As in the above cases, we introduce the variation process Zεt , 0 ≤ t ≤ T of

149

Xεt , 0 ≤ t ≤ T as follows

dZεt = b′(Xε

t )Zεt dt+ σ′(Xε

t )Zεt dWt +

∫R0

(γ′(z,Xεt ) + εγ′(z,Xε

t ))Zεt N(dt, dz)

+

∫R0

γ(z,Xεt )N(dt, dz)

with Zε0 = 0. The derivative of the option price with respect to the amplitude is obtained in

the same way as V . Therefore we have the following proposition.

Proposition 6.4.4 Assume that b, σ, γ and γ (in Equation (6.78)) are continuously dif-

ferentiable with bounded partial derivatives and that the matrix σ is uniformly elliptic (see

3.6 in Chapter 3). Let βti = ZtiY−1ti , i = 1, . . . , n be such that σ−1(Xt)Ytβt belongs to the

domain of the Skorohod integral for all t ∈ [0, T ]. Then, for any a(t) ∈ Υn and for any

Φ : Rn → R of polynomial growth, we have

V1 :=∂

∂εuε(x) |ε=0= E[e−rTΦ(Xt1 , · · · , Xtn)δ0

(σ−1(Xt)Y βt

)| X0 = x] (6.79)

where βt =∑n

i=1 a(t)(βti − βti−1)1ti−1≤t≤ti.

Proof

The proof follows the same as in Proposition 3.1.5. We omit the details. 2

6.5 Greeks for the Heston model with jumps

We extend the Heston model discussed in Chapter 3 to include jumps in both the stock and

the variance. The jumps in the variance being of positive deterministic size. We show how

to calculate ∆ of the Heston model with jumps. We incorporate the jumps into the stock

price as follows

dSt = bStdt+√vtStdW

(1)t +

∫RzSt−N(dt, dz) (6.80)

where W(1)t is a standard Brownian motion and N(dt, dz) is a compensated Poisson random

measure. Using Equation (3.84) in Chapter 3 and letting σ =√vt we can write Equation

(6.80) as

dSt = bStdt+ ρσsStdW(2)t + σsSt

√1− ρ2dZt +

∫RzSt−N(dt, dz). (6.81)

We incorporate the jumps into the variance process as follows

dvt = κ(θ − vt)dt+ ν√vtdW

(2)t +

∫RγN(dt, dz). (6.82)

150

where γ is the constant of jump size in the volatility. Recall that

σ :=√vt.

Applying Ito’s formula on σ =√vt we obtain

dσt = 0 +1

2v− 1

2t [κ(θ − vt)dt+ ν

√vtdW

(2)t ] +

1

2ν2vt(−

1

4)v

− 32

t dt

+

∫|z|<R

√vt + γ −

√vt −

1

2v− 1

2t γν(dz)dt+

∫R√vt + γ −

√vtN(ds, dz) (since

√vt > 0 a.s.)

=

((κθ

2− ν2

8

)1√vt−κ√vt

2+

∫|z|<R

√vt + γ −

√vt −

1

2v− 1

2t γν(dz)

)dt

+ν2

2dW

(2)t +

∫R

(√vt + γ −

√vt)N(ds, dz)

=

((κθ

2− ν2

8

)1

σt− κσt

2+

∫|z|<R

√σ2t + γ − σt −

γ

2σtν(dz)

)dt

+ν

2dW

(2)t +

∫R(√σ2t + γ − σt)N(ds, dz). (6.83)

As in the case without jumps, we work with logarithmic price Xt = logSt rather than the

actual price. Applying Ito’s formula on Xt = logSt we obtain

dXt = (b− 1

2σ2t )dt+

√1− ρ2σtdZt + ρσtdW

(2)t +

∫RzN(dt, dz). (6.84)

The two Equations (6.83) and (6.84) can be thought of as a single two dimensional stochastic

differential equation(Xxt

σt

)=

(log x

σ0

)+

∫ t

0

(b− 12σ2s)

(κθ2− ν2

8

)1σs− 1

2κσs +

∫|z|<R

√σ2s + γ − σs − γ

2σsν(dz)

ds

+

∫ t

0

( √1− ρ2σs ρσs

0 ν2

)(dZs

dW(2)s

)+

∫ t

0

∫R

(z√

σ2s + γ − σs

)N(ds, dz).

We assume that σt is Malliavin differentiable. The arguments on how to compute Greeks in

the case without jumps in Chapter 3 Section 3.5 can be extended in a straight-forward way

to computing Greeks for Heston models with jumps. Therefore, by applying Proposition

6.4.1 with

a(t) =1

T, Yt =

(1x

0

)and σ−1(Xt) =

1√1−ρ2σs

− 2ρ

ν√

1−ρ2

0 2ν

we obtain the following result:


∫ T

0

1

xT√

1− ρ2σsdZs] (6.85)

where x = S0 is the initial stock price.

151

6.6 Greeks for Levy process

In the last stages of writing this thesis we came across a recent paper [12] that contains

similar results to those that are presented in this section. We mention that the authors

consider the case where the Levy process Xt is separable and the payoff functional belonging

to L2(Ω). Apart from the use of Malliavin calculus, the paper considers the Fourier approach

which is not used in our case.

We are motivated by ideas in [4] where the authors studied the approximation of small jumps

in a Levy process by a Brownian motion. We extend the ideas and consider computation of

Greeks.

Let X(t), 0 ≤ t ≤ T be a Levy process. From the Levy-Ito decomposition, X(t) can

be represented as a sum of a deterministic drift, a Brownian motion, a compound Poisson

process and an almost sure limit of compensated compound Poisson process (see [23])

X(t) = bt+ σW (t) +X l(t) + limε↓0

Xε(t) (6.86)

with

X l(t) :=∑s≤t

∆X(s)1|∆X(s)|≥1 =

∫|z|≥1

zN(t, dz) (6.87)

and

Xε(t) :=∑s≤t

∆X(s)1ε≤|∆X(s)|<1 − t

∫ε≤|z|<1

zν(dz) =

∫ε≤|z|<1

zN(t, dz)− t

∫ε≤|z|<1

zν(dz)

(6.88)

where W (t), 0 ≤ t ≤ T is a standard Brownian motion, N(dt, dz) is a Poisson process

and N(dt, dz) is a compensated Poisson random measure of X(t) and b, σ ∈ R are two

constants with σ > 0. W (t), X l(t) and Xε(t) are independent of each other. The convergence

on the right hand side of Equation (6.86) holds a.s uniformly in t ∈ [0, T ]. The term∑s≤t ∆X(s)1|∆X(s)|≥1 in (6.87) describes all the jump sizes greater than or equal to 1 while

the term limε↓0 Xε(t) in Equation (6.86) describes all jump sizes less than 1. We can rewrite

Equation (6.86) as

X(t) = bt+ σW (t) +

∫ε≤|z|<1

z[N(t, dz)− tν(dz)] +

∫|z|≥1

zN(t, dz). (6.89)

We want to find an approximation of a Levy process Xε(t), 0 ≤ t ≤ T which approximates

the Levy process X(t), 0 ≤ t ≤ T in some sense to be specified below. There are several

ways of doing this (see [4]). Here we focus on the approximation by a Brownian motion.

152

Define

σ2(ε) :=

∫|z|<ε

z2ν(dz). (6.90)

The function σ(ε) represents the standard deviation of the jumps smaller than ε of the Levy

process X(t). By dominated convergence theorem, σ2(ε) converges to 0 as ε tends to 0.

We approximate the small jumps of a Levy process X(t) by a Brownian motion as follows

Xε(t) = µεt+ (σ2 + σ2(ε))12W ε(t) +

∫ε≤|z|<1

z[N(t, dz)− tν(dz)] +

∫|z|≥1

zN(t, dz) (6.91)

where

µε = b−∫ε≤|z|<1

zν(dz) and W ε(t) =1√

σ2 + σ2(ε)σW (t) + σ(ε)B(t) (6.92)

with B(t) a new Brownian motion independent of W (t) and σ2(ε) is given in Equation (6.90).

The drift term µε keeps the overall mean of Xt unchanged, that is, E[Xε(t)] = E[X(t)] for

all t. We note that a Brownian motion appears even if the original process does not have

one (σ = 0).

The following proposition says that the approximation of a Levy process Xε(t) converges in

distribution to the original Levy process X(t) (see [13]).

Proposition 6.6.1 Let the Levy processes X(t) and Xε(t) be defined as in Equations (6.86)

and (6.91) respectively. Then, for every t, we have

limε→0

Xε(t) = X(t) in L1. (6.93)

Proof

We have

E[| Xε(t)−X(t) |] = E[| (σ2 + σ2(ε))12W ε(t)− σW (t)− t

∫ε≤|z|<1

zν(dz) |].

From the approximation (6.92), we have

(σ2 + σ2(ε))12W ε(t) = σW (t) + σ(ε)B(t).

Therefore, we have

E[| Xε(t)−X(t) |] = E[| σ(ε)B(t)− t

∫ε≤|z|<1

zν(dz) |].

153

Using the triangle and Cauchy-Schwartz inequalities we have

E[| Xε(t)−X(t) |] ≤ σ(ε)E[| B(t) |] + E[| t∫ε≤|z|<1

zν(dz) |]

≤ σ(ε)E[| B2(t)]12 + E[

(∫ t

0

∫ε≤|z|<1

zν(dz)dt

)2

]12

≤ σ(ε)√t+

√t

∫ε≤|z|<1

z2ν(dz)

= 2σ(ε)√t.

As ε → 0, σ2(ε) → 0 and hence E[| Xε(t) − X(t) |] → 0. This proves the convergence in

L1. 2

In [4] a rigorous discussion is presented of when the approximation (6.91) is valid. It turns

out that the approximation is valid if and only if for each κ > 0

limε→0

σ(κσ(ε) ∧ ε)σ(ε)

= 1. (6.94)

A sufficient condition for (6.94) is

limε→0

σ(ε)

ε= ∞. (6.95)

It is proven that the two conditions (6.94) and (6.95) are equivalent if the Levy measure ν

has no atoms in some neighborhood of the origin (see [4]).

We consider the jump diffusion stochastic differential equation of the form

dXt = b(Xt)dt+ σ(Xt)dWt +

∫R0

γ(Xt, z)N(dt, dz) (6.96)

where Wt is a Brownian motion and N is the compensated Poisson random measure. We

assume that the coefficients b, σ and γ are continuously differentiable with bounded deriva-

tives. In addition the coefficients b, σ and γ satisfy the linear growth condition and the

Lipschitz condition.

In [12] the authors consider γ to be of the form γ(x, z) = γ1(x)g(z) x ∈ R, z ∈ R0 where

γ1(x) has a linear growth and is Lipschitz continuous and the factor g(z) satisfies∫R0

g2(z)ν(dz) <∞.

154

We approximate the small jumps of the jump diffusion by a Brownian motion as follows

dXεt = b(Xε

t )dt+ σ(Xεt )dWt + σ(ε)dBt +

∫|z|≥ε

γ(Xεt , z)N(dt, dz)

= b(Xεt )dt+ (σ2(Xε

t ) + σ2(ε))12dW ε

t +

∫|z|≥ε

γ(Xεt , z)N(dt, dz) (6.97)

where σ(ε) is defined in Equation (6.90).

The solutions Xt and Xεt exist and are unique (see [50] Theorem 9.1). Assume that Xt and

Xεt belong to D1,2. The first variation process Y ε

t of Xεt satisfies the stochastic differential

equation

dY εt = b′(Xε

t )Yεt dt+

σ(Xεt )σ

′(Xεt )Y

εt√

σ2(Xεt ) + σ2(ε)

dW εt +

∫|z|≥ε

γ′(Xεt , z)Y

εt N(dt, dz) (6.98)

where the primes denote the derivative.

We denote by ∆ε derivative of the option price with the underlying asset Xε(T ) with respect

to the initial price. Applying the Malliavin calculus approach discussed in the previous

sections we have the following result.

Theorem 6.6.2 Let a ∈ L2([0, T ]) be an adapted process such that∫ ti

0

a(t)dt = 1 for all i = 1, .., n. (6.99)

Assume that b, σ and γ (in Equation (6.97)) are continuously differentiable with bounded

partial derivatives and that the matrix σ satisfies the uniform ellipticity condition. Then, for

any Φ : R → R of polynomial growth, we have

∆ε = E[e−rTΦ(XεT )

∫ T

0

a(t)

(Y εt√

σ2(Xεt ) + σ2(ε)

)T

dW εt | X0 = x]. (6.100)

Proof

The proof follows the same arguments as in Proposition 6.4.1. We omit the details. 2

In particular, we consider the stochastic differential equation of the form

dXεt = Xε

t [bdt+ (σ2 + σ2(ε))12dW ε

t +

∫R0

zN(dt, dz)], Xε0 = x (6.101)

where b and σ are constants. Using Proposition 6.6.2 with a(t) = 1T

and Y εt =

Xεt

x, we

calculate ∆ε as follows

∆ε =∂

∂xE[e−rTΦ(Xε(T ))] = E[e−rTΦ(Xε(T ))

W εT

xT√σ2 + σ2(ε)

]. (6.102)

155

We have seen before that ∆ is calculated as follows

∆ = E[e−rTΦ(X(T ))WT

xσT]. (6.103)

The following proposition gives the convergence rate of the approximation ∆ε to ∆.

Proposition 6.6.3 Let the Levy processes X(t) and Xε(t) be defined as in Equations (6.86)

and (6.91) respectively. Assume that the matrix σ satisfies the uniform ellipticity condition.

Let Φ : R → R be of polynomial growth. Then ∆ε converges to ∆.

Proof

Using the triangle and Cauchy-Schwartz inequalities we have

| ∆ε −∆ | = | E[e−rTΦ(Xε(T ))W εT

xT√σ2 + σ2(ε)

]− E[e−rTΦ(X(T ))WT

xσT] |

≤ E[| e−rTΦ(Xε(T ))− e−rTΦ(X(T )) || W εT

xT√σ2 + σ2(ε)

|]

+ E[| e−rTΦ(X(T )) || W εT

xT√σ2 + σ2(ε)

− WT

xσT|]

≤ E[| e−rTΦ(Xε(T ))− e−rTΦ(X(T )) |2]12 E[

(W εT

xT√σ2 + σ2(ε)

)2

]12

+ E[(e−rTΦ(X(T )))2]12 E[| W ε

T

xT√σ2 + σ2(ε)

− WT

xσT|2]

12 .

Let C be the polynomial growth constant which we assume to equal to the Lipschitz constant

of Φ for convenience. Then, we get

| ∆ε −∆ | ≤ Ce−rT

xT√σ2 + σ2(ε)

E[| Xε(T )−X(T ) |2]12 + Ce−rTE[1+ | X(T ) |2]

12

· E[

(W εT

xT√σ2 + σ2(ε)

− WT

xσT

)2

]12 .

We recall that W εT = 1√

σ2+σ2(ε)σWT + σ(ε)BT and that WT is independent of BT . Then

we can show that

E[

(W εT

xT√σ2 + σ2(ε)

− WT

xσT

)2

]12 =

σ(ε)

xσT√σ2 + σ2(ε)

√T

and in the proof of Proposition 6.6.1 we have

E[| Xε(T )−X(T ) |] ≤ 2σ(ε)√T .

156

Therefore, we have

| ∆ε −∆ | ≤ 2Ce−rT

xT√σ2 + σ2(ε)

σ(ε)√T + Ce−rT

σ(ε)

xσT√σ2 + σ2(ε)

√T .

As ε goes to 0, σ(ε) goes to 0 and hence

| ∆ε −∆ |→ 0.

Hence, the result follows. 2

In the case of pure jump case, that is, where there is no Brownian motion in the original

Levy process, we notice that a Brownian motion term appears in our approximation, that

is,

Xε(t) ≈ X(t) + σ(ε)B(t) (6.104)

where B(t) is assumed to be independent of Xε(t).

This is crucial for the application of standard Malliavin calculus approach. Assume that the

approximation Xε(t) in Equation (6.104) belongs to D1,2. An application of the Malliavin

calculus gives the Greek of the approximation of the Levy process ∆ε as

∆ε = E[e−rTΦ(Xε(T ))BT

xσ(ε)T]. (6.105)

Remarks

1. Although BT

σ(ε)may fail to be bounded for some integrable random variables, we can

still obtain convergence.

2. Since we can approximate a Levy process by a jump diffusion processes, we can apply

this method to a wider range of Levy processes.

157

Chapter 7

White noise calculus for Levy

Processes and its Application to the

Calculations of Greeks

In this chapter we first review the extension of white noise analysis developed in Chapter

4 to pure jump case (see [27] and the references therein). We follow the construction in

[27]. The results given here generalize the known results for the Malliavin calculus in the

case of Brownian motion. Most of the results given here are known but we think that it is

of interest to have unified approach based on the white noise analysis for Levy processes.

We mention that the white noise analysis developed in [27] is mainly used to generalize the

Clark-Haussmann-Ocone formula.

Our goal in this chapter is to derive explicit expressions for Greeks in the white noise setting

using Malliavin calculus. We make use of the Wick chain rule (to be defined later) and the

Donsker delta function.

7.1 Basic concepts of Levy white noise analysis

As in the Brownian motion case, we let Ω = S ′(R) be the space of tempered distributions

equipped with its Borel σ-algebra F = B(Ω). The space S ′(R) is the dual of the Schwartz

space S(R) of test functions, that is, the rapidly decreasing smooth functions on R. Then

we define the Levy white noise probability measure µ, which exists by the Bochner-Minlos

theorem (see [47] Appendix A), as the measure dµ defined on the Borel σ-algebra B(Ω) of

158

subsets of Ω by ∫Ω

ei〈ω,f〉dµ(ω) = e∫

R ψ(f(y))dy, f ∈ S(R) (7.1)

where i =√−1 and 〈ω, f〉 = ω(f) denotes the action of ω ∈ Ω = S ′(R) applied to f ∈ S(R)

and ψ is given by

ψ(u) = −iau− 1

2σ2u2 +

∫R(eiuz − 1− iuz1|z|<1)ν(dz). (7.2)

Here a ∈ R, σ2 ≥ 0 and ν is a Levy measure on R0.

We suppose that the Levy measure ν(dz) satisfies the condition (5.10), that is, for every

ε > 0 there exists a λ > 0 such that∫(−ε,ε)c

eλ|z|ν(dz) <∞. (7.3)

The triple (Ω,B(Ω), µ) is called the Levy white noise probability space.

Lemma 7.1.1 Let f ∈ S(R). Then we have

E[〈·, f〉] = 0 and E[〈·, f〉2] = M

∫R0

f 2(y)dy. (7.4)

where M =∫

R0z2ν(dz) <∞.

Proof

The proof follows the same arguments as in Lemma 4.1.1. We omit the details. 2

In this chapter we will denote the Levy process by η. Similar to the Brownian motion case we

can, using the Lemma 7.1.1, extend the definition of 〈ω, f〉 from f ∈ S(R) to any f ∈ L2(R).

We can then construct the Levy process η(t, ω) as the cadlag version of η(t, ω) where

η(t, ω) = 〈ω, χ[0,t](·)〉 (7.5)

which is defined since χ[0,t](·) is in L2(R). This leads to the following theorem (see [27]).

Theorem 7.1.2 The stochastic process η(t), 0 ≤ t ≤ T has a cadlag version denoted by

η. The process η(t), 0 ≤ t ≤ T is a Levy process with Levy measure ν.

The Levy process ηt admits the following stochastic integral representation

ηt = at+ σWt +

∫ t

0

∫R0

zN(ds, dz), 0 ≤ t ≤ T, (7.6)

159

where Wt, 0 ≤ t ≤ T is the standard Brownian motion and N(dt, dz) is a compensated

Poisson random measure associated with ηt. Assuming the integrability condition (7.3), we

consider a Levy process with no drift and σ = 0, that is, the representation (7.6) reduces to

ηt =

∫ t

0

∫R0

zN(ds, dz), 0 ≤ t ≤ T. (7.7)

This is called the pure jump Levy process. The Poisson process is the most important

representative among the pure jump Levy processes and it corresponds to the specific case

in which the measure ν is a point mass at 1.

7.2 Chaos expansion

Let Ft0≤t≤T be the completed filtration generated by the Levy process in (7.7). Fix

F = FT . As in the Brownian motion case, we make use of multi-indices of arbitrary length.

Here we let A be the set of multi-indices α = (α1, α2, . . .) which have only finitely non-zero

values. Further we set Index(α) = maxi : αi 6= 0 and | α |=∑

i αi for α ∈ A.

Next we consider two families of orthogonal polynomials. We follow the construction in [27].

First we consider the complete orthonormal system ξj, j = 1, 2, . . . of L2([0, T ]) consisting

of the Laguerre functions of order 12, that is,

ξj(t) =

(Γ(j)

Γ(j + 12)

) 12

e−tt14L

12j−1(t)1(0,∞)(t), t ∈ [0, T ], j = 0, 1, . . . . (7.8)

where Γ(·) is the Gamma function and L12j , j = 0, 1, 2, . . . are the Laguerre polynomials of

order 12

defined by

e−tt12L

12j (t) =

1

j!

dj

dtj

(e−ttj+

12

), j = 0, 1, 2, . . . . (7.9)

The Laguerre functions satisfy

supt∈[0,T ]

| ξj(t) |= O(1). (7.10)

Further, let lmm≥0 = 1, l1, l2, . . . be the Gram-Schmidt orthogonalization of 1, z, z2, . . .with respect to the inner product of L2(ρ) where ρ(dz) = z2ν(dz). Then define the polyno-

mials pj by

pj(z) :=1

‖ zlj−1 ‖L2(ρ)

z · lj−1(z), j = 1, 2, . . . (7.11)

In particular,

p1(z) =1

‖ zl0 ‖L2(ρ)

zl0(z) =z

‖ z ‖L2(ρ)

. (7.12)

160

Therefore z = m2p1(z) where m2 =‖ z ‖L2(ρ). The polynomials pjj≥1 form an orthonormal

basis for L2(ρ) (see [82]). We then define the bijective map

κ : N× N → N; (i, j) → j + (i+ j − 2)(i+ j − 1)/2. (7.13)

We note that κ gives the Cantor enumeration of the Cartesian product N×N. Moreover, if

k = κ(i, j) for i, j ∈ N, set

ζk(t, z) = ξi(t)pj(z). (7.14)

Now, for any α ∈ A with maxi : αi 6= 0 and | α |:=∑

i=1 αi = n we define the tensor

product ζ⊗α as

ζ⊗α(t1, z1, . . . , tn, zn) := ζ⊗α11 ⊗ · · · ⊗ ζ⊗αk

k (t1, z1, . . . , tn, zn)

= ζ1(t1, z1) · · · ζ1(tα1 , zα1) · · · ζj(tα1+···+αk−1+1, zα1+···+αk−1+1 · · · ζk(tn, zn)

with ζ⊗0k = 1. Finally, we denote the symmetrized tensor product of ζ⊗α by ζ⊗α:

ζ⊗α(t1, z1, . . . , tn, zn) = ζ⊗α11 ⊗...⊗ζ⊗αk

j (t1, z1, . . . , tn, zn).

For α ∈ A we define

Kα := I|α|(ζ⊗α). (7.15)

In particular we note that if α = ε(k) with

ε(k)(j) :=

1 if j = k

0 otherwise,

we have

Kε(k) = I1(ζ⊗ε(k)

) = I1(ζk) = I1(ξi(t)pj(z)) (7.16)

where k = κ(i, j). It can be proved that Kαα∈J are orthogonal in L2(µ) and

‖ Kα ‖2L2(µ)= α!

We note, similar to the Brownian motion case, that if | α |= n

α! =‖ Kα ‖2L2(µ)= n! ‖ ζ⊗α ‖2

L2((λ×ν)n) . (7.17)

This leads to the chaos representation for orthonormal systems Kα (see [27] page 202).

Theorem 7.2.1 Every F ∈ L2(µ) admits the unique representation of the form

F =∑α∈A

aαKα (7.18)

161

where aα ∈ R for all α ∈ A. Moreover, the Ito isometry is valid:

‖ F ‖2L2(µ)=

∑α∈A

a2αα!. (7.19)

It is known that any sequence of functions fn ∈ L2((λ× ν)n), n ≥ 1 such that∑n≥1 n! ‖ fn ‖2

L2((λ×ν)n)<∞ defines a random variable F ∈ L2(µ) by

F =∞∑n=0

In(fn)

where by convention I0(f0) = f0 (constant) (see [27]). By the construction of ζ⊗α, any

fn ∈ L2((λ× ν)n) can be written as

fn =∑|α|=n

aαζ⊗α. (7.20)

Hence

In(fn) =∑|α|=n

aαIn(ζ⊗α) =

∑|α|=n

aαKα.

The connection between the expansions

F =∑n≥0

In(fn) and F =∑α∈Aα

aαI|α|(ζ⊗α)

is given by

fn = aαζ⊗α with aα ∈ R.

Example

Choose h ∈ L2(R) deterministic and let F (ω) =∫ T

0h(s)dη(s). For η(t) =

∫ t0

∫R0zN(ds, dz)

we have dη(t) =∫

R0zN(dt, dz). Therefore

F (ω) =

∫ T

0

∫R0

h(s)zN(ds, dz) = I1(hz). (7.21)

Since h(s) =∑

i≥1(h, ξi)L2([0,T ])ξi(s), we can write (7.21) as

F (ω) = I1

(∑i≥1

(h, ξi)L2([0,T ])ξiz

)=∑i≥1

(h, ξi)L2([0,T ])I1(ξiz)

=∑i≥1

(h, ξi)L2([0,T ])I1(ξim2p1) by (7.12)

=∑i≥1

(h, ξi)L2([0,T ])Kεk(i,1)m2 by (7.16).

162

By setting h = χ[0,t], the random variable η(t) ∈ L2(µ) has the expansion

η(t) = m2

∑i≥1

(∫ t

0

ξi(s)ds

)·Kε(i,1)(ω) (7.22)

where m2 =‖ z ‖L2(ρ).

7.3 The Hida/Kondratiev spaces

We introduce the Levy versions of the Hida/Kondratiev test function space (S)ρ and the

Hida/Kondratiev stochastic distribution space (S)−ρ as in the Brownian motion case.

Definition 7.3.1 For 0 ≤ ρ ≤ 1 the Kondratiev test function space (S)ρ consists of all

F =∑

α∈A aαKα ∈ L2(µ) such that

‖ F ‖2ρ,k:=

∑α∈A

(α!)1+ρa2α(2N)kα <∞ for all k ∈ N (7.23)

where (2N)kα := (2 · 1)kα1(2 · 2)kα2 · · · (2 · j)kαj if kα = (kα1, . . . , kαj) ∈ J .

Definition 7.3.2 For 0 ≤ ρ ≤ 1 the Kondratiev distribution space (S)−ρ consists of all

formal series G =∑

α∈A bαKα ∈ L2(µ) such that

‖ G ‖2−ρ,−q:=

∑α∈J

(α!)1−ρb2α(2N)−qα <∞ for some q ∈ N (7.24)

As in the Brownian motion case (S)ρ is endowed with the projective limit topology and

(S)−ρ is endowed with limit topology induced by the above seminorms. We note that for

any f =∑

α∈J aαHα ∈ (S)ρ and F =∑

α∈J bαHα ∈ (S)−ρ the action

〈F, f〉 :=∑α∈J

aαbαα! (7.25)

is well defined and thus the space (S)−ρ is the dual of (S)ρ

We note that for general 0 ≤ ρ ≤ 1 we have the following inclusions

(S)1 ⊂ (S)ρ ⊂ (S)0 ⊂ L2(µ) ⊂ (S)−0 ⊂ (S)−ρ ⊂ (S)−1. (7.26)

The spaces (S) := (S)0 and (S)∗ := (S)−0 are the Levy versions of the Hida test function

space and the Hida stochastic distribution space, respectively. A useful feature of (S)∗ is

that it contains the white noise of the pure jump Levy process η(t) and of the compensated

Poisson random measure as its elements (see [27]).

163

Definition 7.3.3 The Levy white noise process η(t) is defined by the formal expansion

η(t) := m2

∑i≥1

ξi(t)Kε(i,1) = m2

∑i≥1

ξi(t)I1(ξip1) = m2

∑i≥1

ξi(t)I1(ξiz). (7.27)

Lemma 7.3.4 We have η(t) ∈ (S)∗ for all t.

Proof

We have

‖ η(t) ‖2−q = m2

2

∑i≥1

ξ2i (t)(2N)−qε

k(i,1)

= m22

∑i≥1

ξ2i (t)2

−qk(i, 1)−q. (7.28)

Using the fact that k(i, 1) = 1 + (i− 1)i/2 ≥ i and the well-known estimate function

supt∈R

| ξn(t) |= O(n−112 ) (7.29)

we conclude that Equation (7.28) is finite for q ≥ 2. Hence, η(t) ∈ (S)∗ for all t. 2

Comparing Equations (7.22) and (7.27) we have

η(t) =d

dtη(t).

Lemma 7.3.5d

dtη(t) exists in (S)∗ for all t. (7.30)

Proof

The proof follows by the same arguments as in the Brownian motion case (see Lemma 4.2.10).

We omit the details. 2

Definition 7.3.6 The white noise process˙N(t, z) of the compensated Poisson random mea-

sure N(dt, dz) is defined by the expansion

˙N(t, z) =

∑i,j≥1

ξi(t)pj(z)Kε(i,1)(ω). (7.31)

Lemma 7.3.7˙N(t, z) ∈ (S)∗ for all t, z.

164

Proof

The proof follows the same arguments in Lemma 7.3.4. We omit the details. 2

Lemma 7.3.8 η(t) is related to˙N(t, z) by

η(t) =

∫Rz

˙N(t, z)ν(dz). (7.32)

Proof

We have∫R0

z˙N(t, z)ν(dz) =

∫R0

∑i,j≥1

ξi(t)pj(z)Kεi,jzν(dz) =∑i≥1

ξi(t)I1

(ξi∑j≥1

pj

∫R0

pjzν(dz)

)=

∑i≥1

ξi(t)I1(ξiz) = η(t) by (7.27) 2

For any Borel set Λ ∈ B(R) such that its closure does not contain 0 we have

N(t,Λ) = I1(χ[0,t](s)χΛ(z)) =∑i,j≥1

(χ[0,t], ξi)L2(R)(χΛ, pj)L2(ν)I1(ξipj)

=

∫ t

0

∫Λ

(∑i,j≥1

ξi(s)pj(z)Kεi,j(ω)

)ν(dz)ds by (7.16).

So formally we have

˙N(t, z) =

N(dt, dz)

dt× ν(dz)(7.33)

which is similar to the Radon Nikodym derivative in (S)∗.

7.4 Levy Wick product

We define a Levy Wick product as in Chapter 4 (see [27] page 205).

Definition 7.4.1 The Wick product F G of two elements of (S)−1

F =∑α∈A

aαKα and G =∑β∈A

bβKβ, aα, bβ ∈ R

is defined by

F G =∑α,β∈A

aαbβKα+β =∑γ∈A

( ∑α+β=γ

aαbβ

)Kγ. (7.34)

165

The spaces (S)1, (S), (S)∗ and (S)−1 are topological algebras with respect to the Wick prod-

uct (see [47] page 47 for the Brownian motion case).

Remark

Let fn =∑

|α|=n cαζ⊗α ∈ L2(λ× ν)n and gm =

∑|β|=m bβζ

⊗β ∈ L2(λ× ν)m. Then we have

fn⊗gm =∑|α|=n

∑|β|=m

cαbβζ⊗(α+β) =

∑|γ|=n+m

( ∑α+β=γ

cαbβ

)ζ⊗γ

in L2(λ× ν)n+m. Hence

In(fn) Im(gm) = In+m(fn⊗gm).

Example

Choose h ∈ L2([0, T ]) and define F =∫ T

0h(t)dη(t). Then

F F = I1(h1z1) I1(h2z2) = I2(h1h2z1z2)

= 2

∫ T

0

∫R0

(∫ T

0

∫R0

h(t1)h(t2)z1z2N(dt1, dz1)

)N(dt2, dz2)

= 2

∫ T

0

(∫R0

h(t1)dη(t1)

)h(t2)dη(t2).

By the Ito formula, if we put X(t) =∫ t

0h(s)dη(s),

d(X(t))2 = 2X(t)dX(t) + h2(t)

∫R0

z2N(dt, dz).

Hence

F F = 2

∫ T

0

X(s)dX(s) = X2(T )−∫ T

0

∫R0

h(s)2z2N(ds, dz). (7.35)

In particular, choosing h = 1 we obtain

η(T ) η(T ) = η2(T )−∫ T

0

∫R0

z2N(ds, dz). (7.36)

We define an integral in (S)∗ as follows.

Definition 7.4.2 A function X : [0, T ]× R0 → (S)∗ is (S)∗-integrable if

〈X(·), f〉 ∈ L1(λ× ν) for all f ∈ (S),

where the action 〈·, ·〉 is defined in Equation (7.25). Then the (S)∗-integral of X, denoted by∫ T0

∫R0X(t, z)ν(dz)dt, is the unique element of (S)∗ such that

〈∫ T

0

∫R0

X(t, z)ν(dz)dt, f〉 =

∫ T

0

∫R0

〈X(t, z), f〉ν(dz)dt for all f ∈ (S). (7.37)

166

It is a result of Proposition 8.1 in [46] that Equation (7.37) defines∫ T

0

∫R0X(t, z)ν(dz)dt as

an element of (S)∗.

The next theorem gives the relation between the Skorohod integral and the Wick product.

Theorem 7.4.3 Assume that X(t, z) =∑

α∈A cα(t, z)Kα is a Skorohod integrable stochastic

process with ∫ b

a

∫R0

E[X(t, z)2]ν(dz)dt <∞ (7.38)

for some 0 ≤ a < b. Then X(t, z) N(t, z) is ν(dz)dt-integrable in (S)∗ over [a, b)×R0 and∫ b

a

∫R0

X(t, z)N(δt, dz) =

∫ b

a

∫R0

X(t, z) ˙N(t, z)ν(dz)dt. (7.39)

Proof

The proof follows the same arguments as in the Brownian motion case (see [47] page 52).

We omit the details. 2

7.5 Levy Hermite transform

The Levy Hermite transform is introduced as in the case of Brownian motion (see [27] page

211).


α∈A cαKα(ω) ∈ (S)−1. Then the Levy Hermite transform

of F , denoted by HF or F , is defined by

HF (z) = F (z) =∑α∈A

cαzα ∈ C (7.40)

where z = (z1, z2, . . . , ) ∈ Cn and zα = zα1 , zα2 , . . . , z

αnn , . . . under the assumption that the

series in Equation (7.40) converges.

Example

For

η(x) = m2

∑i≥1

ξi(x)Kε(i,1)

we have

H(η(x))(z) = m2

∑i≥1

ξi(x)zε(i,1) .

The following proposition is an important property of the Levy Hermite transform as it

transforms the Levy-Wick product into an ordinary product.

167

Proposition 7.5.2 If F,G ∈ (S)−1 then

H(F G)(z) = HF (z) · HG(z). (7.41)

for all z such that HF (z) and HG(z) exist.

We mention that the results related to the Hermite transform in the Brownian motion case

are valid for the Levy Hermite transform with minor modifications.

7.6 Levy stochastic derivative

We can extend the stochastic derivative operator Dt,z to the whole space (S)∗ by making

use of the chaos expansion

F =∑α∈A

cαKα.

Definition 7.6.1 For any F =∑

α∈A cαKα ∈ (S)∗ we define the stochastic derivative Dt,zF

of F as

Dt,zF :=∑α∈A

cα∑i,j

αk(i,j)Kα−εk(i,j)ξi(t)pj(z). (7.42)

Example

Let

Y =

∫ ∞

0

∫R0

ξi(t)pj(z)N(dt, dz)

for some i, j ≥ 1. Then

Dt,zY = ξi(t)pj(z).

It can be proved that Dt,zF ∈ (S)∗ (λ× ν) a.e for all F ∈ (S)∗. Furthermore it can also be

shown that if F = limn→∞ Fn in (S)∗ then there exists a subsequence Fnk∈ (S)∗ such that

Dt,zF = limn→∞

Dt,zFnkin (S)∗ λ× ν a.e.

Definition 7.6.1 of the stochastic derivative Dt,zF coincides with the Definition 5.4.1 if

F ∈ D1,2 ⊂ (S)∗. This follows with the help of the closability of the operator Dt,z. The

stochastic derivative Dt,z does not satisfy the usual chain rule as in the case of Malliavin

derivative for the Brownian motion setting. Nevertheless a chain rule can still be formulated

in terms of the Wick product (see [29]).

168

Proposition 7.6.2 Let F ∈ (S)∗ and let g(z) =∑

n≥0 anzn be an analytic function in the

whole complex plane. Then∑

n≥0 anFn is convergent in (S)∗. In addition, for g(F ) =∑

n≥0 anFn the following identity holds

Ds,zg(F ) =

(d

dzg

)(F ) Ds,zF. (7.43)

Proof

The convergence of∑

n≥0 anFn in (S)∗ can be derived following similar arguments as in

Theorem 2.6.12 and Theorem 2.8.1 in [47]. The chain rule can be easily be shown that it

holds for polynomials. Then the result follows by the closeness of Dt,z and the continuity of

the Wick product. 2

The following lemma gives an integration by parts formula for Dt,z (see [27] page 226).

Lemma 7.6.3 Let F ∈ (S)∗ and let∫ T

0

∫R ‖ X(t, z) ‖2

−q ν(dz)dt <∞ for some q ≥ 0. Then

the integration by parts formula for D1,2 is given by∫ T

0

∫R0

〈X(t, z), Dt,zF 〉ν(dz)dt = 〈∫ T

0

∫R0

X(t, z) ˙N(t, z)ν(dz)dt, F 〉. (7.44)

As in Gaussian white noise case we let P (x) =∑

α∈A cαxα, x ∈ RN , cα ∈ R, be a

polynomial where xα = (xα11 x

α22 , . . .) and x0

j = 1. Then we can define its Wick version at

X = (X1, . . . , Xn) ∈ (S)∗ by

P (X) =∑α∈A

cαXα.

In the following the derivative of a process X : [0, T ] → (S)∗ is understood in the sense of

the topology of (S)∗. We denote the derivative of Xt by ddtXt. Define

X(t)i,j =

∫ t

0

∫R0

ξi(s)pj(z)N(ds, dz), i, j ≥ 0. (7.45)

We can write this as X(t)i,j =

∫ t0

∫R0ξi(s)pj(z)

˙N(s, z)ν(dz)ds. It follows from Lemma 2.8.4 in

[47] that the derivative of X(t)i,j exists and

d

dtX

(t)i,j =

∫R0

ξi(t)pj(z)˙N(t, z)ν(dz) (7.46)

where we have used the Bochner integral with respect to ν. The following Wick chain rule

then follows by induction (see [27] page 214− 215).

169

Lemma 7.6.4 Let P (x) =∑

α∈A cαxα be a polynomial in x = (x1, . . . , xn) ∈ Rn

0 . Suppose

ik, jk ≥ 1 for all k = 1, 2, . . . , n and let X(t) = (X(t)i1,j1

, . . . , X(t)in,jn

) with X(t)i,j as in Equation

(7.45). Then

d

dtP (X(t)) =

n∑i,j=1

∫R0

(∂P

∂xj

)(X(t)) ξi(s)pj(z)

˙N(t, z)ν(dz). (7.47)

7.7 Donsker delta function of a Levy process

Put

ηt =

∫ t

0

∫R0

zN(ds, dz). (7.48)

In this section we present the Donsker delta function δx(ηt) of ηt. It is a generalized white

noise functional (see [46]). Our presentation follow the work in [27].

Definition 7.7.1 Suppose that X : Ω → R is a random variable belonging to the Levy-

Hida distribution space (S)−1. The Donsker delta function of X is a continuous function

δ·(X) : R → (S)−1 such that ∫Rh(x)δx(X)dx = h(x) (7.49)

for all measurable functions h : R → R for which the integral is well-defined in (S)−1.

We want to represent a certain class of pure jump Levy processes in terms of the Donsker

delta function. We assume that the pure jump Levy process satisfies the condition:

There exists ε ∈ (0, 1) such that, for u ∈ R,

lim|u|→∞

| u |−(1+ε) Re

(∫R(eiuz − 1− iuz)ν(dz)

)= ∞ (7.50)

where Re(∫

R(eiuz − 1− iuz)ν(dz))

denotes the real part of∫

R(eiuz − 1− iuz)ν(dz).

Remark

The condition (7.50) implies that the probability law of ηt, t ≥ 0 is absolutely continuous

with respect to the Lebesgue measure (see [27] page 226).

We need the following lemma (see [27] page 227).

Lemma 7.7.2 Let u ∈ R and t ≥ 0. Then

exp(iuηt) = exp(∫ t

0

∫R0

(eiuz − 1)N(ds, dz) + t

∫R0

(eiuz − 1− iuz)ν(dz)

). (7.51)

170

Proof

Define

Yt = exp

(iuηt − t

∫R0

(eiuz − 1− iuz)ν(dz)

). (7.52)

Then an application of Ito’s formula gives

dYt = Yt−

∫R0

(eiuz − 1)N(dt, dz), Y0 = 1.

Using Equation (7.39) we have

d

dtYt = Yt−

∫R0

(eiuz − 1)˙N(t, z)ν(dz)dt. (7.53)

By means of a version of the Wick chain rule, the solution to (7.53) is given by

Yt = exp(∫ t

0

∫R0

(eiuz − 1)˙N(t, z)ν(dz)dt

)= exp

(∫ t

0

∫R0

(eiuz − 1)N(ds, dz)

). (7.54)

This solution is unique. Comparing Equations (7.52) and (7.54) we obtain the desired re-

sult. 2

We have the following important result (see [27] page 227).

Theorem 7.7.3 The Donsker delta function δx(ηt) of ηt exists in (S)−1 and it admits a

representation of the form

δx(ηt) =1

2π

∫R0

exp(∫ t

0

∫R0

(eiuz − 1)N(ds, dz) + t

∫R0

(eiuz − 1− iuz)ν(dz)− iux

)du

(7.55)

for u ∈ R and t ∈ [0, T ].

Proof

The proof is based on the application of the Levy-Hermite transform and the use of Fourier

inversion formula. A detailed proof can be found in [27] page 227. We omit the de-

tails. 2

7.8 Application: Computing Greeks

Suppose we have a financial market, where the bond price S0(t) and the stock price S(t) are

modelled as follows

171

1. bond price:

S0(t) = 1, 0 ≤ t ≤ T (7.56)

2. stock price:

dS(t) = S(t)dηt, S(0) = x > 0, 0 ≤ t ≤ T (7.57)

where ηt is a Levy process of the form (7.48). Assume that z > −1+ ε for a.a z with respect

to ν for some ε > 0. This ensures that S(t) > 0 for all 0 ≤ t ≤ T .

Using the Ito formula for Levy processes the solution to Equation (7.57) is given by

S(t) = x exp∫ t

0

∫R0

(log(1 + z)− z)ν(dz)ds+

∫ t

0

∫R0

log(1 + z)N(ds, dz). (7.58)

Here we apply the concept of the white noise analysis together with the Donsker delta

function to compute ∆ of a digital option. We consider the digital option of the form

χ[K,∞)(ST ) (7.59)

with strike price K. Similar to the pure Brownian motion case, we apply the concepts of the

white noise analysis together with the Donsker delta function of the Levy process St. We

will only illustrate the computation of ∆.

As in the pure Brownian motion case, we represent f in terms of the Donsker delta function

δx(ST ) =1

2π

∫R0

exp(∫ T

0

∫R0

(eiuz − 1)N(ds, dz) + T

∫R0

(eiuz − 1− iuz)ν(dz)− iux

)du

as

f(ST ) =

∫R0

f(y)δy(ST )dy

=

∫R0

1

2π

(∫R0

f(y) exp(−iuy)dy)

exp∫ T

0

∫R0


+ T

∫R0

(eiuz − 1− iuz)ν(dz)du.

We mention that, for f ∈ L1(R) with compact support the integral above converges in the

distribution space (S)−1. Thus, the option price of the digital option takes the form

u(x) = E[e−rTf(ST )]

= E[e−rT∫

R0

1

2π

(∫R0

f(y) exp(−iuy)dy)

exp∫ T

0

∫R0


+ T

∫R0

(eiuz − 1− iuz)ν(dz)du].

172

Using Lemma 7.7.2 we can write the option price as follows

u(x) = E[e−rT∫

R0

1

2π

(∫R0

f(y) exp(−iuy)dy)

exp(iuST )du]. (7.60)

We now state the following result.

Theorem 7.8.1 Let f be a function of polynomial growth and let the integral∫

R0f(y) exp(−iuy)dy

belong to L1. Then

d

dxE[e−rT

∫R0

1

2π

(∫R0

f(y) exp(−iuy)dy)

exp(iuST )du]

= E[e−rT∫

R0

1

2π

(∫R0

f(y) exp(−iuy)dy)

exp(iuST )iuSTxdu].

Proof

The proof follows the same arguments as in Theorem 4.8.1. We omit the details. 2

Remark

We mention that similar results in the case of jump diffusion were obtained in [28]. However,

in this paper the authors did not use the Malliavin calculus; they use ideas related to the

likelihood method.

We also mention that we have only compute ∆ in this section, the other Greeks are left for

future work. This is because it requires more time.

173

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