Fractional Volatility Models and Malliavin Calculus

NG Chi-Tim

A Thesis Submitted in Partial Fulfilment of the Requirements for the Degree of Master of Philosophy in Risk Management Science

Supervised by

Prof. CHAN Ngai-Hang

©The Chinese University of Hong Kong June 2004

The Chinese University of Hong Kong holds the copyright of this thesis. Any person(s) intending to use a part or whole of the materials in the thesis in a proposed publication must seek copyright release from the Dean of the Graduate School. A/

’ UNIVERSITY 7^/

.一 Abstract

The purpose of this thesis is to develop European option pricing formulae for fractional market models. Although there have been previous papers discussing the option pricing problem for a fractional Black Scholes model using Wick calculus and Malliavin calculus, the formula obtained is similar to the classical Black Scholes formula, which cannot explain the volatility smile pattern that is observed in the market. In this thesis, a fractional version of the Constant Elasticity of Volatility (CEV) model is developed An European option pricing formula similar to that of the classical CEV model is also obtained and a volatility skew pattern is revealed.

Keywords : fractional Brownian motion, Wick Calculus, Malliavin Calcu- lus, fractional Black Scholes model, fractional Constant Elasticity of Volatil- ity model, no arbitrage, volatility skew.

2 摘要

本文旨在探討分數維市場假設下的歐式期權定價模型•近年，有不少文獻討論分數維布舒二氏模型•運用Wick微積分和Malliavin微積分兩套理論，人們求得了分數維的布舒二氏方程式•該方程樣子跟傳統的布舒二氏方程相近，而且都不能有效解釋在市場實踐上發現的“波動微笑”現象•為改善理論與實踐之間的不協調現象，本文提出一個新模型——“分數維常彈性波動模型”，跟非分數維的情形一樣，該模型仍能給出簡潔的歐式期權定價公式•雖然無法預測“波動微笑”現象，但它仍然能夠顯示“波動率偏頗”.

^ a __ CONTENTS

Page

Chapter 1 Introduction 4

Chapter 2 Mathematical Background 7 2.1 Fractional Stochastic Integral 8 2.2 Wick's Calculus 9 2.3 Malliavin Calculus 19 2.4 Fractional Ito's Lemma 27

Chapter 3 The Fractional Black Scholes Model 34 3.1 Fractional Geometric Brownian Motion 35 3.2 Arbitrage Opportunities 38 3.3 Fractional Black Scholes Equation 40

Chapter 4 Generalization 43 4.1 Stochastic Gradients of Fractional Diffusion Processes 44 4.2 An Example : Fractional Black Scholes Mdel with Varying Trend and Volatility 46 4.3 Generalization of Fractional Black Scholes PDE 48 4.4 Option Pricing Problem for Fractional Black Scholes Model with Varying Trend and Volatility 55

Chapter 5 Alternative Fractional Models 59 5.1 Fractional Constant Elasticity Volatility (CEV) Models 60 5.2 Pricing an European Call Option 61

Chapter 6 Problems in Fractional Models 66

Chapter 7 Arbitrage Opportunities 68 7.1 Two Equivalent Expressions for Geometric Brownian Motions 69 7.2 Self-financing Strategies 70

Chapter 8 Conclusions 72

2 Page

Appendix A Fractional Stochastic Integral for Deterministic Integrand 75 A.l Mapping from Inner-Product Space to a Set of Random Variables 76 A.2 Fractional Calculus 77 A.3 Spaces for Deterministic Functions 79

Appendix B Three Approaches of Stochastic Integration 82 B.l S-Transformation Approach 84 B.2 Relationship between Three Types of Stochastic Integral 89

Reference 90

rinted with FinePrint _ purchase at www.fineprjnt.com L, 1 Introduction

Classical mathematical finance, ranging from Bachelier(1901) to Black and

I Scholes, assumes that stock price processes follow Brownian motions or Ge-

ometric Brownian motions. The most important feature of these stochastic

processes is the Markov property, which asserts that knowing the historical

prices up to now, the probability distribution of future stock prices depend

only on current price. All information about the characteristics of a stock

is totally reflected in the current stock price and historical prices are irrele-

vant for predicting future stock prices. Such models are consistent with the

Weak Form Efficient Market Hypothesis (EMH). Under this hypothesis, the

percentage changes in stock price, or returns, must be due to newly arrived

information, not the past behavior. As a result, the returns over time are

not auto-correlated. But, is the financial market efficient? Recent evidence

shows that stock returns are auto correlated and the autocorrelations are

“long-run" as well. The empirical evidence of "long range dependence" is

summarized in Cutland, et al. (1995). They suggest using a "fractional

Brownian motion" (See also Mandalbrot and Van Ness (1968)), which is a

generalization of Brownian motion, to model the stock price behavior.

The fractional Brownian motion Bf is expressed in terms of an improper Ito-integral with respect to the standard Brownian motion Bt as

BtH = ch{ r [{t —扩— + [\t —力丑- J-oo Jo

where ch is a normalizing constant so that the fractional Brownian motion

rinted with FinePrint - purchase at www.fineprint.com I i has unit variance at time Z = 1. For details of fractional Brownian motion i and definitions of long range dependance, see Chapter 7 of Samorodnitsky and Taqqu(1994) or Cutland, et al. (1995).

Rogers (1997) shows that the fractional Brownian motion is not a semi- martingale, and the elegant theory developed by Harrison and Pliska (1981), Harrison and Kreps (1979)，and Delbaen and Schacherrnayer (1994) cannot be directly applicable. To complicate matters, fractional Brownian Motion introduces arbitrage opportunities and as a result, the "no arbitrage pricing principle,，cannot be applied to price derivatives. iV i i ___ Roger suggests two approaches to circumvent this difficulty. The first one is to introduce a “ kernel function" into the setup of fractional Brownian motion, the so-called regularized fractional Brownian Motion. The regularized fractional Brownian motion has the form

Xt= [(p{t — s) — (p{-s)]dBs. J —oo By taking (fnii)=(力+)丑全，^t becomes the fractional Brownian motion. However, Cheridito (2002) shows that when the kernel function satisfies some smoothing conditions and <^(0) + 0, there exists an equivalent martingale measure such that the regularized fractional Brownian motion

is equivalent to a standard Brownian motion. So, an European call option pricing formula similar to the classical Black Scholes formula is obtained.

rinted with FinePrint - purchase at www.fineprint.com L This formula is just the classical one with a being replaced by j^f^，where IIX1II2 is the normalizing constant so that the regularized fractional brownian I motion has unit variance at time t = I. Cheridito's theory does not provide a different method to solve the option-pricing problem. Also, Cheridito shows in his paper that for a given non-zero value of (^(0), a kernal function can be constructed so that the autocorrelations of the regularized fractional Brown- ian Motion can be uniformly approximated by a fractional Brownian motion. With such a choice, the value of I \Xi | U is close to 丄.Since the option pricing C77 formula involves the term j^Jf^，the value of is important to pricing the option. But the choice of (/?(0) is arbitrary. As a result, the theoretical pric- } ing problem is ill-conditioned. Knowing the autocorrelations does not give an accurate estimation of the option price. One way to apply Cheridito's result is to use implied parameters from currently traded option prices. It is clear that the implied value of is just the same as the implied volatility from the classical Black Scholes model, so this method is exactly the implied volatility method of the classical model.

I:秘 Roger's second suggestion is to develop a new stochastic calculus to rede- fine the stochastic differential equation governing the stock prices. Due to J the work of Hu and 0ksendal (2000), the "fractional" version of Wick-Ito I I integral theory (see Holden, et al. (1996)) and Malliavin calculus (see Malli- I 心 I avin (1997) and Nualart (1995)) can be applied. The theoretical foundation I is summerized in Biagini, et al. (2002) and Necula (2002). Applications I of classical Wick-Ito integral and Malliavin calculus to the model based on I Brownian motion is given in 0ksendal (1997). Similar to Cheridito (2002)， I Hu and 0ksendal (2000) provide an option pricing formula similar to the

7 I I f- rinted with FinePrint _ purchase at www.fineprjnt.com L, classical one. As will be discussed later, the new European call option pric-

ing formula obtained does not account for the "implied volatility smile" phe-

nomenon (see Duan (1995)). One way to account for the difference in market

and theory is to introduce non-constant volatility. One popular model of non-

constant volatility is the Constant Elasticity of Variance Model (CEV. see

Cox (1976) and Cox and Ross (1976)). A major objective of this paper is

to extend the work of Rogers (1997)，Hu and 0ksendal (2000), and Necula

(2002) to CEV models. To this end, a fractional extension for CEV model

will be established. Although the proposed fractional CEV model cannot

account for the "volatility smile" pattern, the "implied volatility skewness"

phenomenon can be revealed.

• rtnted with FinePrint - purchase at www.fineprint.com 2 Mathematical Background

I In Hu and 0ksendal (2000), the statistical model for stock price is defined by ！ I a stochastic integral equation with the integration defined in the Wick-Ito's • sense. Wick-Ito integral is the limit sum of Wick's product between the inte- If ；grand and changes in the integrator (i.e. the Brownian motion). Classically,

I stochastic integral defined in Wick-Ito sense is equivalent to the Ito one using I i. ordinary product, so the Wick-Ito integral is just an alternative expression

r of the Ito integral. There are some conveniences by using Wick's product. Ir One main advantage is many stochastic differential equations encountered i j in physics and finance can be converted into ordinary differential equations s i by means of a Hermite transformation, which is related to the definition of Wick's product (see Holden, et al. (1996) for detail). However, it should be pointed out that integrations with Wick's products are not the same as integrations with ordinary products for the fractional case. This discrepancy j can be remedied by means of Malliavin Calculus. What follows is a brief introduction of concepts of Hu and 0ksendal (2000)，Biagini, et al. (2002), f Duncan, et al. (2000) and Lin (1995).

• rtnted with FinePrint - purchase at www.fineprint.com 2.1 Fractional Stochastic Integral

To begin, consider stochastic integrals of deterministic functions. As pointed

out in Lin (1995), integrations in Riemann-Ito sense for predictable process

Xt with respect to a fractional Brownian motion,

may not always converge in probability by virtue of the Bichteler-Dellacherie 1'

theorem. Lin (1995) gives some examples when f XfdB^ converge, one of

j them is a deterministic function. For any bounded Lebesgue measurable func- ；] i tion /(.), the integral f^ f{u)dB^ can be defined in Ito sense(see Lin (1995) Theorem 3.2). In this case, the autocorrelation function of the stochastic

process Y{t) = f^ f{u)dB^ is given by

二 H{2H -I) f [ f{u)f{v)\u - V广2dudv. 1： Jo Jo

The Proof, of the above result can be found in Gripenburg and Norros (1996).

•rtnte dwit h FinePrint - purchase at www.fineprint.com 2.2 Wick's Calculus

For a given sample path of a fractional Brownian motion and a deterministic integrand /(t), one can regard the stochastic integral as an linear operator on f{t). Holden, et al. (1996) use this functional approach to define White Noise. Specifically, let f{t) G S{R) be a rapidly decreasing function (i.e., S{R) is a Schwarz space) and cj G fi == a bounded linear functional I on the Schwarz space S{R). Note that u is an operator on a deterministic function. The set ft not only includes stochastic integrals, but also integrals

of the form J尺 /⑴却⑴ for a given (fi{t). We simply use the notation (p to represent the integral transformation. That is, the same notation represents 1 both the function (p{t) and the operation of (f on f{t).

According to Bochner-Minlos Theorem (see Holden, et al. (1996) Appendix A), there exists a unique probability measure /i^ on B{Q) with respect to the weak star topology such that the characteristic function of / E S{R) is given by

f e,fi4iu�=e-i萬, Jn where

11/11�=/ / f{u)f{v)(l){u,v)dudv, JRJR and

0("U，v) 二 H(2II -1)1^-叫2丑_2.

We define L^ as the space containing all functions such that the square norm

• rtnted with FinePrint - purchase at www.fineprint.com 11/11^ exists. Note that S{R) is a subspace of L^. For f G I/》，we can always

choose a sequence of /几 E S{R) so that —^ / in i含.Then u{f) can be

defined as lima;(/n) as this limit exists in L�.Now , f{t)dBl^ is defined

as a;(/), and the indefinite integral at time s, or f二 f{t)dB� is taken as

Jj^f{t)I[t < s]dB^. By the form of the characteristic function, one can show

that the covariance between u;{f {.)![{.) < 5]} and a;{/(.)/[(.) < t]} is

nf{u)f{v)^{u,v)dudv. As a special case, the fractional Brownian motion is now defined as B^ =

uj{I[{.) < t]}, has autocovariance + — I力—In this way,

the functioanl approach definition is equivalent to the Ito-type definition of

stochastic integral for deterministic functions.

Convention : Throughout this thesis, that the Hurst parameter H G

(1/2,1) is always assumed.

The next step is to define the Wick's product. Wick's product is a binary operator on two functionals F,G •• Q R. Hu and 0ksendal (2000) define Wick's product by the approach of Holden, et al. (1996) which makes use of the Wiener-Ito Chaos Decomposition Theorem.

Theorem 2.2.1 (Wiener-Ito Chaos Decomposition Theorem)

If F e then F{uj) has a unique representation

F{uj) = [CaHjuj),

• rtnted with FinePrint - purchase at www.fineprint.com where a is any finite integers sequence (Ofi, •••，o^i)，c^ are real coeffi-

cients and Ha{uj) = hn{x) are Hermite

polynormials and e^ is an orthonormal set in

Furthermore, the I? norm of the functional F{uj) is given by

i 血 L a here,

Remark : The basis {Ha{(jo) : a} is orthogonal with respect to the inner

product E{XY) in The variance of HA{UJ) is AL. Ho{U) is

taken as constant 1, so the expectation of Ha{ou) except Ho{u;) is E{Ha{u)Ho{uj))= 0. As a result, the term cq is the expectation of the functional.

Now, the Wick's product between two functional F{u) and G{uj) is defined

as follows.

Definition 2.2.2

The Wick's product for two functionals having Wiener-ltd Chaos Decompo-

sition

• rtnted with FinePrint - purchase at www.fineprint.com and

\ P :is defined as

F{(jj) O G{uj) = ^ , a,13 The addition of indices refers to pairwise addition.

The closedness of Wick's product is shown in the following theorem. Ip Theorem 2.2.3

Let (S) be a subset of I?队 B{p), ii^p) consisting of functionals with Wiener- •I i ltd Chaos decomposition such that J！

sup[cla\]J{2jrA I ^ I jeN J for all k < oo and that (5)* consists of all expansions, not necessarily be-

longing to such that

“I jeN

for some q < oo, then, the spaces (S) and (5)* are called Hida test function

space and Hida distribution space respectively.

The Wick operator is close in (S) and (5)*. Note that closedness does not

always hold for /i^).

Some useful results on Wick's product are given below.

• rtnted with FinePrint - purchase at www.fineprint.com Theorem 2.2.4 (see Holden，，et al. (1996) and Biagini” et al. (2002))

(a) If at least one of F{uj) and G{uj) is deterministic，then Wick，s multipli-

cation is the same as ordinary multiplication:

FoG = FG.

(b) For any two real functions

•�� =g ) — (/，g�4>.

where \I

f [ f{s)g{t)cj>{s,t)dsdL JrJr (c) For a real function / G ―仇 11/11�=工舰^ a positive integer n，the Wick's power, i.e. Wick multiplication of n times，of uj{f) is given by

u;{fr = hMf)).

Example 2.2.5

Here, an example (see Biagini, et al. (2002)) for Wick's product is demostrated. Consider the functional B�=a;(l[o，t](.) )where t is a given constant. The function l[o,t](.) can be rewritten in Fourier expansion form:

• rtnted with FinePrint - purchase at www.fineprint.com CO l[o，t]�=Y^{Mo,t]{s),ei{s))^ei{s) i=0

—‘！ / / ei{u)(l){u^ v)dudv > ei{s). i=0 Jr J

=[ r]i{v)dv\ei{s). i=0 LJo J

i Since uj is linear, it can be written as i

�( 一1 = ^ j / Vz{^)dv\uj{ei) i=o 口 0 J

i=0 ) =y^^CgHaju). a •

In this example, c^, = J^r]i{v)dv for a of the forms e{i) = {0…，1,0,...}, i.e., one at position i and zero otherwise. It is tempting to write

J oo

^B? 二 TM特》, i=0 which is illegitimate from a traditional point of view since the Brownian or fractional Brownian motion is nowhere differentiable. With chaos expansion form, differentiation and integration with respect to time t can be defined, but they may not always be square integrable. Such type of operation is

rinted with FinePrint _ purchase at www.fineprjnt.com L, �

-called integration or differentiation in (5)* (see Holden，，et al. (1996)). In

[Hu and 0ksendal (2000) and Biagini,, et al. (2002)，

‘ i=0 i is called derivative of the fractional Brownian motion and is denoted by I ‘W^{t). It can be shown that the sum

：竹,丑⑴ I > \ Now, it is reasonable to define the stochastic integration of a functional f Ztiuj) with respect to the fractional Brownian motion as the integration with 丨 respect to time t of the Wick's product between Zt{u) and Under

the Wiener-Chaos decomposition framework, if the decomposition exists, a functional Zt{uj) o can be written as

a It is natural to think that the integration is

just making an assumption that summation and integration are exchangeable. But in fact, if the integration with respect to time is pathwise classical Riemann integral, it is not clear whether the summation and integration are exchangeable. Such inconveniences can be • eiîeîieî a new definit on o integration with respect to time as stated in Definition 2.2.6, and the integration with respect to the fractional Brownian motion is defined in Definition 2.2.7. 17

rinted with FinePrint _ purchase at www.fineprjnt.com L, Definition 2.2.6

(a) Element in (S)* as an operator : Let F{cjj) = Xla G {Sy and

f{oo) = ^^ baHa{io) G (S), then F can be regarded as an operation on f

< FJ >=Y^baCaa!. a (b) Time integration : If Ft{ui) = J2a are elements in (5)* for

all positive real number t, and that < F^, f > are integrable with respect to t

for all f e {S), then the integral Jj^Ft{io)dt is defined as the unique element

in {Sy, I{uj) such that

= [ dt Jr for all f e (5).

j Remark : It can be shown that the quanity < F, / > in part (a) exists and

is finite under the condition given in the definition. It can be regarded as the

expectation of the product between F{uj) and f{u).

Definlton 2.2.7

Let Z{t) = ^ (5)* foT any given t and there exists a finite q

such that

sup \ a! [ < oc, « [ jeN “ then, the Wick's integral of Z{t) is defined as

• rtnted with FinePrint - purchase at www.fineprint.com [Z{t)odB^{t) = [ {Z{t)oW^{t)}dt. JR J R

Theorem 2.2.8 (see Holden, et al. (1996) Lemma 2.5.6)

Let Z{t) : R (5)*； with Wiener Chaos decomposition

a such that

E洲c為n(2力， a jeN I for some q < oo, then Z{t) is (S)*-integrable, also, integration and summa- i tion are exchangeable, i.e.,

� [Zifydt 二 f Ca(t)dtHa(UJ). JR CT JR

In the following example, the similarity of Wick's integral and Ito's integral

is revealed.

Example 2.2.9

Consider Jj^B^{t) odB"{t) (see Biagini, et al. (2002)). The calculations are

shown below.

rinted with FinePrint _ purchase at www.fineprjnt.com L, rj^ rjn [BH(t) • dBH(t) = [ Jo Jo dt

=[£ Iviii) [ m(i’)dv \ H明+化)dt Jo L 知 J

This summation, by symmetry, can be rewritten as

二 ^ 77“一义 r]j{u)du^ >

丨二 -w(l[o，T])ou(l[o,r])- i According to Theorem 2.2.4 (b)，this yields

rT 1 J^ BH(t) • dBH(t) 二 3Ml[o’T]Ml[o,r]) —(l[O’T]，l[O,t])0}

2 •

Example 2.2.10 (see Hu and 0ksendal (2002))

In this example, the concept of exponential is introduced. Let F(u) E (5)*, then the Wick's exponential is defined as

• rtnted with FinePrint - purchase at www.fineprint.com i

OO 1 expoF 二:E •广• n=0 • Consider F{uj) = uj(J), where f G the exponential is given by

OO 1 expoF = n=0 CO ^ � ( p \ 1 on i-! ^ n vii/iiJj

n=0 丫 '

The last step is derived from Theorem 2.2.4 (c). By using the fact that the

Hermite polynormials hn{x) are the coefficients in the Taylor series of

i 1 I exp(-#2+力工） z with respect to the variable t, the Wick's exponential becomes

exp (陳⑷—*輯） / 1 \ =exp \^uj(f) -

For X,Y E (5)*, the Wick multiplication of two Wick exponentials exp•叉

and exp^ V is

exp^ X o exp^ F

^ ^ n\m\ n m 21

Tinted with FinePrint - purchase at www.fineprint.com I [

I •

i j=0

i 二 exp�(; +t lO-

•

For functions f : K^ R that have Taylor expansion near the point C,

where Ci is the coefficient of Fi{uj) at a = 0，one can define the Wick version

function in a way using Taylor's expansion.

Definition 2.2.11

I Let Fi,F2”..,Fn e {Sy with expectation Ci,C2,."，Cn respectively and f : R几 4 R have Taylor expansion of the form

f (工1，工2,…，Xn) = 以工-（广，

V where v are finite integer indices，then,

tinted with FinePrint - purchase at www.fineprint.com 2.3 Malliavin Calculus

Malliavin calculus, also known as stochastic calculus of variations, was estab-

lished by Malliavin (1978). It extends the concepts of directional derivative,

gradient and divergence to the function space C[0，1] (i.e., Wiener Space).

Before the development of fractional stochastic integration theory, efforts

have already been made to provide a broader definition of stochastic integral

by means of divergence. Ito's theory deals only with the integrand that is

adaptable and progressively measurable. Divergence is a generalization of

Ito's integral and is also called Skorohod integral. Introduction of Malli-

avin calculus can be found in Nualart (1995), Watanabe (1984), Aase, et al.

(2000) and Ustunel (1995). i Decreusefond and C/sti/nel(1999) defines stochastic integral with respect to I fractional Brownian motion by means of the divergence of a kernal transformation for H > They show that it is different from the pathwise Riemann-Stieltjes integral and the difference can be expressed in terms of a pathwise integration of a directional derivative with respect to time. The integral defined by Decreusefond and Ustiinel (1999) is, in fact, equivalent to the Wick's integral if some smoothing conditions are satisfied. The related results can be found in Duncan, et al. (2000). In this section, concepts of directional derivative and stochastic gradient is introduced. There are several papers (e.g. Duncan, et al. (2000), Hu and 0ksendal (2002)，and Biagini, et al. (2002)) that introduce the concepts of Malliavin calculus and its applications to fractional stochastic integration, but the definitions used among different papers are not exactly the same. Herein notations used in Duncan,

• rtnted with FinePrint - purchase at www.fineprint.com ：et al. (2000) are followed.

I In classical mathematical analysis, the directional derivative of an n-dimensional

！ function / : i?^ —i? in direction a is given by the limit, if exists,

n ""^� T /(茫 + 紀）

and the gradient is defined as a vector Vf such that for all non-zero direction

a, the directional derivative is given by

D,m = (•/) • n.

It is natural to extend the concept of vector in Euclidean spaces to that in a

more general vector space. Consider the dual of the Schwarz space Vt = S"(i?).

The direction a and the variable x become elements in Q = S'{R). The n-

dimensional function f : R^ ^ R becomes a square integrable functional

F .M R. Roughly speaking, the directional derivative along the direction

7 can be defined as

从,� 二 lim + 57) — F(u;)}.

Let 屯g be an integral transformation with kernel function (j){t, s), i.e.,

(击…⑷=[(l>{t, s)g{s)ds. JR To define stochastic gradient, dot product is replaced by integration of prod-

uct. Then the unique stochastic process D'^F such that the following equality

holds for all g G L^:

tinted with FinePrint - purchase at www.fineprint.com Lk —^ ： = [ {D^F)s9{s)ds, 丨 JR 1 is called the stochastic gradient, f I Remark 1: Beware that the term (屯g) refers to an operation, not a function.

Remark 2: Many times, we handle not only one functional F(cj), but a

continuous time series of Ft(cj). In this case, for any given time t, e.g. t = 0.3,

will then be a functional mapping from fi to R. Then the derivative

originally written as D'^Fs will be written as D^ ^Fg. If t is set to s, then

we will use the notation DfFg. Keeping in mind that the derivative is with

respect to the path cu. not the time t or s.

Remark 3: It should be note that F{uj + 5^) is not always differentiable

with respect to 5.

Below, a special type of functional, called stochastic polynomial, is introduced. As a consequence, definitions for directional derivative and gradient : are provided for this special case. Stochastic polynomial is chosen as the starting point of the theory due to its simplicity and the fact that the class of stochastic polynomial is dense in ^(Q),/i^)).

Definition 2.3.1 (see Watanabe (1984))

Stochastic Polynomial: A functional F : Q ^ R that can be expressed in the

form of

•rt ntedwit h FinePrint - purchase at www.fineprint.com where F : R^ ^ R is a n-dimensional polynomial and /i, A, fn : R — R G 5(i?)； are one-dimensional rapidly decreasing functions. The set of all stochastic polynomial is denoted by P.

Example 2.3.2 (see Duncan, et al. (2000))

Let f : R R E S{R) be a real deterministic function，then f …M/))厂（$/)�

and

Solution

Using the linearity of the operator u and 屯g, this yields

D电g�(f)

S-^O 0

5->0 0

=[f 9{s)f{t)cl>{s,t)dsdt JR JR =[{ [ f[t)cKs,t)dt�g[s�ds Jr JR

=[^f{s)g{s)dsJR

rinted with FinePrint _ purchase at www.fineprjnt.com L, the results follows immediately after interchanging the order of integration.

For exponential,

D$,exp�(/)

=limUexp + — exp[a;⑴一-11/11 扑 V L. J L. -J ^

S^O 0 =exp�(/)(御)(/).

Note that does not depend on any time variables and the function

•

More generally, a simplified chain rule can be obtained.

Example 2.3.3

In this example, directional derivatwes and stochastic gradients for stochastic

polynomail will be given. The stochastic gradient of

is given by

rinted with FinePrint _ purchase at www.fineprjnt.com L, Solution

Consider the Taylor expansion of F : R^ R which is differentiable to any

order and to anv variable. Bv Definition 2.3.1, the directional derivative can ty / be written as

； =lim - F(u;)} S-^O 0 =lim ^{F(u;) + S^g{h)Fl{u;) + …+ 5^g{fMuj) + — F{u;{f))} “0 d =如+ …+ 劫(/n)�M.

Using similar arguments as in Example 2.3.2, the result follows. •

Example 2.3.3 now serves as the foundation of Malliavin calculus (Watanabe

(1984)).

Definition 2.3.4

Let F{uj) e P be a stochastic polynomial of the form

F(a;(/i)，a;(/2),…，a;(/n)).

• rtnted with FinePrint - purchase at www.fineprint.com The stochastic gradient {D^F)s, s > 0 (s is a dummy variable) is

and the (j)-derimtive，also called the directional derivative in the direction of

龟g，for a given g 6 丄^ is given by

D^gF = {fug)^F[{uj) + …+

It is not difficult to show that the chain rule and product rule are valid for

stochastic polynomials.

Theorem 2.3.5 (Chain Rules)

(a) Chain rule I: Let Fi, F2,be stochastic polynomials and f : K^ ^ R

I be a polynomial, then,

[肉讽,札…，Fn)]s = d驰,F2,…，

(b) Chain rule II: A Wick version stochastic polynomial /•(i^i，i^2，is

/(Fi,F2, Fn) with powers and multiplications in f replaced by Wick's powers and multiplications. The stochastic gradient for Wick version stochastic

polynomial is

•••’ Fr,)]s = …，K) O [D^F,],.

Theorem 2.3.6 (Product Rules)

rinted with FinePrint _ purchase at www.fineprjnt.com L, Let F{iu) and G{uj) be two stochastic polynomials, then,

D'^FGl 二 FlD'f'Gl +

and

\D 小 F oG]s = Fo [D^G], + Go [D'^Fl.

An example involving Wick's product is illustrated below. This example

helps extending concepts of Malliavin calculus to more general cases, e.g.,

1 the space (5)*, by means of Wiener-Chaos decomposition.

I ！

Example 2.3.7 (See Biagini, et al. (2002))

Find (D�a;(/)•”s for f : RR e Ll and hence, evaluate {D'^Ha{oj))s.

Solution Using Theorem 2.2.4 (c), Definition 2.3.4, and the fact that

4-hn{x) = nhn-i{x), ax we have,

(DMfDs

二 Wm-j-nhn-l J ct> \ \ J 4>/ / 30

rinted with FinePrint _ purchase at www.fineprjnt.com L, / f \ o(n-l)

\ J 4>/ =—/)•(“)$/�.

Again, applying Definition 2.3.4 to the stochastic polynomial

X2，Xn) = hcci (^2) •••/^Qn (^n)

‘with fi 二 ei, the orthonormal basis for L^, we obtain

n 厂

i=l L 计、 -

i=l •

With the preceding example, it is natural to define (D印)s for functional

F(LJ) with Wiener-Chaos Decomposition

a as the sum of stochastic gradients of Ha- It is not necessary to assume that the decomposition is square integrable, defining (D'^F)s in the Hida distribution space (5)* is possible.

• rtnted with FinePrint - purchase at www.fineprint.com r

Q I f I = ^Ca <âiêi{s)Ha-e{i){uj) > I CC I i > it f X =^ j 好e�( A+ Hîu). i3 I i >

Definition 2.3.8 (See Biagini, et al. (2002))

For F E {Sy, the stochastic gradient is defined as

Remark 1 : The stochastic gradient {D^F{u;))s belongs to (5)* also.

Remark 2 : Note that if the functional F{uj) is square integrable, i.e., F G 召(f]), A於)，the stochastic gradient may not necessarily be square I integrable.

One can show that the product rules are valid for the Hida distribution space (5)* and thus, inductively, it is true that the n-th Wick's power of a functional F{u;) is differentiable in (5)*. The chain rule is true in (5)* for functionals of the type /•(Fi(cj),厂2(0；)，••.，Fn(a;))，where f is an n-dimensional function that have Taylor expansion near the expectation of Fi{uj), F2(u;), Fn((^). But the chain rule cannot be extended to the case of /(Fi(a;), ^2(0;),…，Fn(cj)) since ordinary multiplication may not be closed in the space (S)*.

•rtnte dwit h FinePrint - purchase at www.fineprint.com The space (5)* is too large for the ordinary chain rule to hold, but ordi-

nary chain rule can still be applied in a smaller space. Duncan, et al. (2002)

.pointed that by virtue of the Stone-Weierstrass Theorem, the class of stochas-

！ tic polynomials is dense in /i^). So, if a square integrable func- i tional can be approximated by a sequence of stochastic polynomials and the

• sequence of stochastic gradients are Cauchy in the space B(Q), //诊)

then, the directional derivative and stochastic gradient are well defined also.

“This domain of gradient is denoted by Dom(V) in (jstilnel (1995) and Watan-

！ abe (1984). Below, a chain rule is given.

Theorem 2.3.9 (See Malliavin (1997))

Let Fi，F2, •••，Fn E Dom(V) and f : R^ ^ R be a function having polynomial growth and having derivatives with respect to each variables, then

I ‘.

rinted with FinePrint _ purchase at www.fineprjnt.com L, 2.4 Fractional Ito's Lemma

After introducing the concepts of Malliavin calculus, linkage between two

types of integral, Stratonovich integral and Wick integral can be illustrated.

This linkage is important when discussing the Ito's lemma in terms of Wick

integral. Consider a simple example.

Example 2.4.1 (see Lin (1995))

Consider a stochastic process F{B^{t)), where F : R R has derivatives up to second order and the second derivative is bounded on [0，T]. Let 0 = to < ti < ... < tn-i < tn = T he di partition of the inteval [0, T], then a fractional Ito's lemma can be derived as follows.

i=l

=f {F'OB^训B叩糾）-B^iU)] + - B^itm i=0 By assumption, F"(� i) is bounded and it is known that fractional Brownian motion with H > ^ has zero quadratic variation (see Lin (1995)), so, the second term must converge to zero. This yields

n-l

i=0

When the mesh of the partition tends to zero, the right hand side becomes a Riemann-Stieltjes integral, i.e., Stratonovich integral

• rtnted with FinePrint - purchase at www.fineprint.com F{B"{T)) = F(BH(0)) + [ F'{B''{t))dB^. Jo In this situation, the Riemann-Ito integral f^ F'{B^(t))dB^ is well defined

and it is equal toF{B^(T)) - F{B^{T)) is now written in terms

of a Stratonovich integral, but not a Wick integral. •

Below, two types of integrals are studied under Wiener Chaos decomposition

framework. Thev are

limE Xt“a;) o (BH(t…—(1)

and

lim；^^ 不计 1 —B 丑⑷))• ⑵

The limits here refer to convergence in (5)*, i.e., is defined as

the unique element L in (5)* such that i

for all / G (5). The second integral (2) requires that Xt to be square inte-

grable for all t in the integration region.

It can be shown that the result of Example 2.4.1 is also true for the integral

defined under Wiener Chaos framework. This can be seen in the Lemma

2.4.2 and Theorem 2.4.3 below.

rinted with FinePrint _ purchase at www.fineprjnt.com L, Lemma 2.4.2

Let F ： ftRe (a n Dom{V) and h •• R 4 R e then

Fou;{h) = Fu{h) — D 补F,

and the Wick product is square integrable.

Proof.

Let F(u/) = CaHa(uj). Theii，

F(u;)ou;(h)

二 E Ca(h, e山丑a+e �M- a,i Using the recursive relation

hn+i(x) = xhn(x) — nhn-i(x),

we have,

a,i

=- a此— a,i =^Ca(h,ei)^u(ei)IIa 一 Y] Cg ("，(CJ) a,i a，i 二 F(u;)uj(h) - D^hF.

rinted with FinePrint - purchase at www.fineprint.com

‘ — • i Proposition 2.4.3

,If H > ^ and Xt satisfies the condition stated in Definition 2.2.6, then, the

integral defined in Riemann-Stieltjes sense is the same as that in (1), i.e.,

rjn [Xt O Wf rft = o (B丑(“+i — Jo The limits here refer to convergence in (5)*； i.e.，cu) is defined as

the unique element L in (5)* such that

I < L J >= \im< Xn{t,u) J >

for all f e {S).

Proof.

Using

Xrrit,��= i as an approximation to X(t,uj), it is not difficult to see that the first conclu-

sion holds. 丨 • iI

! The following Theorem follows from Proposition 2.4.3 immediately.

Theorem 2.4.4, see Duncan, et al. (2000)

I: •rtnted wit h FinePrint - purchase at www.fineprint.com Let H > The symbol L(0, T) is used to donote the domain of stochastic

process Xf such that both integrals，

and

f XsdBf = limEx“c^)(B� —+i BH嫌 Jo exist in Then for Xt G 1/(0,T), the equality

r XJBf 二 f XsO dBf + [ Dfx,\t=sds 丨 Jo Jo Jo

:is satisfied almost surely.

Proof.

Applying Lemma 2.4.2.

丨二 Xtt (a;) o - Bf] + 广 1 DfXtA I Jti

i ij •

“ 38

•rt ntedwit h FinePrint - purchase at www.fineprint.com Remark ： It should be noted that when H =DfXt-, where (l){s,t) is

considered to be a Dirac function 6{t — s), is not continuous. When Xt is

adaptable, DtXt- = 0 for t > U (see Nualart (1995)). Hence,

rU+i / DfXt,dt = 0. Jti ‘ Thus, integrations defined in terms of

lim[似付(力仔1 —B丑�)）

and

lim (炉丑�)）

are the same.

The mean and variance of Wick integral, if exists, are given in the following

theorem.

Theorem 2.4.5 ( Duncan, et al. (2000) p-591 )

LetX G L(0,T)，then the hrrnt J^ XtodB^ = limE Xs，（Bff+i -Bff) exist in Also, the first and second moment is given by

E J XtodB? = Q,

and

E 1: Xt • dB�=E ((J: DfXtdt) +义义 XM{s,t)dsdt >.

• rtnted with FinePrint - purchase at www.fineprint.com •

Recall Example 2.4.1. By Theorem 2.4.4,

=[m好 X

rji rp =[nBf)odBf+ [ DtF'{B^)ds. Jo Jo Using chain rule (Theorem 2.3.9), we have

DtnBf)

The next theorem follows from Theorem 2.4.5 immediately.

Theorem 2.4.6

If f ： R ^ R e C^ has bounded derivatives up to order two，then

fiB?) — /«) = r f{Bf) o dB^ + H� Jo Jo

a.s. and

^rtnted with FinePrint - purchase at www.fineprint.com i mt^. This result can be found in Duncan, et al. (2000) Theorem 4.1 and Lin (1995) Theorem 2.3.

There is no quadratic term in the formula for 'Stratonovich' type integration.

The above theorem is used to derive a fractional Ito's Lemma for more general

situations. We state the theorem below without proof.

Theorem 2.4.7 (Duncan, et al. (2000) Theorem 4.5)

Let

r]t = C+ f Gudu+ [ Fu o dB^ Jo Jo 1. Let Fu,0 < u

condition, i.e., there exists a > I - H such that E\Fu - < C\u -

where < 6 for some 5 > 0 such that E\D^{Fu--> 0 when \u-v

tends to 0.

2. Let Gu satisfy, E supo<5

3. Let f{t,x) \ B? ^ R have first continuous derivative in t and second

continuous derivative in x. These derivatives are bounded.

I Let E J^ \FsDtiis\ds < oo and € L(0,T).

•rtnted wit h FinePrint - purchase at www.fineprint.com Then

fit.Vt) = /(O, C) + J^ ^{s, Vs)ds + ^{s, rjs)Gsds +

I:藍(�Vs)Fs o dBf + j:岛(s, r^s)FsD%ds

I ！i fI

rinted with FinePrint _ purchase at www.fineprjnt.com L, 3 The Fractional Black Scholes Model

By applying the fractional Ito's Lemma, Necula (2002) derives the fractional Black-Scholes PDE governing the derivative price for the fractional Black- Scholes model. Here the fractional Black-Scholes model assumes that the market consist of two securities, stock and a risk free asset, which are specified by the following fractional stochastic diffusion equations:

Xt = [ iiXds+ [ aX o dBf Jo Jo

lit = f rUds. (3) Jo

In this model, X governs the stock price process while U governs the bond or risk-free asset price, /i and a can be interpreted as the rate of return and the volatility of the stock. Model (3) is similar to the fractional version Black- Scholes model studied in Cutland, et al. (1976), but integration is defined in Wick's sense. Hu and 0ksenda (2000) stated that under this model, no arbitrage opportunities exist. In this paper, a fractional Black-Scholes formula for European options is given. Necula (2002) is a generalization to Hu and 0ksendal (2000), With the fractional Black-Scholes PDE, one can give an option pricing formula for a derivative with payoff at maturity depending on the underlying asset price at maturity characterized by (3).

i 43 j

—ed with FinePrint - purchase at 3.1 Fractional Geometric Brownian motion

This section aims at giving an explicit solution to the fractional stochastic differential equation

Xt = R M^SDS + R (JXS O DB?, (4) Jo Jo

in the space (5)*. In order to obtain a solution, a technique, Hermite trans-

form, is introduced. The importance of Hermite transform is that it converts

a {Sy functional to a continuous real function and the Wick product to

ordinary product. Hence, a stochastic differential equation becomes a deter-

ministic differential equation. The concepts here mainly come from Holden,

et al. (1996).

Definition 3.1.1

The Hermite Transform of F{uj) is defined as

Q where z = {zi,Z2, ts a sequence of complex number and z"^ = z^'z^'',

provided that the summation exists.

Some important features of Hermite transform is summarized in the following

Theorem.

Theorem 3.1.2

• rtnted with FinePrint - purchase at www.fineprint.com (a) Let F{uj) and G{UJ) be two (Sy functionals, then，

HF{z)HG{z) = H{FoG).

(b) Let f : R ^ R be a smooth function such that a Taylor expansion exists

near the expectation of F{uj), then，

H{r{F)} = f{HF).

Ca{t) are all continuous. Then for those z such that the sequence

sup {ca{t}) a 力对仅力] is absolutely convergent，the Hermite transform of Xt{uj) is continuous.

(d) If Xt satisfies the condition given in Definition 2.2.9, with integration

region [a, h], and the condition in part (c) is satisfied, then Hermite transform

and integration are interchangeable.

Proof.

(a) and (b) follows immediately by Definition 2.2.2 and Definition 3.3.1.

If conditions in part(c) are satisfied, the Hermite transform is uniformly

convergent by virtue of Weierstrass M-test theorem, (c) and (d) follow. •

•rtnted wit h FinePrint - purchase at www.fineprint.com We return to diffusion equation (4). Taking Hermite transform on both sides,

the equation becomes

HXt{z) = Xo + [ fi{HXs){z)ds^ f a{HXs){z){HW,''){z)ds Jo Jo

For a given sequence z� itis just a differential equation in classical calculus

and the solution is

HXtiz) =Xoexp(/i^ + (j [ {HWf){z)ds). Jo Using the result of Theorem 3.1.2 (c) and (d), this yields

Xt{z) = Xo + (j^f)

=Xo exp(/x力) ( 1 \ 二 Xo exp(//t) exp oBf — -o^t^^ \ 2 y

=Xo exp (/if - 丑 + oBf ^

From Theorem 2.4.7 and the fact that DfB? = Hs训it follows that

is a solution to equation (4).

Mnted with FinePrint - purchase at www.fineprint.com i . ；3.2 Arbitrage Opportunities

Before proceeding, concepts of self-financing and arbitrage are introduced

first. The definition is based on a more generalized model,

Xt = [ fi{s,X)ds-}- [ a{s,X)odB^ Jo Jo

Ut = [ rUds. Jo

For simplicity, notations in classical Ito calculus (see 0ksendal (1998)) is

adopted. The equation can be written in differential form,

dXt = iJi{t,X)dt + a(t,X)odB^

dUt = rlldt.

Definition 3.2.1 (see Hu and 0ksendal(2OOO) Section 5)

(a) A trading strategy consists of a quantity {ut, Vt) of bonds and stocks is

called self-financing if the infintestimal change in the portfolio value at time

t is given by

dZt = d{utUt + VtXt)

=rUtUtdt + Xt)vtdt + Xt)vt] • dB^ + dA

where dA is an infintestimal dividend payment term.

• rtnted with FinePrint - purchase at www.fineprint.com (b) The market is said to be having arbitrage opportunity if there is a self- financing trading strategy with initial investment Z(0) < 0 but at time T, the

portfolio have value Z(T) > 0 and there is positive probability that Z{T) > 0.

any contingent claim, there is always a self-financing strategy that provide

the same payoff as the the claim. And the arbitrage price of the contingent

claim is the value of the portfolio at any time.

For the origin of continuous trading theory and related definitions, please

refer to Harrison and Kreps(1979) and Harrison and Pliska(1981).

Remark 1: Beware that the Wick integral in part (a) means Wick integral

of ordinary product between a{s, Xg) and Vs i.e.,

Jo which should be distinguished from

Jo These two types of integral lead to different conclusions, which will be dis-

cussed in Section 6.

Convention : Throughout this thesis, when the ordinary products are in-

volved, the two operands are considered to be square integrable so that the

ordinary products are well defined.

• rtnted with FinePrint - purchase at www.fineprint.com 3.3 Fractional Black Scholes Equation

Hu and 0ksendal (2000) and Necula (2002) derive an option pricing formula

for the fractional Black-Scholes model that is similar to the classical one. In

fact, the formula can be obtained by the following fractional Black-Scholes

PDE proved in Necula (2002).

Theorem 3.3.1 (Fractional Black-Scholes PDE)

The price of a derivative on the stock price with a bounded payoff f{X{T)) is given by P{t, X)，where X) is the solution of the PDE:

f + + —…=�

with boundary condition

i Ii i P{T,X) = f(X).

Remark : For an European call option, the payoff function is given by

max(0, Xt).

To make this result useful, two analytical methods for solving the PDE are introduced in this section. One way is to use a suitable substitution and the equation becomes a heat diffusion equation. The solution can be found by standard methods. Another way is solving by constructing an auxiliary non-fractional stochastic differential equation.

Theorem 3.3.2 (Fractional Black-Scholes Formula)

•rtnte dwit h FinePrint - purchase at www.fineprint.com The price of an European call option with strike price K, mature at T at

current time t is given by

P{t, X) = XN{di) - Ke-r�T-t�N^

where

,ln#+r(T —+ 付—力2 丑） di = , 二 (j v/T2^ - t姊 and

d2 = di- O^JT^H -

The function iY(-) refers to the cumulative distribution function of a standard

normal random variable.

Method of Substitution :

Using substitution

丨 t = u I( 1 2 9 f{1 I

X 二 exp ,

the PDE becomes a heat diffusion equation

&•二 0 du dv] Q(T, v) = e-rT max (exp + 臺一严}，0)，

• rtnted with FinePrint - purchase at www.fineprint.com where Q 二 e'^'^P is the discounted option price. Applying standard method

(see Lewis (2000)) to solve this PDE yields the call price formula at time t. •

Method of Auxiliary Equation :

It is useful to think the solution of a PDE at as the expected value

of the payoff function under the following diffusion process with respect to a

classical Brownian motion

dX' = rX'dt + V^at^-'^X'dBu

X'{h) = Xo, (5)

with t�t o(see Milstein(1995)). Applying classical Ito's Lemma to logX',

the solution of (5) is

X；二 exp |r(t - to) _ 丢一(力2 付—tf) + •

The term

-臺一(力2�r)

is deterministic while the Ito's integral

ft / V^as^-'^dBs J to is normal with mean zero and variance cr2(户丑—力吕丑)by Ito isometry. It is the same as the classical Black-Scholes formula with volatility

• rtnted with FinePrint - purchase at www.fineprint.com Hence the fractional Black-Scholes formula is obtained. •

• rtnted with FinePrint - purchase at www.fineprint.com 4 Generalization

In general, the fractional Ito's Lemma cannot be applied easily to an arbitrary given fractional diffusion process

= Xo + [ [ (6) Jo Jo

because the Ito's Lemma involves the term ^a{t,X)DfXt. The fractional stochastic gradient depends not only on the value of X at time t. but also the whole or part of the path of X. This makes the fractional Black-Scholes equation problematic. Furthermore, even the pricing formula exists, the option price process may not possess the Morkov property and thus the assumption that the formula can be expressed in terms of time t and stock price at t may not hold. In general, the stochastic gradient depends on the

whole path of X，but it can be shown that /Ji{t,X) and a{t,X) can be chosen so that the stochastic gradient D^Xt can be expressed in terms of t, Xt, and a dummy variable u. Thus, the fractional Black-Scholes equation is still meaningful.

•rinted with FinePrint - purchase at www.fineprint.com — ————^―^―^―——— 4.1 Stochastic Gradients of Fractional Diffusion Processes

Below, stochastic process controlling the Malliavin derivatives are presented

first.

Theorem 4.1.1

Assume that Xt governed by (6) belongs to L(0,T) and /i(t, X), a{t, X) are

smooth mappings: E? R. then the Malliavin derivative Yt{(jj^ u) 二 D^Xf

satisfies the stochastic differential equation:

Yt{u;, u) = j:嘉/i(s，Xs)Y,{uj; u)ds + L [嘉外，^s)ys{uj； u)] o dB^

+ / (j�s,Xs)(Ku,s)ds . (7) Jo

This result follows directly from the chain rule (Theorem 2.3.9) for stochastic

gradient and the following two lemmas.

Lemma 4.1.2

Let Zt ： R {Sy and D^Zt (u is regarded as a given number here) he stochastic processes saUsfing the conditions of Theorem 2.2.8, then, stochastic gradient operation and integration is interchangeable，i.e.，

Dt f X{uj;s)ds)= f DtX{uj;s)ds. Jo Jo

This is a direct consequence of Theorem 2.2.9 and Definition 2.3.8.

rinted with FinePrint _ purchase at www.fineprjnt.com L, Lemma 4.1.3

Let Zt and D^Zt (u is regarded as a given number here) be stochastic processes

satisfying the conditions in Definition 2.2.8, then

Dt [ Z(uj;s)odBf = [ DtZ{uj;s)odBf [ Z{u;; s)ds. Jo Jo Jo

Proof.

Apply product rule (Theorem 2.3.6) to o W^,

DtZs o Wf

二 o Dtllf + o DtZs

二 ZSO

The last identity follows from the fact that 0(5, w) is deterministic and The-

orem 2.2.4 (a). The conclusion follows Lemma 4.1.2.

•

Remark : For results in 巾�,see Duncan, et al. (2000) Theorem

4.2.

•rtnted wit h FinePrint - purchase at www.fineprint.com 4.2 An Example : Fractional Black Scholes Model with Varying Trend and Volatility

In this section, fractional Black-Scholes stock price model with deterministic

but varying trend and volatility rate is studied. Consider the following model

Xt = Xo+ [\{s)Xsds^ f a{s)XsOdB^, (8) Jo Jo where /i(t) and a{t) are deterministic. Such model is a natural extension

of the original fractional Black-Scholes model. Take ii{t, X) = iJ.{t)X and

.a(t,X) = (j{t)X, that is, the trend and volatility of stock are deterministic.

It is not difficult to solve the equation (8) and find the stochastic gradient of

the solution. By applying Hermite transform to equation (8) and equation

(7) in Theorem 4.1.1, we have

=场麗+ •吻)， at ^^ = f如)(冲)+ a�对0；;彻(i/，外 at

The above equations are just first order ordinary differential equations. The

solution for Xt{(^) is given by

Xt(a;) 二 Xo exp (/x. + asWH)ds^

Performing inverse transform,

Xt = H-\Xt}

• rtnted with FinePrint - purchase at www.fineprint.com -XoH-' exp {义+

=Xo {义(jUs + ajyf )^/^!

=Xo exp (J^sds 一臺义义 cr内0(C, r])dCdri + 义 cr.rffif | •

And for

Yt{u) = JCoexp {义…滅”斗

J s)小(u, s) exp 一 J^少 + aW^)d{.) | ds

=Xt ds小� u,s)ds. Jo

This yields

DtXt = Yt{uj)

=H-'{Xt [ Jo

=Xt s)ds. Jo

In this example, the stochastic gradient depends on Xt, but not the whole

path of X, so steps similar to Necula (2002) can be adopted to obtain the

Black-Scholes PDE.

•rtnte d with FinePrint - purchase at www.fineprint.com 4.3 Generalization of Fractional Black Scholes PDE

Now, consider the stochastic gradient of Xt. Assume that D^Xt has the form

f{t,Xt; u), which is an B? ^ R mapping with derivative up to second order

with respect to t and X and bounded second derivative with respect to X.

\ 丨 Under this assumption, Black-Scholes equations can be obtained as in Necula i (2002). The purpose of this section is to find some conditions on the drift

term /^(t, X) and the volatility term a{t, X) so that the identity

DtXt = f{t’Xt.,u� (9)

holds for three dimensional real functions /.

Assuming (9), by fractional Ito's Lemma,

f{t,Xuu) 二 Ij^ + X) + X)f{s, X- s)]ds

Using Theorem 4.1.1,

DtXt = lj^f�s,X� +u ) a�s,X�(Ku,s�Ads

+ j:备f�8,X.,u)odB:.

Comparing these two expressions, we conclude that if,

i !, • rtnted with FinePrint - purchase at www.fineprint.com 餐+ “盖-德+咖,邪- 力)二 0 (11)

are satisfied, then

Y如•,!!、二

solves the stochastic diffusion equation (8).

Let us obtain the explicit relationship between fi, a and f first which is a

necessary condition for which the identity (7) holds. With this relationship,

Xt can be solved.

For given (j{t,Xt), (10) implies

f(j:,Xt.,v) = C{t,u)a[t,Xt), (12)

where C{t,u) is an arbitrarily chosen deterministic function that satisfies

C(0,ii) = 0. The last condition holds because at time t = 0, Xq is a given constant that does not depend on the path cj and so is deterministic. This leads to D^Xo = 0. Now, put (12) into (11),

dC{t,u) , X da a/i (J^^ + 以)液-冗(力,以)砍 +

a^C{t,t)C{t,u)^ — (J 小 M + C��u) = S0.

• rtnted with FinePrint - purchase at www.fineprint.com Rearranging it,

。[华-+ C� t+, 盖—• + 孜0.(13)

Since a and the square bracket in the second term of (13) do not involve u

while u) and the square bracket in the first term do not involve X, there

must be a g{t) such that

1 ,dC(t,u) “ �i 兩-射)j 1 da dfji 2 � �92(7 d(j =一少庇—J砍际+瓦] E g{t), (14)

for all t > 0. Dividing (13) by we get

/1 w … �d^o - 1 , � 1 da

Integrating,

fX 1 da

二 M(，X) (15)

h{t) in the last expression is an arbitrary function of t. The next task is to obtain possible solutions for Xt. It is useful to construct a two dimensional function a{t, X) so that the stochastic innovation of a(t,Xt) does not

•rtnte dwit h FinePrint - purchase at www.fineprint.com involve Xt explicitly. After that, the stochastic diffusion equation (6) can be

simplified and hence Xt can be solved. Let

fX 1 a(t,X) = / dx, Jo cr{t,x) the integral here is defined in terms of classical calculus. Further assume that

冲 1入)has a bounded derivative with respect to X. Such choice of X)

renders the stochastic term not involving X. It is shown below by using

fractional Ito's Lemma.

/ ft da da ,, 广 da ^^rr .. a(t，X) = j)瓦+ P丽丽W + l^a冠•财.(16)

Substituting (9), (12)，(15) and a{t,X) = ff ^dx into (16),

a(t,X)=义“⑷& +义补)[义 cr{l X严、IQ dB:

=[h{s)ds + [ g(s)a(s,Xs)ds+ [ dBf Jo Jo Jo

Again, use Hermite transformation and Wick's calculus,

at a{t,X) = eio'"’� + / 厂 e_f�礼g)dBf] Jo Jo

The above result is summarized in part (a) of the following theorem. Theorem 4.3.1 61

rinted with FinePrint _ purchase at www.fineprjnt.com L, Let fj,{t,X)j a{t, X) be two smooth mappings: B? R satisfying relation- ships (11) and (12).

(a) If Xt governed by (6) has stochastic gradient D^Xt = f{t,Xt;u) which

is a B? R mapping twice differentiable with respect to t and X as well

bounded derivatives up to second order, then, Xt and D^Xt are given by

a{t,X) 二 e九、[ao+�eM�h�ds+�e� (.�(17dBf) � Jo Jo DtXt = C(tu)a(t,X) (18)

where g(t) and h{t) are arbitrary continuous functions and C{t, u) is given

ft C{t,u) = e^osds^ / (19)

(b) Also, Xt in Equation(17) satisfies stochastic diffusion equation(6) and

the stochastic gradient of Xt is given by (18) and (19)

Proof.

(a) equation (17) and equation (18) are shown before, equation (19) is the

solution of ordinary differential equation (14).

(b) By the chain rule and the fact that for any k{s) G I^，^t fo k�s�dB?= f^k{s)(t){u, s)ds , DtXt = C(t,u)a(t,X) immediately follows. That (17) solves the stochastic diffusion equation is shown below. Let b(t,Z) be a

• rtnted with FinePrint - purchase at www.fineprint.com R^ R function determined by a{t, b{t, Z)) = e^o'"⑷心Making thetrans-

formation Z = e—几⑷心X), after some calculations,

瓦二一 Jo •瓦，

da — 1

d^a — -1 da 际=7玩） and

dt ^dx> [y dt^'

dx —�dx) ‘ 炉6 = da , d'a dz^ —�dx) •

Using fractional Ito's Lemma,

dX = db{t,Z) =db(t,ao+ ft h�s)ef“礼)ds +�“ef礼)dBf) Jo Jo

=+ heft一. + ef“礼、{t, s)dBf]^}dt

oZ =\age� Z + a L ^^x + ah r e" fo s)ds]dt -i-ao dB^ dX Jo NX 1 PX 1 QJ QA =a\g / -dx-v / — —6/x + + t)]dt + a o dB^ Jo cr Jo cH dt dX 63

rinted with FinePrint _ purchase at www.fineprjnt.com L, •

Now, an extended fractional Black-Scholes PDE with continuous dividend stream is presented.

Theorem 4.3.2 (Extended Fractional Black-Scholes Equation)

Suppose that the market consists of two securities, a risk-free bond and a

stock. Here, the stock provides dividend continuously with rate S and the

drift ijL{t,X)，volatility a{t, X) satisfy equations (10) and (11). Then the

price of a derivative on the stock price with a bounded payoff f{X{T)) is

given by P(t, X)，where P(t, X) solves the PDE:

f)P d'^P dP + a2[t,X)C 拟+ (r - -rP = 0， (20)

with boundary condition

P(T,X) = m. (21)

Proof.

Consider a solution P{t,X) given by (20) and (21). Applying fractional Ito's Lemma and Theorem 3.3，if Xt is a stochastic process governed by (6)，then

p\TD PIP fp- P BP dP[t, Xt) = l苦 + 叉）+ 际一(力,XW, mt + 砍冲,X) o dB^

%\e6 with FinePrint _ purchase at www.fineprint.com - Form a trading strategy by dynamically adjusting a portfolio consisting a

varying quantity v{t) of stocks and u{t) of bonds. By choosing

冲）=M 1 dP 姻=n;(户—叉蓝)，（22)

then the portfolio value at time t is Pt and

ru{t)Utdt + v{t)iJ.{t, Xt)dt + [v{t)a{t, Xt)] o dB^ + 5v{t)Xtdt QP gp QP QP =[rP - rXjAdt + p-dt + (a-) •财 + -5Xdt f)P f)p ff p =力+ (力拳 fjp f)p f)p 玩出作冠）玩腳亡 fiP f)p ；92 p f)p =I盖 + ；+ it, X)C{t^ t)]ds + —X) o dB^

=dP{t,XT).

By Definition 3.2.1(a), {u{t),v{t)) is a self-financing strategy. It can be shown

that such strategy hedges the derivative. The portfolio value at time t is

given by u{t)Iit + v(t)Xt and it is equal to P(t,Xt). At b 11x10 of mati_irit�，

the portfolio value is just P{T,XT). By assumption, the function P{t,X)

satisfies the boundary condition (21), so P(T,XT) = /(^T)- NOW we have proved that (u(t),v(t)) hedges the derivative and F(t, X) is the "no arbitrage price". •

• rtnted with FinePrint - purchase at www.fineprint.com 4.4 Option Pricing Problem for Fractional Black Scholes Model

with Varying Trend and Volatility

Here, we illustrate the application of results in Section 4.3 to the fracional

Black Scholes model with varying trend fit and volatility cr,. Further, assume

that the stock provides a continuous flow of dividend at a constant rate 6.

Assume further that there is a risk-free asset with continuous interest rate r.

The market model can be summarized in the following stochastic differential

equations

Xt = Xo+ [ fi{s)Xsds + [ a{s)XsOdD^, Jo Jo

Ut = [ rUsds. (23) Jo

The option pricing formula for an European call option will be given in this

section.

From the discussion in Section 4.3，it is known that if

+ 广 (24) Jo dt

for some chosen g{t) and ⑴，then, the extended fractional Black Scholes equation (Theorem 4.3.2) can be used to give option pricing formulae for derivatives that has payoff at a fixed maturity that depends only on the

• rtnted with FinePrint - purchase at www.fineprint.com value of the underlying asset price at maturity. In the case of varying trend and volatility with the form

f^it.X)三

a{t,X) = a{t)X,

they satisfiy equation (24) by choosing

(� G\i)

h(t) = ^一 ⑴，

where C(t,t), from (19), is given by

1 广 C(t,t) 二一 / (750(t, s)ds. (^t Jo According to Theorem 4.3.2, the PDE governing the European call option

price is given as below.

Proposition 4.4.1

The price of an European call option in the varying trend and volatility model is given by P{t,X), where P{t,X) is the solution of the PDE:

Qp pt f)2 p gp 菩 + 冲)[y�魂 s)ds]X'^ + (r — -rP = 0,

with boundary condition

• rtnted with FinePrint - purchase at www.fineprint.com P(T, X) = max(X-A^ 0)，

where K is the strike price.

As in Section 3.3, two analytical methods, method of substitution and method

of auxilliary equation can be used to solve the PDE.

Proposition 4.4.2

The price of an European call option with strike price K, mature at T at

current time t is given by

P = SN{di)-Ke-''N{d2),

where

= ， / \ T (k = —(CT, a)"

and

(cr, = It

The function N{') refers to the cumulative distribution function of a standard

normal random variable.

Method of Substitution :

I i

^finted with FinePrint _ purchase at www.fineprint.com Using substitution

t = u,

X = exp[^ + (r - 5)u — -{a, a)^. 2

The PDE becomes a heat diffusion equation

�+�=0 du 彻2

Q(T, v) = e-rT max exp + rT —丢(a，o")�j>，0夕，

where Q 二 e—即，which is the discounted option price. Applying standard

method (see Lewis (2000)) to solve this PDE yields the call price formula at

time t. •

Method of Auxiliary Equation :

It is useful to think the solution of a PDE at as the expected value

of payoff function under the following diffusion process with respect to a

classical Brownian motion

dX' = {r- 6)X'dt + {2(Jt [ as小(t, s)dsy^^X'dBt, (25) Jo

with t�t o(see Milstein(1995)). Applying classical Ito's Lemma to logX',

the solution of (25) is given by

〜如ted with FinePrint - purchase at www.fineprint.com ifliL.. - - X; 二 Xoexp[(r-5)力—S [ [ � / “)圳�“心丄 Jo Jo Jo Jo

The term

(r — - I 义 L CT一(C,執dr]

is deterministic while the Ito's integral

[t{2a� /\0(C，”)d")i�Bc Jo Jo is normal with mean zero and variance (cj, a)^ by It6 isometry. Hence the fractional Black-Scholes formula is obtained. •

^finted with FinePrint _ purchase at www.fineprint.com 5. Alternative Fractional Models

The study of stochastic volatility can be traced back to two papers Black (1975) and Black (1976). They provide a model called Constant Elasticity of Variance model, to allow stochastic volatility. It is expressed in terms of stochastic diffusion process with respect to classical Brownian motion

dX = fiXdt + aX3 丨 2dBt,

where 0 < /3 < 2 is a constant. If /? = 2，such model degenerates to a geometric Brownian motion. This model is characterized by the dependance of the volatility rate (i.e. aX卢on the stock price when (3^2. When the stock price increasess, the instantaneous volatility rate decreases. This is thought to be reasonable because the higher the stock price, the higher the proportion of equity market value and thus the lower the proportion of liability. The risk of ruin decrease as a result. So, the volatility rate or the risk measure should decrease. Cox (1976) studies the CEV model and gives an option pricing formula which involves a noncentral x^ distribution function. In this section, we extend the CEV model to capture long-range dependence and derive an option pricing formula for the CEV Model.

• rtnted with FinePrint - purchase at www.fineprint.com [

5.1 Fractional Constant Elasticity Volatility (CEV) Models

In order to extend the model into the fractional setting, a new definition for

fractional CEV model is given first. Consider the model

dX = flit, X)dt + dX…2 o dBf,

where X) is chosen such that Equations (11) and (12) are satisfied. Here

we do not choose 冲，X)三 jiX because g{{), h(t) cannot be found so that

lji{t,X)三 iiX when a{t,X)三 aX附.From equation (15). fi has the form

M 力，= Y^x + a"� X明 + 令 x"c�t ,t).

Let g{t)三 //(I - f) and h{t)三 0. Then

C{t,t) = egt [ s)ds Jo

=H{2H —1)[ Jo The drift term becomes

and the stochastic gradient of X is

Though still looks complicated, by using the relationship between Stratonovich type and Wick-Ito type integral (i.e. Theorem 2.2.4) and the chain rule (Theorem 2.3.9), the diffusion process can be rewritten as

• rtnted with FinePrint - purchase at www.fineprint.com dX = iiXdt + dX 附 dB?

which looks very similar to the classical model. As well when (3-^2, the

model becomes the one given in the Cutland, et al. (1976)，i.e.，a geometric

fractional Brownian motion with drift fj, and volatility rate a.

-at

• rtnted with FinePrint - purchase at www.fineprint.com 5.2 Pricing an European Call Option

The Black Scholes PDE (Theorem 4.3.2) of this model is now given by

By putting Y' = X" (see Cox (1976)) and Q = e—”中，this equation

becomes

尝+ +鄉 t)]祟+ aCM|| = 0

where a = - pf, b = (r - S){2 - /3) and c = - d){l - p).

The approach of Cox and Ross (1976) using Feller's result (see Feller (1951a,b))

can be adopted. First the solution for this equation at (to, lo) is the expec-

tation of payoff function under the SDE

dY = [bY + cC{t, t)]dt + s/ma{2 - (3)C{t, t) VYdBu (26)

with Yt�二 lo (see Milstein (1995)). To solve this SDE, we follow Feller

(1951b). First, a useful theorem is stated below.

Theorem 5.2.1(Kolmogorov(1931))

The probability density function of a diffusion process Xt driven by classical

Brownian motion

dXt - f^t, Xt)dt + a(t, Xt)dBt

•rtnte dwit h FinePrint - purchase at www.fineprint.com is given by the PDE

In our case, because of (26), the Kolmogorov's equation becomes

ut = [aC{t, t)YU{t, Y)]YY - [{bY + cC(t, t))u{t, Y)]Y-

We follow Feller (1951b) to derive u{t, X). Assume that the Laplace Trans-

form of u{t, Y) with respect to Y exists and equals to u{t, s). Since the value

of Y at time 力。is given, so 1^� ideterministis c and thus u{to. Y) = 6{Y — Yq),

the Dirac function. Also, a;(to, s) = e—$补=7r(s) and the equation becomes

the boundary value problem

Ut 十 s{aC{t, t)s — b)(Js = -cC{t, t)su + /� (27)

u;(to,s) = e—s补. (28)

In equation (27), /(•) is called the flux of u at the origin (see Feller (1951b)) which is to be determined later. Now, we find the characteristic curve of the first order PDE (27). The characteristic curve is given by

rfq dt

The solution for this equation is

s = - aj{t)]-\

where 7� is defined as

Printed with FinePrint - purchase at www.fineprint.com —^―——^ 7⑴二 f T)dT. J to On this curve, uj{t, s{t)) satisfies

盒如⑴)二 f{t)-cC{t,t)sit)u{t,s{t)).

Solving it, we get

uKt,s(t)) = [Ci — a7⑷严[C2 + f f{T)\Ci — a7(T)|-c�T]. J to The given point (t, s) passes through the characteristic curve with Ci —

aj{t) + • = s). By putting t = to, we have

C2 二 s)).

The Laplace transform of u{t, Y) is thus

J to se oMt-to) pt ^ , =[霞％ + 1 广�(• 1)�� + I [縦况+(7t - Ir) + 1]力⑷办• （29)

Following the argument of Feller (1951b)，when u(t,0) is finite and c < 0 or 0 < c < a, then f(t) is given by the integral equation

+ f /(tK^^C〜T 二 0. Vy � J to It-IT To solve this equation, applying the substitutions 2: 二女 and = ；；^,

•rt ntedwit h FinePrint - purchase at www.fineprint.com I邏—‘广� c =兀(•)=兀(7) xz =exp{—一)， (30) Qj

where x 二 e—况。:T(力。）and g� Q= —/MC�"奢.

The solution of (30) is

y V 1 \ —C + a , X V

fir)=水 —1 . 1 . . X ^c+a X djr =救)a e於力I

Substituting this result back into (29), after simplification, we get

u;(t，s) = + + — a'仅7“寶％ + 1))

The next step is to perform an inverse Laplace transform. To do this, let

A =丄， OTft z = sae^^jt + 1-

One can verify that equation (6.5) in Feller (1951b) is still valid after these substitutions. u{t,s) can now be rewritten as

•rtnte dwit h FinePrint - purchase at www.fineprint.com Using the fact that Laplace Transform of ig e~f we have

u(t,Y) = 守 [丄(e-� ^ , �ae讨 7/� Y ) L ae切 7t � �jt In the above equations, JA(-) is the first type Bessel function with order A.

Since

P{t, X) = e�{t X), = - K, 0)],

using the solution of u{t, F), we have

广 1 P{t,X) = - K)u{T,y)dy JK" ^ 1 1 XT 1 =e-y 工点丄广G(r + 1 + ；— r\ ajT 2 - p ae坑It r=0 ^ 1 X r 1 Ke—八T-t) y --^e—;(丄广+ 1，台 r(r + l —点） �T ) ��) Since x is the discounted Yt, i.e. therefore

x^ = e'^Xt = e 令的 tXt.

So the option pricing formula is

~ 1 XT 1 P(t, XO 二 e-収—V mr + 1 + - \ ‘ tj j^i ^ajT 2-/3 ae^^jT

•rt ntedwit h FinePrint - purchase at www.fineprint.com c I

jI i,

: Ke-八T—t) f； ‘ 1�e-念(丄由G( r+ 1， I + ORFT A一TT r F ft

1 Ij1 .

>r：

• rtnted with FinePrint - purchase at www.fineprint.com 6 Problems in Fractional Models

We have seen some fractional market models above. All these models look

similar to their corresponding models for classical Brownian motions. As

studied in Duan (1995), classical models, especially, the original Black Scholes

model, introduces a phenomenon called 'volatility smile' in practice. This

gives motivation to develop new model to overcome this problem. So, it is

reasonable to expect that the fractional models can account for the volatility

smile. But, the models studied in this thesis derive option-pricing formulae

just similar to the classical one, will it give better explaination for such

phenomenon than classical models?

For the sake of simplicity, Black Scholes formula is discussed first. If the

fractional model is correct, one can see directly from the fractional Black

Scholes formula that the implied volatility from the classical Black Scholes

formula is

a lt2H - tl^ 1 t-to • Such implied volatility does not depend on the strike price K, so, the new model does not account for the volatility smile observed in practice. The situation for fractional CEV model is similar, it cannot give better result then the classical CEV model. The new model can still reveal volatility skew, but not smile.

Though the new formula is very similar to the classical one, new difficulties arise. One of the problems is that the fractional Black Scholes model introduces non-stationarity. If fJi and a are just considered as mean return and

%ited with FinePrint - purchase at www.fineprint.com i • «« standard deviation of return as usual, non-stationaritv is introduced bv the

fractional Black Scholes diffusion equation. As we have discussed in Section

3.1, the solution of the equation (4)

= [ I^Xsds + [ aXs o dBf, Jo Jo is given by the fractional geometric Brownian motion

/ 1 \ Xo exp jjit — -ah训 + (jJ5f • V 2 y To make this stochastic process make sense, the unit of a must not be per-

centage change per unit time, but somewhat depends on the Hurst parameter

H. Then, a cannot be simply regarded as the standard deviation of the re-

turn.

• rtnted with FinePrint - purchase at www.fineprint.com 7 Arbitrage Opportunities

A difficulty in pricing an asset using Wick calculus approach is discussed in

this section. To illustrate the problem, consider the following two models.

Xt = Xo + [ + f aXs o dBf, Jo Jo and

Xt = Xo+ [ fiXsds + [ aXsdB^. Jo Jo It is shown in Section 6.1 that the two stochastic differential equations are equivalent. The first equation satisfies the condition for Black Scholes PDE, and therefore no arbitrage opportunities exist. The second stochastic differential equation gives the fractional geometric Brownian motion model discussed in Cutland, et al. (1976) and according to Rogers (1997), this introduces arbitrage opportunities. This discrepancy arises from the definition of "self-financing strategies", which is explained in Section 7 .2.

tinted with FinePrint - purchase at www.fineprint.com I ••MIIMIMI.IIII.IIWII —••im_i •• 7.1. Two Equivalent Expressions for Geometric Brownian Motions

It is discussed in Section 2.4 that Stratonovich integrals and Wick integrals

are related and the linkage is established by the formula

r FsdBf = f FsO dB^ + r DfFsds. (31) Jo Jo Jo

Now, this relationship is applied to a fractional diffusion process satisfies

conditions (11) and (12), then,

X = f fi{s,X)ds+ [ a{s,X)odBf Jo Jo rt f)^ ft =Xo+ / [^{s,X)-—DtXs]ds + y�a[s,X)dBf

t ft /

If we choose g(t)三 0 and h{t)三，—crHt饥it is known in Section 4 that the Black-Scholes model of Hu and 0ksendal (2000) is obtained. But herein,

h(t)三尝 and g{t) is still chosen to be zero. The new model is thus

X = l\ + (j2lIs2H-i�Xds+ [ aX o dBf Jo Jo =[fiXdt-^ I (jXdBf. Jo Jo

According to Theorem 4.3.1, the explicit solution of new model is

Xt 二 exp (/it + crBf),

•rtnte dwit h FinePrint - purchase at www.fineprint.com which is geometric Brownian motion given in Cutland, et al. (1995).

Since the Black-Scholes PDE (Theorem 4.3.2) does not depend on the choice of h{t), so both models of Hu and 0ksendal(2OOO) and Cutland, et al. (1995) yield the same option pricing formula and have no arbitrage opportunities.

rinted with FinePrint _ purchase at www.fineprjnt.com L, —„

I'N 丨�Fro ma classical point of view, such as Delbaen and Schacherrnayer (1994), i I the market has no arbitrage opportunities if and only if the stochastic process

I. ：-governing the asset price is a semi-martingale. Rogers (1997) proves that I I fractional Brownian motion is not a semi-martingale and thus the market model has arbitrage opportunities. So, no arbitrage pricing theory cannot be i used to price the option for such an asset. But Hu and 0ksendal (2000) show i that under fractional Black-Scholes model defined using Wick-It6 integral, i there is no ” arbitrage opportunities" • This discrepancy exists because of the

I definitions of self-financing strategies used by Hu and 0ksendal is different

I from that of Cutland, et al. (1995) and Rogers (1997).

I In Hu and 0ksendal (2000), a trading strategy consisting of a quantity (ut, tk)

I of bonds and stocks is called self-financing if the infinitestimal change in the

I portfolio value at time t is given by

I； 4 I dZt 二 diutUt + VtXt)

I 二 rUtUtdt + Xt)vtdt + [vta{t, Xt)] o dB^ + dA,

I where dA is an infinitestimal dividend payment term.

It should be noted that the Wick integral in Hu and 0ksendal (2000) is the ：Wick integral of ordinary product between a{s, Xs) and Vs, i.e.,

：

[\vsa{s.Xs)]odB^ = limE[�(7(s”Xj o] (B: — (32) Jo

I it I •rtnte dwit h FinePrint - purchase at www.fineprint.com A more natural way to describe the infinitestimal change is

义力尤’她幻•dBf] 二 limE�—B� ]. (33)

The integral in (32) can be roughly interpreted as the sum of Wick product

between Vs(j{s, Xg) and the change in fractional Brownian motion while the

one in (33) refers to the sum of product between number of stocks held and

the infinitestimal change in stock price. The purpose of this section is to

show the difference between two types of integrals. The discrepancy between

two integrals can be seen through Theorem 2.4.4.

=[{vsa{s,Xs))dB^ - [ Dt{vsa{s,Xs))ds. Jo Jo Because the Stratonovich integral is the limit sum of ordinary product, so,

[\vsa{s,Xs))odB^ Jo

=I^Vs[a{s,Xs)dB^]- Dtivs•聯s

=1: Vs[a{s, Xs) o dBf + Dt(j{s, Xs)ds\ — ^ Df—Qs, Xs))ds.

Using Product Rule of Malliavin calculus,

[\vsa{s,Xs))odBl' Jo

=f Vs[a{s,Xs)odB^]- [ a{s,Xs)Dtvsds. Jo Jo

rinted with FinePrint _ purchase at www.fineprjnt.com L, The two integrals are differenced by the term

[a{s,Xs)Dtvsds. Jo

m�

.i^B

�

d •ar : 87

T t I

•rtnte dwit h FinePrint - purchase at www.fineprint.com 8. Conclusion

This thesis illustrates the usefulness of Malliavin calculus and Wick product in analyzing long-range dependent financial models. We have briefly describe the work of Hu and 0ksendal (2000) and Necula (2002). They assume that the stock price Xt and the risk free asset price follow

Xt = [ ^iXds^ [ aXodBf Jo Jo

lit = L r肠.

The integration here is defined in Wick-Ito sense. There are some conveniences using Wick's product. One of the advantages is that the Wick integral has mean zero. The solution to the fractional Black Scholes model is

1 Xt = exp (/it — -t^^ + gB^). At Hu and 0ksendal (2000) provide an option pricing formula similar to the classical one. A Black Scholes PDE, with independent variables i and X, is given in Necula (2002).

The first contribution of this thesis is to extend the above model to a more general setting. The fractional Black Schole PDE does not always exist for more general situations such as

Xt= [ i4s,X)ds+ [ a{s,X)odBf. Jo Jo

• rtnted with FinePrint - purchase at www.fineprint.com Conditions on fi{t,X) and a{t,X) can be found, however, such that the

PDE approach is still applicable to solve the option pricing formula. As

an example, it can easily be shown that the fractional Black Scholes model

satisfies such conditions. Two methods of solving the PDE are demostrated.

Also, an extended model, fractional Black-Scholes stock price model with

deterministic but varying trend and volatility rate, is studied. The model is

formulated bv k!

Xt = XQ+ [ fi{s)Xsds + [ a{s)XsOdBf. Jo Jo A second contribution is to illustrate how fractional model can be used to

model volatility. It is well known that European call option pricing formulae

does not account for the "implied volatility smile" phenomenon, see Duan

(1995). One way to account for the difference is to introduce stochastic

volatility. One popular model is the Constant Elasticity of Variance Model

(CEV, see Cox and Ross (1976) ). Here, we study a fractional CEV model

given by

dX 二 i^Xdt + aX.叫dBf.

Here 0 < ；0 < 2 is a constant. If /? = 2, such model degenerates to the

fractional Geometric Brownian motion proposed by Cutland, et al. (1995).

Although the proposed fractional CEV model cannot account for the "volatil-

ity smile" pattern, the "implied volatility skewness" phenomenon can be re-

vealed.

The third topic studied in this thesis is the refinement of "self-financing

^rinted with FinePrint - purchase at www.fineprint.com f /V . — __ ... strategies". It can be shown that the following two stochastic differential equations are equivalent,

= [ [fiXs + + [ aXs o dBf, Jo Jo and

=為 + [ iiXsds + [ aXsdB^. Jo Jo The first equation satisfies the condition for Black Scholes PDE. and therefore no arbitrage opportunities exist. The second stochastic differential equation gives the fractional geometric Brownian motion model discussed in Cutland, et al. (1976) and according to Rogers (1997), this introduces arbitrage opportunities. This discrepancy arises from the definition of "self-financing strategies" used. In Hu and 0ksendal (2000), a trading strategy consisting of a quantity [Ut’ Vt) of bonds and stocks is called self-financing if the infinitestimal change in the portfolio value at time t is given by

dZt 二 d{utUt + VtXt)

=rUtUtdt + /i(t, Xt)vtdt + [a{t, Xt)vt] • dB^ + dA,

where dA is an infinitestimal dividend payment term.

It should be noted that the Wick integral in Hu and 0ksendal (2000) is the

Wick integral of ordinary product between and Vs, i.e.,

[[vsa{s,Xs)]odB^. Jo 90

Vinted with FinePrint - purchase at www.fineprint.com •^^^^^MMMMr—wi—iwiiiiiiiiiii 111 liiMii mil llllll A more natural way to describe the infinitestimal change is

[Vs[a{s,Xs)odB^l Jo

which is consistent with settings in Cutland,, et al. (1995). In this case, the

two integrals are related by

I [ {Vs(j{s, Xs)) O dB^ = [ Vs[a{s, X,) • dB^] — [ a{s, Xs)Dtvsds. Jo Jo Jo

Consequently, using the latter definition, the no arbitrage condition becomes

void and the resulting valuations is consistent with Rogers's finding.

I I 1 ！i

r. 1.‘ !•

•rtnte dwit h FinePrint - purchase at www.fineprint.com Appendix A. Fractional Stochasitic Integral for Deterministic In- tegrand

The purpose of this appendix is to compare different approaches that define stochastic integration with respect to fractional Brownian motion for deterministic integrand. According to Lin (1995), if the integrand is a real deterministic and Lebesgue measurable function, then the stochastic integration can be defined in Ito sense. In particular, if the integrand is an elementary function of the form,

n /n ⑴=[‘红+1)， k=0 where {ak} and {t^} are finite sequences of real numbers, then, the stochastic

integration of / � is defined as

[/n � < = <� J R k=0 Suppose that f{t) is a Lebesgue measurable function which can be approx-

imated by a sequence of elementary functions {fn{t)}- Passing to the limit in L^-sense, the integration is

[/ �<)• J R k=o This approach requires that the Hurst parameter H > In order to release the restriction, several methods can be employed. Some of them are summerized and studied in Pipiras and Taqqu (2000) and Pipiras and Taqqu (2003). Such methods build up mappings from some spaces of deterministic real

•rtnte dwit h FinePrint - purchase at www.fineprint.com functions to a set of random variables (the deterministic fractional stochastic integrals). In some particular methods, the mappings can be taken as the classical Ito integral of the integrand after performing transformation. Other approaches, such as the Malliavin calculus approach and S-transformation approach are more general and they are discussed in Appendix B.

！i

i I ！ i I >:i .V I

il I ^ 93 I；

• rtnted with FinePrint - purchase at www.fineprint.com A.l. Mapping from Hilbert Space to a Set of Random Variables

Here, an important theorem is introduced. It builds up the mapping from

deterministic functions to a random variable which is refered to the stochastic

integral for that function. The set of elementary functions, i.e., real functions

of the form

^ k=0 where {ak} and {t^} are finite sequences of real numbers, is denoted by E.

For a function f e E, the stochastic integration of f with respect to the

fractional Brownian motion refers to the random variable

[fn{t)dBf 二 (34) J R k=0

The probability space for fractional Brownian motions can be chosen to be

(Q, i.e., the one defined in Section 2.2. It is a convenient choice

if improper integral of infinite region is Brownian motion, which is a parallel analogy for the fractional case, is well developed (see Kuo (1996) and Holden,，et al. (1996)) in this space. But the ^ choice of sample space is not unique. An alternative is to use C[0,1], the set of continuous real functions, which is the setting used in Decreusefond and Ustuuel (1999). Let Sp be the closure of the set of random variables in the form of expression (34) in

Theorem A.1.1 (see Pipiras and Taqqu (2000))

•.imr

• rtnted with FinePrint - purchase at www.fineprint.com Suppose that 0 < H < I and C is a set of real deterministic functions

equipped with inner product (•, ')c- C contains all elementary functions and

the inner product on C is an extension of that in E. Here the inner product

in E is the covariance of two stochastic integrals (i.e., the one defined in

(34)). Also, E is dense in C. Then, there is an isometry between C and a

linear subspace of Sp. And C �S pif and only if C is complete.

S • ；

TV 1 ^ >

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.會*

：

inted with FinePrint - purchase at www.fineprjnt.com m ••••III••III III•nil••«•II醐 _ I mill A.2. Fractional Calculus

Before we proceed, concepts of fractional integration and differentiation are

introduced first. They can be regarded as operators on a deterministic func-

tion. These concepts will be used to define inner products later.

Definition A.2.1

Let f E b] and real number a�0，th eleft fractional integral of order a

：is

\ (C/)�=f ^ 馳—0“成

and the right fractional integral is

(/?-/)W = Y^) f: fm — ty-'dt^x < b.

If a = -oo orb = -foo, the notations are {I^f){x) and {I^f){x) respectively.

Remark: Fractional integral can be regarded as extension for iterated integral. If a is an integer, the fractional integral is the same as iterated integral (e.g., see Pipiras and Taqqu (2003)), and the following relationship holds.

(/：+/)(x)=厂厂 1 …厂 mdtdh • . • dtn-i. J a J a J a Definition A.2.2

Fractional derivatives are defined as

tinted with FinePrint - purchase at www.fineprint.com i r —_ . {D:.f){x) 二 (去)二-

= •[咖(/i—�/)� r I Remark: Fractional differentiation is inverse operation of fractional Integra- ii j1 I tion. 5

For more details about fractional calculus, one can refer to Samko, et al (1993).

tinted with FinePrint - purchase at www.fineprint.com f J A.3. Spaces for Deterministic Functions

In this thesis, the space for deterministic functions is i.e., the space

consisting of functions that have finite norm for the inner product

= / / f{u)g{v)(l){u,v)dudv, JRJ R where

(l){u, v) = H{2H - l)\u — if 开-2,

or equivalently,

["2/)⑷(广"2州咖& JR This space only valids for H > Pipiras and Taqqu (2000) shows that E is

dense in L^, but L^ is not complete and so, by Theorem A.1.1, an isometry

between L^ and a proper subspace of Sp is established. The isometric image

of a function / G is refered to the stochastic integral with respect to

fractional Brownian motion, i.e.,

[ f{t)dB^- JR In case ot H < A16, et al. (2000) suggests another inner product , that is Zd

(/，")c= f�D]/2-Hf)(s)(Di�2—Hg)[s)ds. JR The corresponding function space thus consists of those ^ f G L^. This

function space is complete, however. So, an one-one correspondance between

this space an Sp itself can be established.

•rtnte dwit h FinePrint - purchase at www.fineprint.com Due to the one-one correspondance between I? and the set of random variables spanned by increaments of classical Brownian motion (see 0ksendal (2003)), one can map the fractional stochastic integral to a stochastic integral which belongs to the closure of set of linear combination of increaments of classical Brownian motion, i.e.,

[f{s)dBf ^ [(广⑷讽 Jr JR and

丨 [/(柳f 4 [ pi"-丑/)(批， Jr JR here, we ignored the constant multipliers. The left hand side and the right hand size differ by a constant multiplier.

Consider a simple example that f{s) = l[o，t](力 and H > 1/2. The stochastic integral with respect to the fractional Brownian motion can be expressed in terms of an improper integral with respect to classical Brownian motion with infinite interval.

J[f{s)dB^R 二 JRf l[o’t](视丑

=[(料 JR

It is consistent with the definition of fractional Brownian motion provided by Mandelbrot, B. and Van Ness, J. (1968). Besides this, there is a canon-

• rtnted with FinePrint - purchase at www.fineprint.com ical form of the fractional Brownian motion that does not involve improper

integral. Norros, et al (1999) give the following representation,

B''{t)=CH�K�t,s�dBs, Jo where, CH is a normalizating constant and the kernal function

K{t,s) = slA—Hj:uH-L/2� U— S)H-^du.

The kernal can also be expressed in terms of fractional integral,

It is natural to represent the finite region integration of a deterministic func-

tion f as

ff{s)dBf〜广"2_丑(/厂〜丑-1/2,⑷)⑷啦 Jo Jo Similarily, the representation for integrals in case H < 1/2 is

ft f(S)dBf〜〜付-1/2,⑷)⑷吼， Jo Jo see Pipiras and Taqqu (2003) for detail discussions of these integrals.

100

•rtnte dwit h FinePrint - purchase at www.fineprint.com Appendix B. Three Approaches of Stochastic Integration

In Section 2.2, integration by the Wick calculus approach is illustrated in detail. This model assumes that the Hurst parameter H is given and fixed, so, only one fractional Brownian motion can be involved. In case of derivatives depending on more than one underlying assets, each asset may be driven by a different fractional Brownian motion with different Hurst parameter. To tackle such cases, the approach of Elliott and Van de Hoek (2003) can be adopted. Instead of directly choosing fractional white noise as in Hu and 0ksendal (1999), they propose using classical white noise as the probability setting. The integration of deterministic function with respect to fractional

‘Brownian motion is then regarded as integration of the transformed function with respect to classical Brownian motion. Besides using classical Brownian motion as the starting point, Elliott and van de Hoek (2003) do not differ much from Hu and 0ksendal (1999). Both papers define stochastic integration by means of Wick product. Such methods are refered as the white noise approach (see Bender (2003 a,b)).

Bender (2003 b) studies stochastic integration with respect to fractional Brownian motion in three ways. They are all based on classical Brownian motion settings. These approaches are white noise approach, 5-Transform approach and Malliavin calculus approach. Here the Malliavin calculus approach is the one originated from Decreusefond and t)stiinel (1999), but the settings are modified by Bender (2003 b). Bender (2003 b) shows the relationship between these three types of integrals.

101

• rtnted with FinePrint - purchase at www.fineprint.com Before proceeding, the setting which is parallel to the one in section 2.2 is

stated first. Let f{t) G S{R) be a rapidly decreasing function (i.e., S{R) is

a Schwarz space) and co E Q = S"(i?), a bounded linear functional on the

Schwarz space S{R). Note that u is an operator on a deterministic function.

According to Bochner-Minlos Theorem (see Holden, et al. (1996) Appendix

A), there exists a unique probability measure fi on with respect to the

weak star topology such that the characteristic function of / G S{R) is given

Jn where

||/||2 二 [ f{t)dt. JR That is, the norm for L^. For / E we can always choose a sequence of

f几 e S{R) so that /n / in lA Then �(J ca) n be defined as lima;(/„) as

this limit exists in I?. Now, Jj^f{t)dBt is defined as uj(J), and the indefinite

integral at time s, or f{t)dBt is taken as fj^f{t)I[t < s]dBt.

In the following, different definitions for stochastic integration and Bender's

results are introduced.

102

• rtnted with FinePrint - purchase at www.fineprint.com B.l. 5-Transformation Approach

5-Transform is a tool to convert a random variable into a deterministic func-

tional. With this, we can analyze stochastic processes in a non-stochastic

way. The definition is given below. The symbols are originated from Kuo

(1996).

Definition B.1.1

Let X{uj) e {Sy, where (5)* is defined in Theorem 2.2.3. The S-Transform

of X is defined as

SX{r]) =< X{uj),exp{uj{r])—去丨丨“們〉，

for all T] G S{R). See Definition 2.2.6. for the usage of angular bracket If

X is square integrable，this operation is just the expectation of products.

Remark: 5-Transform maps a random variable X to a functional and the

domain of this functional is the Schwarz space. Bender (2003 b) proves that

5-Transform is injective.

Example B.l.2 (see Lemma 2.4 and Section 3.1 of Bender (2003

b))

Let / be a deterministic real function and t is a given real number. Evaluate

5(/t)(77) and S{u垂•

Sftiv) 二五[/CO.exp(a;(") —

103

• rtnted with FinePrint - purchase at www.fineprint.com 二 m . exp ^ /^exp - • (If^) } “

=m.

S{ujf){r]) can be found similarily by noting that ujf and ur] are jointly normal

with covariance given by the inner product

(/, v) = [ f{t)vm. JR

After some manupulations, the expectation is

S{cjf){v) = E S^ujf . exp | 二 —->2}‘/J严 + 為 + 品二歸 An alternative method is provided by Bender (2003 b) which uses Girsanov change of measure. •

In Appendix A, we see how a fractional stochastic integral of deterministic

real function f corresponds to a classical Ito integral. If H < 1/2, it refers

to the Ito integral of {D]/'^"^f){s), and for H > 1/2, it is the Ito integral

of Let us use the same symbol M呈f (see Bender (2003 b)),

as in Elliott and van de Hoek (2003) to denote such transformations. The

operator M^f is normalized so that the covariance between Mfl[o,s](.) and

Mf l[o,t](-) equals to

104

•rtnte dwit h FinePrint - purchase at www.fineprint.com When H = 1/2, M^ f is the identity operator.

Remark : It should be noted that M^ f in Bender (2003 b) and Elliott

and van de Hoek (2003) are not exactly the same. They differ not only by

a constant multiplier, in fact, the 'fundamental operator' in Elliott and van

de Hoek (2003) is

M^f 〜（/付⑷ + (I”2腦,H�1/2

M” �{D]/'-''f){s) + (Dif-Hf)(s),H< 1/2.

Here, notation of Bender is adopted. Below, stochastic integrations with

respect to classical and fractional Brownian motion are defined by means of

5-Transform and M-operator.

Definition B.1.3

Let Xt be a stochastic process. If the product of the S-Transform of X，i.e.,

SX{r}) and M^r] is integrable, then, the stochastic integral of X with respect

to the fractional Brownian motion is the unique element in $ G {S)* such

that

例(77) = I {SXtMiMi^Vmdt

105

• rtnted with FinePrint - purchase at www.fineprint.com To be consistent，$ is denoted by

I XtodB^.

Using fractional integration by parts (see Samko, et al. (1993)), i.e., for any

two real functions, f and g,

[m(MHg)�t)dt 二 [ g{t)M^f){t)dt, Jr J R

and Example B.I.2., it can be easily checked that

^ [讽丑=•/!/). JR

Three important properties of 5-Transform are stated without proof.

Theorem B.1.4 (see Bender (2003 a) and Kuo (1996))

1. S-Transform is injective.

2. Let ^ E {Sy, the space defined in Theorem 2.2.3. There exists a unique element in (5)* so that for all r] G S{R), the S-Transform of this element is

⑷•(例)⑷.

Such element is the Wick product between $ and 屯.

3. For a stochastic process Xt with range (5)*； S-Transform and differenti-

ation are interchangeable.

106

Printed with FinePrint - purchase at www.fineprint.com k 丨••丨丨丨圓咖 By means of Transform, Bender (2003 a) derives an Ito's formula for 0 <

H < 1, compared with Duncan, et al. (2000) which only deals with the case

for H > 1/2. But the application of this formula is much more restricted

because it requires the function F to be a function involving both t and B^,

not an arbitrary fractional diffusion processes.

Theorem B.1.5 (see Bender (2003 a))

Let 0 < H < 1 and F{t,x) be a twice differentiable function, then the follow-

ing Itd,s formula holds，

F{T,B?) = F(0,0) + ^^ B^dt + £ B^dB^

Bender (2003 b) attempts to extend his previous result Bender (2003 a) and

studies the functionals of form, F{t,Xt), where

Xt 三 fasdBf Jo and a is continuous deterministic function. The result for H > 1/2 is a special case for the Ito's formula in Duncan, et al. (2000) and there are no extension for H < 1/2 because it requires cr to be a constant function.

Theorem B.1.6 (see Bender (2003 b))

Let Q < H < 1 and F{t,x) be a twice differentiable function. Let a be

continuous for H > 1/2 and a be a constant for H < 1/2，then the following

107

• rtnted with FinePrint - purchase at www.fineprint.com I ij i Ito's formula holds,

‘ Fif.Xr) 二 F(0,0) + J^ J^ at^F{t,Xt)dXt

j I ！

1 J ,1 t I 1

.1 I

i •i

108

• rtnted with FinePrint - purchase at www.fineprint.com B.2. Relationship between Three Types of Stochastic Integral

Here, we end this appendix with a theorem proved in Bender's papers.

Definition B.2.1 (see Bender (2003 a,b))

M-operator for the stochastic process :

It is similar to that for deterministic real functions except that the integration i with respect to time t refers to Pettis integral，i.e.，the one defined in Section

2, Definition 2.2.6(b).

Theorem B.2.2 (see Bender (2003 a,b)) i Wick approach and S-Transform approach : The stochastic integral with re-

spect to fractional Brownian motion in Wick calculus approach extends the

S-Transform approach stochastic integral. For a square integrable random

variable X G f^), if it is integrable in Wick sense, then U is integrable

in S-Tranform sense.

Malliavin approach and S-Transform approach : Assume that X E LP(Rxft) for some p. Here，when H < 1/2，p > 1 is required and when H > 1/2, 1 < r) < 2 is required. Then，Maliavin approach and S-Transform approach 尸 2.H一 1 give the same integral if the following condition is satisfied,

M^X e L'^iRxQ).

109

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