Risk analysis and hedging in incomplete markets

DISSERTATION

Presented in Partial PulIillment of the Requirements for

the Degree Doctor of Philosophy in the

Graduate School of The Ohio State University

By

George Argesanu, M.Sc.

$\mathrm{*}\mathrm{*}\mathrm{*}\mathrm{*}\mathrm{*}$

The Ohio State University

2004

Dissertation Committee:

Approved by

Prof Bostwick Wyman, Ph. D., Adviser

Prof Robert Brown, Ph. D.

Prof Richard Evans, FSA

Adviser

Department of Mathematics

$\mathrm{eggc}$ Copyright by

George Argesanu

2004

ABSTRACT

Variable annuities are in the spotlight in today's insurance market. The tax de-

ferral feature and the absence of the investment risk for the insurer (while keeping

the possibility of investment benefits) boosted their popularity. They represent the

sensible way found by the insurance industry to compete with other stock market

and Iinancial intermediaries. A variable annuity is an investment wrapped with a life

insurance contract. An insurer who sells variable annuities bears two different types

of risk. On one hand, he deals with a Iinancial risk on the investment. On the other

hand there exists an actuarial (mortality) risk, given by the lifetime of the insured.

Should the insured die, the insurer has to pay a possible claim, depending on the op-

tions elected (return of premium, reset, ratchet, roll-up). In the Black-Scholes model,

the share price is a continuous function of time. Some rare events (which are rather

frequent lately), can accompany jumps in the share price. In this case the market

model is incomplete and hence there is no perfect hedging of options. I considered a

simple market model with one riskless asset and one risky asset, whose price jumps

in different proportions at some random times which correspond to the jump times

of a Poisson process. Between the jumps the risky asset follows the Black-Scholes

model. The mathematical model consists of a probability space, a Brownian motion

and a Poisson process. The jumps are independent and identically distributed. The

approach consists of defining a notion of risk and choosing a price and a hedge in

ii

order to minimize the risk. In the dual market (insurance and financial) the risk-

minimizing strategies defined by Follmer and Sondermann and the work of Moller

with equity-linked insurance products are reviewed and used for variable annuities,

with death or living benefits.

The theory of incomplete markets is complex and intriguing. There are many in-

teresting connections between such models and game theory, while the newest and

maybe the most powerful research tool comes from economics, the utility function

(tastes and preferences).

iii

This is dedicated to my family.

iv

ACKNOWLEDGMENTS

First of all I want to thank my advisor, Professor Bostwick Wyman, for his con-

stant help and guidance. His support and encouragement came always at the right

time.

Secondly, I want to thank Professor Boris Mityagin for his introductory classes to

the Iield of Iinancial mathematics. I have beneficiated greatly from the experience in

his classes. I also want to thank Professor Richard Evans for his kindness in opening

the door of the actuarial world and in sharing with us real insurance problems and

Professor Robert Brown for the intriguing experience in his probability class and for

accepting to be on this Committee.

I am also very grateful for the experience I had related to the annuities market

while working at Nationwide during the summer of 2002 and 2003.

And Iinally I want to thank my wife for her constant love, support, patience and

understanding and for all the help she's given me through the years. I also want

to thank my parents and generally all my family for all their efforts in bringing me

where I am today.

$\mathrm{v}$

VITA

1975. . . . . . Born in Ramnicu Valcea, Romania

1998. . . . . . $\mathrm{B}\mathrm{.}\mathrm{S}$. University of Bucharest

2000. . . . . . $\mathrm{M}\mathrm{.}\mathrm{S}$. joint University of Bucharest and Free University of Berlin

2002. . . . . . $\mathrm{M}\mathrm{.}\mathrm{S}$. The Ohio State University

Publications

1. On the representation type of Sweedler's Hopf algebra, Mathematical Reports

$\mathrm{2}\mathrm{\left(}\mathrm{5}\mathrm{2}\mathrm{\right)}$, 263-268, 2000

2. Taft algebras are cyclic serial (with Csabo Szanto), Mathematica 44(67), 11-17,

2002

3. Risk analysis, ARCH 2004.1, Actuarial Research Clearing House, Risk Theory

section: 1-7, 2004

Fields of Study

Major Iield: Mathematics

Specialization: Financial Mathematics

vi

TABLE OF CONTENTS

Page

Abstract ii

Dedication. iv

Acknowledgments. $\mathrm{v}$

Vita vi

List of Tables ix

List of Pigures $\mathrm{x}$

Chapters:

1. Introduction 1

1.1 Introduction and General Settings 1

1.2 Discrete time Iinancial mathematics 2

1.3 Continuous time Iinancial mathematics 7

1.4 Options in the Black-Scholes model 10

2. A product space 15

2.1 Introduction to the dual risk in unit-linked insurance products 15

2.2 Product space. 16

2.2.1 Financial backgroud 16

2.2.2 Insurance-Actuarial background 16

2.3 Combining the two markets 18

2.4 Disjoint pricing techniques 21

vii

3.

Risk Analysis

23

3.1 Derivatives in incomplete markets. 23

3.1.1 Super-replication 23

3.1.2 Utility-based indifference pricing 25

3.1.3 Quadratic approaches 25

3.1.4 Quantile hedging and shortfall risk minimization 26

3.2 Description of the GMDB problem 27

3.2.1 The model 27

3.2.2 Other market models with jumps 32

3.2.3 Risk analysis 33

3.3 Game options in incomplete markets 42

3.4 Hedging insurance claims in incomplete markets 45

3.5 The combined model in the GMDB case 47

3.5.1 GMDB with return of premium 52

3.5.2 GMDB with return of premium with interest 53

3.5.3 GMDB with ratchet 53

3.6 Living benefits 57

3.6.1 VAGLB with return of premium 60

3.6.2 VAGLB with return of premium with interest 61

3.7 Discrete time analysis 62

3.7.1 Risk comparison 66

3.8 Multiple decrements for variable annuities 77

4. Results and concluding remarks 80

Appendices:

A. Kunita-Watanabe decomposition 82

Bibliography 84

viii

LIST OF TABLES

Table Page

3.1 Pricing formulas for living benefits contracts 76

ix

LIST OF FIGURES

Figure Page

3.1 Random walk for stock price 34

3.2 Reflection principle for random walks 36

3.3 Sample stock price evolution 55

3.4 Binomial stock price process 67

3.5 Risk-neutral probabilities. 69

3.6 Call Option Price Process 71

3.7 Hedging Process. 72

3.8 Risk-minimizing trading strategy, $\mathit{\mu }\mathrm{=}\mathrm{1}$ 74

3.9 Risk-minimizing trading strategy, $\mathit{\mu }\mathrm{=}\mathrm{0}\mathrm{.}\mathrm{5}$. 75

$\mathrm{x}$

CHAPTER 1

INTRODUCTION

1.1 Introduction and General Settings

The approach throughout this thesis is based on the concept of arbitrage. It is a

remarkably simple concept and it is independent of preferences of the actors in the

Iinancial market.

The basic assumption is that everybody prefers more to less and that any increase in

consumption opportunities must somehow be paid for.

The core background for our exposition is the risk-neutral (probabilistic) pricing of

derivatives securities. A derivative (or contingent claim) is a Iinancial contract whose

value at expiration date $\mathit{T}$ (or expiry) is determined by the price of an underlying

Iinancial asset at time $\mathit{T}$. In this chapter we discuss the basics for pricing contin-

gent claims. The general assumption of this chapter is that we are in the classic

Black-Scholes model, which means that, according to the fundamental theorem of

asset pricing, the price of any contingent claim can be calculated as the discounted

expectation of the corresponding payoff with respect to the equivalent martingale

measure.

1.2 Discrete time financial mathematics

In this section we will consider a discrete-time model.

We consider a Iinite probability space $\mathrm{(}\mathrm{\Omega }\mathrm{,}\mathrm{}{F}\mathrm{,}\mathrm{}\mathit{P}\mathrm{)}$, with $\mathrm{\left|}\mathrm{\Omega }\mathrm{\right|}$ a Iinite number, and

for any $\mathit{\omega }$ $\mathrm{\in }\mathrm{\Omega }$ $\mathit{P}\mathrm{\left(}\mathrm{\right\{}\mathit{\omega }\mathrm{\left\}}\mathrm{\right)}\mathrm{>}\mathrm{0}$. We have a time horizon $\mathrm{T}$, which is the terminal

date for all economic activities considered. We use a Iiltration $\mathrm{F}$ of $\mathit{\sigma }$ -algebras

${{F}}_{\mathrm{0}}\mathrm{\subset }{{F}}_{\mathrm{1}}\mathrm{\subset }\mathrm{\text{...}}\mathrm{\subset }{{F}}_{\mathit{T}}$ and we take ${{F}}_{\mathrm{0}}\mathrm{=}\mathrm{\{}\mathrm{\varnothing }\mathrm{,}\mathrm{}\mathrm{\Omega }\mathrm{\}}$ and ${{F}}_{\mathit{T}}\mathrm{=}{F}\mathrm{=}\mathit{P}\mathrm{\left(}\mathrm{\Omega }\mathrm{\right)}$ the power

set of $\mathrm{\Omega }$. This Iinancial market contains $\mathit{d}\mathrm{+}\mathrm{1}$ Iinancial assets. One is a risk-free

asset (a bond or a bank account for example) labeled 0 and $\mathit{d}$ are risky assets (stocks)

labeled 1 to $\mathit{d}$. The prices of these assets at time $\mathit{t}\mathrm{:}{\mathit{S}}_{\mathrm{0}}\mathrm{(}\mathit{t}\mathrm{,}\mathrm{}\mathit{\omega }\mathrm{)}$, ${\mathit{S}}_{\mathrm{1}}\mathrm{(}\mathit{t}\mathrm{,}\mathrm{}\mathit{\omega }\mathrm{)}$, $\mathrm{\text{...}}$, ${\mathit{S}}_{\mathit{d}}\mathrm{(}\mathit{t}\mathrm{,}\mathrm{}\mathit{\omega }\mathrm{)}$

are non-negative and ${{F}}_{\mathit{t}}$ -measurable. Let $\mathit{S}$ $\mathrm{\left(}\mathit{t}\mathrm{\right)}\mathrm{=}\mathrm{\left(}{\mathit{S}}_{\mathrm{0}}\mathrm{\right(}\mathit{t}\mathrm{)}\mathrm{,}\mathrm{}\mathrm{\text{...}}\mathrm{,}\mathrm{}{\mathit{S}}_{\mathit{d}}\mathrm{(}\mathit{t}\mathrm{\left)}\mathrm{\right)}$ denote the

vector of prices at time $\mathit{t}$. We assume ${\mathit{S}}_{\mathrm{0}}\mathrm{\left(}\mathit{t}\mathrm{\right)}$ is strictly positive for all $\mathit{t}\mathrm{\in }\mathrm{\{}\mathrm{0}\mathrm{,}\mathrm{1}\mathrm{,}\mathrm{}\mathrm{\text{...}}\mathrm{,}\mathrm{}\mathit{T}\mathrm{\}}$

and also assume that ${\mathit{S}}_{\mathrm{0}}\mathrm{\left(}\mathrm{0}\mathrm{\right)}\mathrm{=}\mathrm{1}$. We define $\mathit{\beta }\mathrm{\left(}\mathit{t}\mathrm{\right)}\mathrm{=}\frac{\mathrm{1}}{{\mathit{S}}_{\mathrm{0}}\mathrm{\left(}\mathit{t}\mathrm{\right)}}$ as a discount factor.

We have then constructed a market model $\mathrm{\Lambda }\mathrm{4}$ consisting of a probability space

$\mathrm{(}\mathrm{\Omega }\mathrm{,}\mathrm{}{F}\mathrm{,}\mathrm{}\mathit{P}\mathrm{)}$, a set of trading dates, a price process $\mathit{S}$, and the information structure $\mathit{F}$.

Definition 1.2.1 A trading strategy (or dynamic portfolio) $\mathit{\phi }$ is a ${\mathit{R}}^{\mathit{d}\mathrm{+}\mathrm{1}}$ vector

stochastic process $\mathit{\phi }\mathrm{=}$ $\mathrm{\left(}{\mathit{\phi }}_{\mathrm{0}}\mathrm{\right(}\mathit{t}\mathrm{,}\mathrm{}\mathit{\omega }\mathrm{)}$, ${\mathit{\phi }}_{\mathrm{1}}\mathrm{(}\mathit{t}\mathrm{,}\mathrm{}\mathit{\omega }\mathrm{)}$, $\mathrm{\text{...}}$, ${\mathit{\phi }}_{\mathit{d}}\mathrm{(}\mathit{t}\mathrm{,}\mathrm{}\mathit{\omega }\mathrm{)}{\mathrm{)}}_{\mathit{t}\mathrm{=}\mathrm{1}}^{\mathit{T}}$ which is predictable: each

${\mathit{\phi }}_{\mathit{i}}\mathrm{\left(}\mathit{t}\mathrm{\right)}$ is ${{F}}_{\mathit{t}\mathrm{-}\mathrm{1}}$ -measurable for $\mathit{t}\underset{\mathrm{‾}}{\mathrm{>}}\mathrm{1}$, where ${\mathit{\phi }}_{\mathit{i}}\mathrm{\left(}\mathit{t}\mathrm{\right)}$ denotes the number of shares of

asset $\mathit{i}$ held in the portfolio at time $\mathit{t}$ and which is to be determined on the basis of

information available before time $\mathit{t}$ (predictability).

Definition 1.2.2 The value of the portfolio at time $\mathit{t}$ is the scalar product

${\mathit{V}}_{\mathit{\phi }}\mathrm{\left(}\mathit{t}\mathrm{\right)}\mathrm{=}\mathit{\phi }\mathrm{\left(}\mathit{t}\mathrm{\right)}\mathrm{ċ}\mathit{S}\mathrm{\left(}\mathit{t}\mathrm{\right)}\mathrm{=}{\displaystyle \sum }_{\mathit{i}\mathrm{=}\mathrm{0}}^{\mathit{d}}{\mathit{\phi }}_{\mathit{i}}\mathrm{\left(}\mathit{t}\mathrm{\right)}{\mathit{S}}_{\mathit{i}}\mathrm{\left(}\mathit{t}\mathrm{\right)}$ $\mathit{t}\mathrm{=}\mathrm{1}$, 2, $\mathrm{\text{...}}$, $\mathit{T}$

and ${\mathit{V}}_{\mathit{\phi }}\mathrm{\left(}\mathrm{O}\mathrm{\right)}\mathrm{=}\mathit{\phi }\mathrm{\left(}\mathrm{1}\mathrm{\right)}\mathit{S}\mathrm{\left(}\mathrm{0}\mathrm{\right)}$

2

The process ${\mathit{V}}_{\mathit{\phi }}\mathrm{(}\mathit{t}\mathrm{,}\mathrm{}\mathit{\omega }\mathrm{)}$ is called the wealth or value process of the trading strategy

$\mathit{\phi }$. We call ${\mathit{V}}_{\mathit{\phi }}\mathrm{\left(}\mathrm{0}\mathrm{\right)}$ the initial investment of the investor (endowment).

Definition 1.2.3 The gains process ${\mathit{G}}_{\mathit{\phi }}$ of a trading strategy $\mathit{\phi }$ is given by:

${\mathit{G}}_{\mathit{\phi }}\mathrm{\left(}\mathit{t}\mathrm{\right)}\mathrm{=}{\displaystyle \sum }_{\mathit{x}\mathrm{=}\mathrm{1}}^{\mathit{t}}\mathit{\phi }\mathrm{\left(}\mathit{x}\mathrm{\right)}\mathrm{\left[}\mathit{S}\mathrm{\right(}\mathit{x}\mathrm{)}\mathrm{-}\mathit{S}\mathrm{(}\mathit{x}\mathrm{-}\mathrm{1}\mathrm{\left)}\mathrm{\right]}$, $\mathit{t}\mathrm{=}\mathrm{1}$, 2, $\mathrm{\text{...}}$, $\mathit{T}$

If we define $\mathit{S}\mathrm{-}\mathrm{\left(}\mathit{t}\mathrm{\right)}\mathrm{=}\mathrm{(}\mathrm{1}\mathrm{,}$ $\mathit{\beta }\mathrm{\left(}\mathit{t}\mathrm{\right)}{\mathit{S}}_{\mathrm{1}}\mathrm{\left(}\mathit{t}\mathrm{\right)}$, $\mathrm{\text{...}}$, $\mathit{\beta }\mathrm{\left(}\mathit{t}\mathrm{\right)}{\mathit{S}}_{\mathit{d}}\mathrm{\left(}\mathit{t}\mathrm{\right)}\mathrm{)}$ the vector of discounted prices

we also have the discounted value process ${\overline{\mathit{V}}}_{\mathit{\phi }}\mathrm{\left(}\mathit{t}\mathrm{\right)}\mathrm{=}\mathit{\phi }\mathrm{\left(}\mathit{t}\mathrm{\right)}\overline{\mathit{S}}\mathrm{\left(}\mathit{t}\mathrm{\right)}$ for $\mathit{t}\mathrm{=}\mathrm{1}$, 2, $\mathrm{\text{...}}$, $\mathit{T}$ and

we can see that the discounted gains process ${\overline{\mathit{G}}}_{\mathit{\phi }}\mathrm{\left(}\mathit{t}\mathrm{\right)}\mathrm{=}{{\displaystyle \sum }}_{\mathit{x}\mathrm{=}\mathrm{1}}^{\mathit{t}}\mathit{\phi }\mathrm{\left(}\mathit{x}\mathrm{\right)}\mathrm{\left[}\overline{\mathit{S}}\mathrm{\right(}\mathit{x}\mathrm{)}\mathrm{-}\overline{\mathit{S}}\mathrm{(}\mathit{x}\mathrm{-}\mathrm{1}\mathrm{\left)}\mathrm{\right]}$

reflects the gains from trading with assets 1 to $\mathit{d}$ only.

Definition 1.2.4 The strategy $\mathit{\phi }$ is self-fifinancing, $\mathit{\phi }\mathrm{\in }\mathrm{\Phi }$, if

$\mathit{\phi }\mathrm{\left(}\mathit{t}\mathrm{\right)}\mathit{S}\mathrm{\left(}\mathit{t}\mathrm{\right)}\mathrm{=}\mathit{\phi }\mathrm{(}\mathit{t}\mathrm{+}\mathrm{1}\mathrm{)}\mathit{S}\mathrm{\left(}\mathit{t}\mathrm{\right)}$, $\mathit{t}\mathrm{=}\mathrm{1}$, 2, $\mathrm{\text{...}}$, $\mathit{T}\mathrm{-}\mathrm{1}$

This means that when new prices $\mathit{S}\mathrm{\left(}\mathit{t}\mathrm{\right)}$ are quoted at time $\mathit{t}$ the investor adjusts

his portfolio from $\mathit{\phi }\mathrm{\left(}\mathit{t}\mathrm{\right)}$ to $\mathit{\phi }\mathrm{(}\mathit{t}\mathrm{+}\mathrm{1}\mathrm{)}$, without bringing in or consuming any wealth.

To prove the fundamental theorem of asset pricing we need the following results,

which are also interested and important by themselves:

Proposition 1.2.1 A trading strategy $\mathit{\phi }$ is self Iinancing with respect to $\mathit{S}\mathrm{\left(}\mathit{t}\mathrm{\right)}$ if

and only if $\mathit{\phi }$ is self-Iinancing with respect to $\tilde{\mathit{S}}\mathrm{\left(}\mathit{t}\mathrm{\right)}$.

Proposition 1.2.2 A trading strategy $\mathit{\phi }$ is self Iinancing if and only if

$\tilde{\mathit{V}}\mathrm{\left(}\mathit{t}\mathrm{\right)}\mathrm{=}{\mathit{V}}_{\mathit{\phi }}\mathrm{\left(}\mathrm{0}\mathrm{\right)}\mathrm{+}{\tilde{\mathit{G}}}_{\mathit{\phi }}\mathrm{\left(}\mathit{t}\mathrm{\right)}$

The well-being of any market is given by the absence of arbitrage opportunities

arbitrage $\mathrm{=}\mathrm{f}\mathrm{r}\mathrm{e}\mathrm{e}$ lunch).

Definition 1.2.5 Let ${\mathrm{\Phi }}_{\mathrm{0}}\mathrm{\subset }\mathrm{\Phi }$ be a set of self-Iinancing strategies. A strategy

$\mathit{\phi }\mathrm{\in }{\mathrm{\Phi }}_{\mathrm{0}}$ is called an arbitrage opportunity or arbitrage strategy with respect to ${\mathrm{\Phi }}_{\mathrm{0}}$ if

3

$\mathit{P}\mathrm{\left\{}{\mathit{V}}_{\mathit{\phi }}\mathrm{\right(}\mathrm{0}\mathrm{)}\mathrm{=}\mathrm{0}\mathrm{\}}\mathrm{=}\mathrm{1}$ and the terminal wealth satisfies

$\mathit{P}\mathrm{\left\{}{\mathit{V}}_{\mathit{\phi }}\mathrm{\right(}\mathit{T}\mathrm{\left)}\underset{\mathrm{‾}}{\mathrm{>}}\mathrm{0}\mathrm{\right\}}\mathrm{=}\mathrm{1}$ and $\mathit{P}\mathrm{\left\{}{\mathit{V}}_{\mathit{\phi }}\mathrm{\right(}\mathit{T}\mathrm{)}\mathrm{>}\mathrm{0}\mathrm{\}}\mathrm{>}\mathrm{0}$

We say that a security market $\mathrm{\Lambda }\mathrm{4}$ is arbitrage-free if there are no arbitrage oppor-

tunities in the class $\mathrm{\Phi }$.

Next we introduce the notion of "risk-neutral probability" which also also central

in Iinancial mathematics:

Definition 1.2.6 A probability measure ${\mathit{P}}^{\mathrm{*}}$ on $\mathrm{(}\mathrm{\Omega }\mathrm{,}\mathrm{}{{F}}_{\mathit{T}}\mathrm{)}$ equivalent to $\mathit{P}$ is called

a martingale measure for $\overline{\mathit{S}}$ if the process $\overline{\mathit{S}}$ follows a ${\mathit{P}}^{\mathrm{*}}$ -martingale with respect to

the Iiltration F. We denote ${\mathrm{7}}^{\mathrm{\supset }}\mathrm{\left(}\overline{\mathit{S}}\mathrm{\right)}$ the class of equivalent martingale measures.

One proposition that follows quickly and is useful in proving Theorem 1.2.1 is:

Proposition 1.2.3 Let ${\mathit{P}}^{\mathrm{*}}\mathrm{\in }\mathit{P}$ $\mathrm{\left(}\overline{\mathit{S}}\mathrm{\right)}$ and $\mathit{\phi }$ a self-Iinancing strategy. Then the

wealth process $\overline{\mathit{V}}\mathrm{\left(}\mathit{t}\mathrm{\right)}$ is a ${\mathit{P}}^{\mathrm{*}}$ martingale with respect to the Iiltration P.

The no-arbitrage theorem describes the necessary and sufficient conditions for

no-arbitrage and makes the connection between the real world (finacial market) and

theory (martingales):

Theorem 1.2.1 (No-Arbitrage Theorem) The market $\mathrm{\Lambda }\mathrm{4}$ is arbitrage-free if

and only if there exists a probability measure ${\mathit{P}}^{\mathrm{*}}$ equivalent to $\mathit{P}$ under which the

discounted $\mathit{d}$ -dimensional asset price process $\overline{\mathit{S}}$ is a ${\mathit{P}}^{\mathrm{*}}$ -martingale.

The question now is how we use this theorem to price contingent claims. We start

with a definition:

Definition 1.2.7 A contingent claim $\mathit{X}$ with maturity date $\mathit{T}$ is an arbitrary

non-negative ${{F}}_{\mathit{T}}$ -measurable random variable.

4

We say that the claim is attainable if there exists a replicating strategy $\mathit{\phi }\mathrm{\in }\mathrm{\Phi }$

such that

${\mathit{V}}_{\mathit{\phi }}\mathrm{\left(}\mathit{T}\mathrm{\right)}\mathrm{=}\mathit{X}$

The following theorem is the Iirst theoretical approach to pricing contingent

claims:

Theorem 1.2.2 The arbitrage price process ${\mathit{\pi }}_{\mathit{X}}\mathrm{\left(}\mathit{t}\mathrm{\right)}$ of any attainable contingent

claim $\mathit{X}$ is given by the risk-neutral valuation formula:

${\mathit{\pi }}_{\mathit{X}}\mathrm{\left(}\mathit{t}\mathrm{\right)}\mathrm{=}\mathit{\beta }\mathrm{\left(}\mathit{t}{\mathrm{)}}^{\mathrm{-}\mathrm{1}}{\mathit{E}}^{\mathrm{*}}\mathrm{\right(}\mathit{X}\mathit{\beta }\mathrm{\left(}\mathit{T}\mathrm{\right)}\mathrm{\left|}{{F}}_{\mathit{t}}\mathrm{\right)}$ for any $\mathit{t}\mathrm{=}\mathrm{0}\mathrm{,}\mathrm{1}$, $\mathrm{\text{...}}$, $\mathit{T}$

where ${\mathit{E}}^{\mathrm{*}}$ is the expectation taken with respect to an equivalent martingale measure

${\mathit{P}}^{\mathrm{*}}$

Theorem 1.2.2 says that any attainable contingent claim can be priced using the

equivalent martingale measure. So, clearly, "attainability" would be a very desirable

property of any market. So the next definition follows naturally:

Definition 1.2.8 The market $\mathrm{\Lambda }\mathrm{4}$ is complete if every contingent claim is attain-

able.

The following theorem gives a nice characterization of a complete market:

Theorem 1.2.3 (Completeness Theorem) An arbitrage-free market $\mathrm{\Lambda }\mathrm{4}$ is com-

plete if and only if there exists a unique probability measure ${\mathit{P}}^{\mathrm{*}}$ equivalent to $\mathit{P}$ under

which discounted asset prices are martingales.

Let's summarize what we have seen so far.

Theorem 1.2.1 tells us that if the market is arbitrage-free, equivalent martingale

measures ${\mathit{P}}^{\mathrm{*}}$ exist. Theorem 1.2.3 tells us that if the market is complete, equivalent

martingale measures are unique. Putting them together we get:

5

Theorem 1.2.4 (Fundamental Theorem of Asset Pricing)

In an arbitrary-free

complete market $\mathrm{\Lambda }\mathrm{4}$, there exists a unique equivalent martingale measure ${\mathit{P}}^{\mathrm{*}}$

Finally, Theorem 1.2.2 gives us in the complet market setting the following:

Theorem 1.2.5 (Risk-Neutral Pricing Formula) In an arbitrage-free complete

market $\mathrm{\Lambda }\mathrm{4}$, arbitrage prices of contingent claims are their discounted expected values

under the risk-neutral (equivalent martingale) measure ${\mathit{P}}^{\mathrm{*}}$

6

1.3 Continuous time financial mathematics

We start with a general model of a frictionless securities market where investors are

allowed to trade continuously up to some Iixed Iinite planning horizon $\mathit{T}$. Uncertainty

in this Iinancial market is modeled by a probability space $\mathrm{(}\mathrm{\Omega }\mathrm{,}\mathrm{}{F}\mathrm{,}\mathrm{}\mathit{P}\mathrm{)}$ and a Iiltration $\mathit{F}$

of $\mathit{\sigma }$ -algebras ${{F}}_{\mathit{t}}$, with $\mathrm{0}\underset{\mathrm{‾}}{\mathrm{<}}\mathit{t}\underset{\mathrm{‾}}{\mathrm{<}}\mathit{T}$, satisfying the usual conditions of right-continuity

and completeness. We assume that ${{F}}_{\mathrm{0}}$ is trivial and that ${{F}}_{\mathit{T}}\mathrm{=}{F}$.

There are $\mathrm{d}\mathrm{+}\mathrm{1}$ primary traded assets, whose price processes are given by stochas-

tic processes ${\mathit{S}}_{\mathrm{0}}$, $\mathrm{\text{...}}$, ${\mathit{S}}_{\mathit{d}}$. We assume that $\mathit{S}\mathrm{=}$ $\mathrm{(}{\mathit{S}}_{\mathrm{0}}\mathrm{,}\mathrm{}\mathrm{\text{...}}\mathrm{,}\mathrm{}{\mathit{S}}_{\mathit{d}}\mathrm{)}$ follows an adapted, right-

continuous with left-limits (RCLL) and strictly positive semimartingale on $\mathrm{(}\mathrm{\Omega }\mathrm{,}\mathrm{}{F}\mathrm{,}\mathrm{}\mathit{P}\mathrm{,}\mathrm{}\mathit{F}\mathrm{)}$.

We also assume that ${\mathit{S}}_{\mathrm{0}}\mathrm{\left(}\mathit{t}\mathrm{\right)}$ is a non-dividend paying asset which is almost surely

strictly positive and use it as a numeraire.

We denote by $\mathrm{M}\mathrm{\left(}\mathrm{P}\mathrm{\right)}$ the Iinancial market described above.

Definition 1.3.1 A trading strategy (or dynamic portfolio) $\mathit{\phi }$ is a ${\mathit{R}}^{\mathit{d}\mathrm{+}\mathrm{1}}$ vector

stochastic process $\mathit{\phi }\mathrm{\left(}\mathit{t}\mathrm{\right)}\mathrm{=}\mathrm{\left(}{\mathit{\phi }}_{\mathrm{0}}\mathrm{\right(}\mathit{t}\mathrm{)}\mathrm{,}\mathrm{}{\mathit{\phi }}_{\mathrm{1}}\mathrm{(}\mathit{t}\mathrm{)}$, $\mathrm{\text{...}}$, ${\mathit{\phi }}_{\mathit{d}}\mathrm{\left(}\mathit{t}\mathrm{\right)}\mathrm{)}$, $\mathrm{0}\underset{\mathrm{‾}}{\mathrm{<}}\mathit{t}\underset{\mathrm{‾}}{\mathrm{<}}\mathit{T}$ which is predictable

and locally bounded.

Here ${\mathit{\phi }}_{\mathit{i}}\mathrm{\left(}\mathit{t}\mathrm{\right)}$ denotes the number of shares of asset $\mathrm{i}$ held in the portfolio at time t-

determined on the basis of information available before time $\mathrm{t}$. This means that the

investor selects his time $\mathrm{t}$ portfolio after observing the prices $\mathit{S}\mathrm{(}\mathit{t}$-$\mathrm{)}$.

Definition 1.3.2 The value of the portfolio $\mathit{\phi }$ at time $\mathit{t}$ is given by the scalar

product

${\mathit{V}}_{\mathit{\phi }}\mathrm{\left(}\mathit{t}\mathrm{\right)}\mathrm{=}\mathit{\phi }\mathrm{\left(}\mathit{t}\mathrm{\right)}\mathrm{ċ}\mathit{S}\mathrm{\left(}\mathit{t}\mathrm{\right)}\mathrm{=}{\displaystyle \sum }_{\mathit{i}\mathrm{=}\mathrm{0}}^{\mathit{d}}\mathit{\phi }\mathrm{\left(}\mathit{t}\mathrm{\right)}{\mathit{S}}_{\mathit{i}}\mathrm{\left(}\mathit{t}\mathrm{\right)}$, $\mathit{t}\mathrm{\in }\mathrm{[}\mathrm{0}\mathrm{,}\mathrm{}\mathit{T}\mathrm{]}$

Definition 1.3.3 The gains process ${\mathit{G}}_{\mathit{\phi }}\mathrm{\left(}\mathit{t}\mathrm{\right)}$ is defined by

7

${\mathit{G}}_{\mathit{\phi }}\mathrm{\left(}\mathit{t}\mathrm{\right)}\mathrm{=}{{\displaystyle \int }}_{\mathrm{0}}^{\mathit{t}}\mathit{\phi }\mathrm{\left(}\mathit{u}\mathrm{\right)}\mathit{d}\mathit{S}\mathrm{\left(}\mathit{u}\mathrm{\right)}\mathrm{=}{\displaystyle \sum }_{\mathit{i}\mathrm{=}\mathrm{0}}^{\mathit{d}}{{\displaystyle \int }}_{\mathrm{0}}^{\mathit{t}}{\mathit{\phi }}_{\mathit{i}}\mathrm{\left(}\mathit{u}\mathrm{\right)}\mathit{d}{\mathit{S}}_{\mathit{i}}\mathrm{\left(}\mathit{u}\mathrm{\right)}$

Definition 1.3.4 A trading strategy is called self-fifinancing if the value process

satisfies

${\mathit{V}}_{\mathit{\phi }}\mathrm{\left(}\mathit{t}\mathrm{\right)}\mathrm{=}{\mathit{V}}_{\mathit{\phi }}\mathrm{\left(}\mathrm{0}\mathrm{\right)}\mathrm{+}{\mathit{G}}_{\mathit{\phi }}\mathrm{\left(}\mathit{t}\mathrm{\right)}$ for all $\mathit{t}\mathrm{\in }\mathrm{[}\mathrm{0}\mathrm{,}\mathrm{}\mathit{T}\mathrm{]}$

We can define the discounted price process, the discounted value process and

discounted gains process with the help of the numeraire ${\mathit{S}}_{\mathrm{0}}\mathrm{\left(}\mathit{t}\mathrm{\right)}$.

Definition 1.3.5 A self-Iinancing strategy is called an arbitrage opportunity or

arbitrage strategy if ${\mathit{V}}_{\mathit{\phi }}\mathrm{\left(}\mathrm{0}\mathrm{\right)}\mathrm{=}\mathrm{0}$ and the terminal value satisfies

$\mathit{P}\mathrm{\left\{}{\mathit{V}}_{\mathit{\phi }}\mathrm{\right(}\mathit{T}\mathrm{\left)}\underset{\mathrm{‾}}{\mathrm{>}}\mathrm{0}\mathrm{\right\}}\mathrm{=}\mathrm{1}$ and $\mathit{P}\mathrm{\left\{}{\mathit{V}}_{\mathit{\phi }}\mathrm{\right(}\mathit{T}\mathrm{)}\mathrm{>}\mathrm{0}\mathrm{\}}\mathrm{>}\mathrm{0}$

Definition 1.3.6 We say that a probability measure $\mathit{Q}$ defined on $\mathrm{(}\mathrm{\Omega }\mathrm{,}\mathrm{}{F}\mathrm{)}$ is a

(strong) equivalent martingale measure if $\mathit{Q}$ is equivalent to $\mathit{P}$ and the discounted

process $\overline{\mathit{S}}$ is a $\mathit{Q}$-local martingale (martingale).

We denote the set of martingale measures by ${\mathrm{7}}^{\mathrm{\supset }}$

Definition 1.3.7 A self Iinancing strategy is called tame if ${\overline{\mathit{V}}}_{\mathit{\phi }}\mathrm{\left(}\mathit{t}\mathrm{\right)}\underset{\mathrm{‾}}{\mathrm{>}}\mathrm{0}$ for all

$\mathit{t}\mathrm{\in }\mathrm{[}\mathrm{0}\mathrm{,}\mathrm{}\mathit{T}\mathrm{]}$. We denote by $\mathrm{\Phi }$ the set of tame trading strategies.

The following proposition assures us that the existence of an equivalent martingale

measure implies the absence of arbitrage.

Proposition 1.3.1 Assume ${\mathrm{7}}^{\mathrm{\supset }}$ is not empty. Then the market model contains no

arbitrage opportunities in $\mathrm{\Phi }$.

8

In order to get equivalence between the absence of arbitrage opportunities and

the existence of an equivalent martingale measure we need some further definitions

and requirements.

Definition 1.3.8 A simple predictable strategy is a predictable process which

can be represented as a Iinite linear combination of stochastic processes of the form

$\mathit{\psi }{\mathrm{1}}_{\mathrm{[}{\mathit{\tau }}_{\mathrm{1}}\mathrm{,}{\mathit{\tau }}_{\mathrm{2}}\mathrm{]}}$ where ${\mathit{\tau }}_{\mathrm{1}}$ and ${\mathit{\tau }}_{\mathrm{2}}$ are stopping times and $\mathit{\psi }$ is an ${{F}}_{{\mathit{\tau }}_{\mathrm{1}}}$-measurable random

variable.

Definition 1.3.9 We say that a simple predictable trading strategy is $\mathit{\delta }$-admissible

if ${\mathit{V}}_{\mathit{\phi }}\mathrm{\left(}\mathit{t}\mathrm{\right)}\underset{\mathrm{‾}}{\mathrm{>}}\mathrm{-}\mathit{\delta }$ for every $\mathit{t}\mathrm{\in }\mathrm{[}\mathrm{0}\mathrm{,}\mathrm{}\mathit{T}\mathrm{]}$.

Definition 1.3.10 A price process $\mathit{S}$ satisfies NFLVR (no free lunch with van-

ishing risk) if for any sequence $\mathrm{\left(}{\mathit{\phi }}_{\mathit{n}}\mathrm{\right)}$ of simple trading strategies such that ${\mathit{\phi }}_{\mathit{n}}$ is

${\mathit{\delta }}_{\mathit{n}}$-admissible and the sequence ${\mathit{\delta }}_{\mathit{n}}$ tends to zero, we have ${\mathit{V}}_{{\mathit{\phi }}_{\mathit{n}}}\mathrm{\left(}\mathit{T}\mathrm{\right)}\mathrm{\to }\mathrm{0}$ in probability

as $\mathit{n}\mathrm{\to }\mathrm{\infty }$.

The following fundamental theorem of asset pricing is proved in [7]:

Theorem 1.3.1 (Fundamental Theorem of Asset Pricing-continuous time) There

exists an equivalent martingale measure for the Iinancial market model $\mathrm{M}\mathrm{\left(}\mathrm{P}\mathrm{\right)}$ if and

only if the condition NFLVR holds true.

For all the proofs of the above theorems refer to [3].

9

1.4 Options in the Black-Scholes model

Let us assume that we have a market with two assets, a riskless (B) and a risky

one (S). The riskless asset (bond or savings account) is modelled by the following

ordinary differential equation

$\mathit{d}{\mathit{B}}_{\mathit{t}}\mathrm{=}\mathit{r}{\mathit{B}}_{\mathit{t}}\mathit{d}\mathit{t}$

where $\mathit{r}\underset{\mathrm{‾}}{\mathrm{>}}\mathrm{0}$ is an instantaneous interest rate (difffferent from the rate in the discrete

models). Without loss of generality, we set ${\mathit{B}}_{\mathrm{0}}\mathrm{=}\mathrm{1}$ and so ${\mathit{B}}_{\mathit{t}}\mathrm{=}{\mathit{e}}^{\mathit{r}\mathit{t}}$ for $\mathit{t}\underset{\mathrm{‾}}{\mathrm{>}}\mathrm{0}$. The risky

asset is a stock (or stock index) whose price is modelled by the following stochastic

differential equation:

$\mathit{d}{\mathit{S}}_{\mathit{t}}\mathrm{=}{\mathit{S}}_{\mathit{t}}\mathrm{(}\mathit{\mu }\mathit{d}\mathit{t}\mathrm{+}\mathit{\sigma }\mathit{d}{\mathit{Z}}_{\mathit{t}}\mathrm{)}$,

where $\mathit{\mu }$ and $\mathit{\sigma }$ are constants and ${\mathit{Z}}_{\mathit{t}}$ is a standard Browninan motion. The model

is described on a probability space $\mathrm{(}\mathrm{\Omega }\mathrm{,}\mathrm{}{F}\mathrm{,}\mathrm{}\mathit{P}\mathrm{)}$ equipped with a Iiltration $\mathrm{F}$ of $\mathit{\sigma }$ -

algebras ${{F}}_{\mathrm{0}}\mathrm{\subset }{{F}}_{\mathrm{1}}\mathrm{\subset }$. . . $\mathrm{\subset }{{F}}_{\mathit{T}}$. We take ${{F}}_{\mathit{t}}\mathrm{=}\mathit{\sigma }\mathrm{\{}{\mathit{S}}_{\mathit{u}}\mathrm{,}\mathrm{}\mathit{u}\underset{\mathrm{‾}}{\mathrm{<}}\mathit{t}\mathrm{\}}$, and so the price

process for the stock is adapted to the Iiltration. We want to prove that there exists

a probability measure ${\mathit{P}}^{\mathrm{\star }}$ equivalent to $\mathit{P}$, under which the discounted share price

${\mathit{S}}_{\mathit{t}}^{\mathrm{\star }}\mathrm{=}{\mathit{e}}^{\mathrm{-}\mathit{r}\mathit{t}}{\mathit{S}}_{\mathit{t}}$ is a martingale. We have:

$\mathit{d}{\mathit{S}}_{\mathit{t}}^{\mathrm{\star }}\mathrm{=}\mathrm{-}\mathit{r}{\mathit{e}}^{\mathrm{-}\mathit{r}\mathit{t}}{\mathit{S}}_{\mathit{t}}\mathit{d}\mathit{t}\mathrm{+}{\mathit{e}}^{\mathrm{-}\mathit{r}\mathit{t}}\mathit{d}{\mathit{S}}_{\mathit{t}}\mathrm{=}{\mathit{S}}_{\mathit{t}}^{\mathrm{\star }}\mathrm{\left(}\mathrm{\right(}\mathit{\mu }\mathrm{-}\mathit{r}\mathrm{)}\mathit{d}\mathit{t}\mathrm{+}\mathit{\sigma }\mathit{d}{\mathit{Z}}_{\mathit{t}}\mathrm{)}$

and if we set ${\mathit{W}}_{\mathit{t}}\mathrm{=}{\mathit{Z}}_{\mathit{t}}\mathrm{+}\frac{\mathrm{(}\mathit{\mu }\mathrm{-}\mathit{r}\mathrm{)}\mathit{t}}{\mathit{\sigma }}$, we get

$\mathit{d}{\mathit{S}}_{\mathit{t}}^{\mathrm{\star }}\mathrm{=}{\mathit{S}}_{\mathit{t}}^{\mathrm{\star }}\mathit{\sigma }\mathit{d}{\mathit{W}}_{\mathit{t}}$.

Now, recall the Girsanov theorem:

Theorem 1.4.1. Let $\mathrm{(}{\mathit{\theta }}_{\mathit{t}}{\mathrm{)}}_{\mathrm{0}\mathrm{\le }\mathit{t}\mathrm{\le }\mathit{T}}$ be an adapted process satisfying ${{\displaystyle \int }}_{\mathrm{0}}^{\mathit{T}}{\mathit{\theta }}_{\mathit{s}}^{\mathrm{2}}\mathit{d}\mathit{s}\mathrm{<}\mathrm{\infty }\mathrm{a}\mathrm{.}\mathrm{s}$.

10

and such that the process $\mathrm{(}{\mathit{L}}_{\mathit{t}}{\mathrm{)}}_{\mathrm{0}\mathrm{\le }\mathit{t}\mathrm{\le }\mathit{T}}$ defined by

${\mathit{L}}_{\mathit{t}}\mathrm{=}\mathit{e}\mathit{x}\mathit{p}\mathrm{(}\mathrm{-}{{\displaystyle \int }}_{\mathrm{0}}^{\mathit{t}}{\mathit{\theta }}_{\mathit{s}}\mathit{d}{\mathit{Z}}_{\mathit{s}}\mathrm{-}\frac{\mathrm{1}}{\mathrm{2}}{{\displaystyle \int }}_{\mathrm{0}}^{\mathit{t}}{\mathit{\theta }}_{\mathit{s}}^{\mathrm{2}}\mathit{d}\mathit{s}\mathrm{)}$

is a martingale. Then, under the probability ${\mathit{P}}^{\mathit{L}}$ with density ${\mathit{L}}_{\mathit{T}}$ relative to $\mathit{P}$, the

process $\mathrm{(}{\mathit{W}}_{\mathit{t}}{\mathrm{)}}_{\mathrm{0}\mathrm{\le }\mathit{t}\mathrm{\le }\mathit{T}}$ defined by ${\mathit{W}}_{\mathit{t}}\mathrm{=}{\mathit{Z}}_{\mathit{t}}\mathrm{+}{{\displaystyle \int }}_{\mathrm{0}}^{\mathit{t}}{\mathit{\theta }}_{\mathit{s}}\mathit{d}\mathit{s}$ is a standard Brownian motion.

For a proof of Girsanov's theorem, see [19].

Using this theorem with ${\mathit{\theta }}_{\mathit{t}}\mathrm{=}\frac{\mathit{\mu }\mathrm{-}\mathit{r}}{\mathit{\sigma }}$ we get that there exists a probability ${\mathit{P}}^{\mathrm{\star }}$ under

which $\mathrm{(}{\mathit{W}}_{\mathit{t}}{\mathrm{)}}_{\mathrm{0}\mathrm{\le }\mathit{t}\mathrm{\le }\mathit{T}}$ is a standard Brownian motion. Under this probability ${\mathit{P}}^{\mathrm{\star }}$, the

discounted price process $\mathrm{\left(}{\mathit{S}}_{\mathit{t}}^{\mathrm{\star }}\mathrm{\right)}$ is a martingale and

${\mathit{S}}_{\mathit{t}}^{\mathrm{\star }}\mathrm{=}{\mathit{S}}_{\mathrm{0}}^{\mathrm{\star }}\mathit{e}\mathit{x}\mathit{p}\mathrm{(}\mathit{\sigma }{\mathit{W}}_{\mathit{t}}\mathrm{-}\frac{\mathrm{(}\mathit{\sigma }{\mathrm{)}}^{\mathrm{2}}\mathit{t}}{\mathrm{2}}\mathrm{)}$.

Let us consider now a standard European call option. The option is defined by a

non-negative, ${{F}}_{\mathit{T}}$-measurable random variable $\mathit{H}\mathrm{=}\mathit{f}\mathrm{\left(}{\mathit{S}}_{\mathit{T}}\mathrm{\right)}\mathrm{=}\mathrm{(}{\mathit{S}}_{\mathit{T}}\mathrm{-}\mathit{K}{\mathrm{)}}_{\mathrm{+}}$, where $\mathrm{K}$ is

the exercise price.

Theorem 1.4.2. In the Black-Scholes model, any option defined by a non-negative,

${{F}}_{\mathit{T}}$-measurable random variable $\mathit{H}$, which is square-integrable under the probability

${\mathit{P}}^{\mathrm{\star }}$, is replicable by a trading strategy and the value at time $\mathit{t}$ of any replicating

portfolio is given by:

${\mathit{V}}_{\mathit{t}}\mathrm{=}{\mathit{E}}^{\mathrm{\star }}\mathrm{\left(}{\mathit{e}}^{\mathrm{-}\mathit{r}\mathrm{(}\mathit{T}\mathrm{-}\mathit{t}\mathrm{)}}\mathit{H}\mathrm{\right|}{{F}}_{\mathit{t}}\mathrm{)}$.

Thus, the option value at time $\mathit{t}$ can be naturally defined by the expression ${\mathit{E}}^{\mathrm{\star }}\mathrm{\left(}{\mathit{e}}^{\mathrm{-}\mathit{r}\mathrm{(}\mathit{T}\mathrm{-}\mathit{t}\mathrm{)}}\mathit{H}\mathrm{\right|}{{F}}_{\mathit{t}}\mathrm{)}$.

Proof: We follow [21]. Let us assume that there is an admissible strategy $\mathrm{(}\mathit{\theta }\mathrm{,}\mathrm{}\mathit{\eta }\mathrm{)}$ repli-

cating the option. The value of the portfolio at time $\mathit{t}$ is given by

${\mathit{V}}_{\mathit{t}}\mathrm{=}{\mathit{\theta }}_{\mathit{t}}{\mathit{B}}_{\mathit{t}}\mathrm{+}{\mathit{\eta }}_{\mathit{t}}{\mathit{S}}_{\mathit{t}}$,

11

and the terminal values are equal: ${\mathit{V}}_{\mathit{T}}\mathrm{=}\mathit{H}$. Defining the discounted value ${\mathit{V}}_{\mathit{t}}^{\mathrm{\star }}\mathrm{=}{\mathit{V}}_{\mathit{t}}{\mathit{e}}^{\mathrm{-}\mathit{r}\mathit{t}}$

we get

${\mathit{V}}_{\mathit{t}}^{\mathrm{\star }}\mathrm{=}{\mathit{\theta }}_{\mathit{t}}\mathrm{+}{\mathit{\eta }}_{\mathit{t}}{\mathit{S}}_{\mathit{t}}^{\mathrm{\star }}$.

The strategy is self-Iinancing and hence

${\mathit{V}}_{\mathit{t}}^{\mathrm{\star }}\mathrm{=}{\mathit{V}}_{\mathrm{0}}\mathrm{+}{{\displaystyle \int }}_{\mathrm{0}}^{\mathit{t}}{\mathit{\eta }}_{\mathit{u}}\mathit{d}{\mathit{S}}_{\mathit{u}}^{\mathrm{\star }}\mathrm{=}{\mathit{V}}_{\mathrm{0}}\mathrm{+}{{\displaystyle \int }}_{\mathrm{0}}^{\mathit{t}}{\mathit{\eta }}_{\mathit{u}}\mathit{\sigma }{\mathit{S}}_{\mathit{u}}^{\mathrm{\star }}\mathit{d}{\mathit{W}}_{\mathit{u}}$.

It can be shown that $\mathrm{\left(}{\mathit{V}}_{\mathit{t}}^{\mathrm{\star }}\mathrm{\right)}$ is a square-integrable martingale under ${\mathit{P}}^{\mathrm{\star }}$ and hence

${\mathit{V}}_{\mathit{t}}^{\mathrm{\star }}\mathrm{=}{\mathit{E}}^{\mathrm{\star }}\mathrm{\left(}{\mathit{V}}_{\mathit{T}}^{\mathrm{\star }}\mathrm{\right|}{{F}}_{\mathit{t}}\mathrm{)}$,

and so

${\mathit{V}}_{\mathit{t}}\mathrm{=}{\mathit{E}}^{\mathrm{\star }}\mathrm{\left(}{\mathit{e}}^{\mathrm{-}\mathit{r}\mathrm{(}\mathit{T}\mathrm{-}\mathit{t}\mathrm{)}}\mathit{H}\mathrm{\right|}{{F}}_{\mathit{t}}\mathrm{)}$.

So, if a portfolio $\mathrm{(}\mathit{\theta }\mathrm{,}\mathrm{}\mathit{\eta }\mathrm{)}$ replicates the option, its value is given by the above formula.

Now it remain to show that the option is indeed replicable, i.e. there exist some

processes $\mathrm{\left(}{\mathit{\theta }}_{\mathit{t}}\mathrm{\right)}$ and $\mathrm{\left(}{\mathit{\eta }}_{\mathit{t}}\mathrm{\right)}$ such that

${\mathit{\theta }}_{\mathit{t}}{\mathit{B}}_{\mathit{t}}\mathrm{+}{\mathit{\eta }}_{\mathit{t}}{\mathit{S}}_{\mathit{t}}\mathrm{=}{\mathit{E}}^{\mathrm{\star }}\mathrm{\left(}{\mathit{e}}^{\mathrm{-}\mathit{r}\mathrm{(}\mathit{T}\mathrm{-}\mathit{t}\mathrm{)}}\mathit{H}\mathrm{\right|}{{F}}_{\mathit{t}}\mathrm{)}$.

The process ${\mathit{M}}_{\mathit{t}}\mathrm{:}\mathrm{=}{\mathit{E}}^{\mathrm{\star }}\mathrm{\left(}{\mathit{e}}^{\mathrm{-}\mathit{r}\mathit{t}}\mathit{H}\mathrm{\right|}{{F}}_{\mathit{t}}\mathrm{)}$ is a ${\mathit{P}}^{\mathrm{\star }}$-square integrable martingale.

Now, using the representation theorem for martingales we obtain that there exists an

adapted process $\mathrm{(}{\mathit{K}}_{\mathit{t}}{\mathrm{)}}_{\mathrm{0}\mathrm{\le }\mathit{t}\mathrm{\le }\mathit{T}}$ such that ${\mathit{E}}^{\mathrm{\star }}\mathrm{\left(}{{\displaystyle \int }}_{\mathrm{0}}^{\mathit{T}}{\mathit{K}}_{\mathit{s}}^{\mathrm{2}}\mathit{d}\mathit{s}\mathrm{\right)}\mathrm{<}\mathrm{\infty }$ and

${\mathit{M}}_{\mathit{t}}\mathrm{=}{\mathit{M}}_{\mathrm{0}}\mathrm{+}{{\displaystyle \int }}_{\mathrm{0}}^{\mathit{t}}{\mathit{K}}_{\mathit{s}}\mathit{d}{\mathit{W}}_{\mathit{s}}\mathit{a}\mathrm{.}\mathit{s}$.

for any $\mathit{t}\mathrm{\in }\mathrm{[}\mathrm{0}\mathrm{,}\mathrm{}\mathit{T}\mathrm{]}$.

The strategy $\mathit{\phi }$ $\mathrm{=}\mathrm{(}\mathit{\theta }\mathrm{,}\mathrm{}\mathit{\eta }\mathrm{)}$ with ${\mathit{\eta }}_{\mathit{t}}\mathrm{=}\frac{{\mathit{K}}_{\mathit{t}}}{\mathrm{\left(}\mathit{\sigma }{\mathit{S}}_{\mathit{t}}^{\mathrm{\star }}\mathrm{\right)}}$ and ${\mathit{\theta }}_{\mathit{t}}\mathrm{=}{\mathit{M}}_{\mathit{t}}\mathrm{-}{\mathit{H}}_{\mathit{t}}{\mathit{S}}_{\mathit{t}}^{\mathrm{\star }}$ is self-Iinancing and its

value at time $\mathit{t}$ is given by

${\mathit{V}}_{\mathit{t}}\mathrm{\left(}\mathit{\phi }\mathrm{\right)}\mathrm{=}{\mathit{e}}^{\mathit{r}\mathit{t}}{\mathit{M}}_{\mathit{t}}\mathrm{=}{\mathit{E}}^{\mathrm{\star }}\mathrm{\left(}{\mathit{e}}^{\mathrm{-}\mathit{r}\mathrm{(}\mathit{T}\mathrm{-}\mathit{t}\mathrm{)}}\mathit{H}\mathrm{\right|}{{F}}_{\mathit{t}}\mathrm{)}$.

12

In our case (European call), the random variable $\mathit{H}$ can be written as $\mathit{H}\mathrm{=}\mathit{f}\mathrm{\left(}{\mathit{S}}_{\mathit{T}}\mathrm{\right)}\mathrm{=}$

$\mathrm{(}{\mathit{S}}_{\mathit{T}}\mathrm{-}\mathit{k}{\mathrm{)}}_{\mathrm{+}}$ and we can express the option value ${\mathit{V}}_{\mathit{t}}$ at time $\mathit{t}$ as a function of $\mathit{t}$ and ${\mathit{S}}_{\mathit{t}}$

as follows:

${\mathit{V}}_{\mathit{t}}\mathrm{=}{\mathit{E}}^{\mathrm{\star }}\mathrm{\left(}{\mathit{e}}^{\mathrm{-}\mathit{r}\mathrm{(}\mathit{T}\mathrm{-}\mathit{t}\mathrm{)}}\mathit{f}\mathrm{\right(}{\mathit{S}}_{\mathit{T}}\mathrm{\left)}\mathrm{\right|}{{F}}_{\mathit{t}}\mathrm{)}$

$\mathrm{=}{\mathit{E}}^{\mathrm{\star }}\mathrm{\left(}{\mathit{e}}^{\mathrm{-}\mathit{r}\mathrm{(}\mathit{T}\mathrm{-}\mathit{t}\mathrm{)}}\mathit{f}\mathrm{\right(}{\mathit{S}}_{\mathit{t}}{\mathit{e}}^{\mathit{r}\mathrm{(}\mathit{T}\mathrm{-}\mathit{t}\mathrm{)}}{\mathit{e}}^{\mathit{\sigma }\mathrm{(}{\mathit{W}}_{\mathit{T}}\mathrm{-}{\mathit{W}}_{\mathit{t}}\mathrm{)}\mathrm{-}\mathrm{\left(}\frac{{\mathit{\sigma }}^{\mathrm{2}}}{\mathrm{2}}\mathrm{\right)}\mathrm{(}\mathit{T}\mathrm{-}\mathit{t}\mathrm{)}\mathrm{)}}\mathrm{\left|}{{F}}_{\mathit{t}}\mathrm{\right)}$.

The random variable ${\mathit{S}}_{\mathit{t}}$ is ${{F}}_{\mathit{t}}$-measurable and ${\mathit{W}}_{\mathit{T}}\mathrm{-}{\mathit{W}}_{\mathit{t}}$ is independent of ${{F}}_{\mathit{t}}$. A

standard result in probability theory allows us to write

${\mathit{V}}_{\mathit{t}}\mathrm{=}\mathit{F}\mathrm{(}\mathit{t}\mathrm{,}\mathrm{}{\mathit{S}}_{\mathit{t}}\mathrm{)}$,

where

$\mathit{F}\mathrm{(}\mathit{t}\mathrm{,}\mathrm{}\mathit{x}\mathrm{)}\mathrm{=}{\mathit{E}}^{\mathrm{\star }}\mathrm{\left(}{\mathit{e}}^{\mathrm{-}\mathit{r}\mathrm{(}\mathit{T}\mathrm{-}\mathit{t}\mathrm{)}}\mathit{f}\mathrm{\right(}\mathit{x}{\mathit{e}}^{\mathrm{-}\mathit{r}\mathrm{(}\mathit{T}\mathrm{-}\mathit{t}\mathrm{)}}{\mathit{e}}^{\mathit{\sigma }\mathrm{(}{\mathit{W}}_{\mathit{T}}\mathrm{-}{\mathit{W}}_{\mathit{t}}\mathrm{)}\mathrm{-}\mathrm{\left(}\frac{{\mathit{\sigma }}^{\mathrm{2}}}{\mathrm{2}}\mathrm{\right)}\mathrm{(}\mathit{T}\mathrm{-}\mathit{t}\mathrm{)}\mathrm{)}}\mathrm{)}$.

As ${\mathit{W}}_{\mathit{t}}$ is a standard Brownian motion, ${\mathit{W}}_{\mathit{T}}\mathrm{-}{\mathit{W}}_{\mathit{t}}$ is a zero-mean normal variable with

variance $\mathit{T}\mathrm{-}\mathit{t}$ and so,

$\mathit{F}\mathrm{(}\mathit{t}\mathrm{,}\mathrm{}\mathit{x}\mathrm{)}\mathrm{=}{\mathit{e}}^{\mathrm{-}\mathit{r}\mathrm{(}\mathit{T}\mathrm{-}\mathit{t}\mathrm{)}}{{\displaystyle \int }}_{\mathrm{-}\mathrm{\infty }}^{\mathrm{\infty }}\mathit{f}\mathrm{\left(}\mathit{x}{\mathit{e}}^{\mathrm{(}\mathit{r}\mathrm{-}\frac{{\mathit{\sigma }}^{\mathrm{2}}}{\mathrm{2}}\mathrm{)}\mathrm{(}\mathit{T}\mathrm{-}\mathit{t}\mathrm{)}\mathrm{+}\mathit{\sigma }\mathit{y}\sqrt{\mathit{T}\mathrm{-}\mathit{t}}}\mathrm{\right)}\frac{{\mathit{e}}^{\mathrm{-}\frac{{\mathit{y}}^{\mathrm{2}}}{\mathrm{2}}}}{\sqrt{\mathrm{2}\mathit{\pi }}}\mathit{d}\mathit{y}$.

Now $\mathit{F}$ can be calculated explicitly for call options. In this case $\mathit{f}\mathrm{\left(}\mathit{x}\mathrm{\right)}\mathrm{=}\mathrm{(}\mathit{x}\mathrm{-}\mathit{K}{\mathrm{)}}_{\mathrm{+}}$ and

$\mathit{F}\mathrm{(}\mathit{t}\mathrm{,}\mathrm{}\mathit{x}\mathrm{)}\mathrm{=}{\mathit{E}}^{\mathrm{\star }}\mathrm{\left(}{\mathit{e}}^{\mathrm{-}\mathit{r}\mathrm{(}\mathit{T}\mathrm{-}\mathit{t}\mathrm{)}}\mathrm{\right(}{\mathit{e}}^{\mathrm{(}\mathit{r}\mathrm{-}\frac{{\mathit{\sigma }}^{\mathrm{2}}}{\mathrm{2}}\mathrm{)}\mathrm{(}\mathit{T}\mathrm{-}\mathit{t}\mathrm{)}\mathrm{+}\mathit{\sigma }\mathrm{(}{\mathit{W}}_{\mathit{T}}\mathrm{-}{\mathit{W}}_{\mathit{t}}\mathrm{)}}\mathrm{-}\mathit{K}{\mathrm{)}}_{\mathrm{+}}\mathrm{)}$

$\mathrm{=}\mathit{E}\mathrm{(}\mathit{x}{\mathit{e}}^{\mathit{\sigma }\sqrt{\mathit{\theta }}\mathit{g}\mathrm{-}{\mathit{\sigma }}^{\mathrm{2}}\frac{\mathit{\theta }}{\mathrm{2}}}\mathrm{-}\mathit{K}{\mathit{e}}^{\mathrm{-}\mathit{r}\mathit{\theta }}{\mathrm{)}}_{\mathrm{+}}$

where $\mathrm{g}$ is a standard Gaussian variable and $\mathit{\theta }\mathrm{=}\mathit{T}\mathrm{-}\mathit{t}$. We define:

${\mathit{d}}_{\mathrm{1}}\mathrm{=}\frac{\mathit{l}\mathit{n}\mathrm{(}\mathit{x}\mathrm{/}\mathit{K}\mathrm{)}\mathrm{+}\mathrm{(}\mathit{r}\mathrm{+}\frac{{\mathit{\sigma }}^{\mathrm{2}}}{\mathrm{2}}\mathrm{)}\mathit{\theta }}{\mathit{\sigma }\sqrt{\mathit{\theta }}}$

and

${\mathit{d}}_{\mathrm{2}}\mathrm{=}{\mathit{d}}_{\mathrm{1}}\mathrm{-}\mathit{\sigma }\sqrt{\mathit{\theta }}$.

13

Then we have,

$\mathit{F}\mathrm{(}\mathit{t}\mathrm{,}\mathrm{}\mathit{x}\mathrm{)}\mathrm{=}\mathit{E}\mathrm{\left[}\mathrm{\right(}\mathit{x}{\mathit{e}}^{\mathit{\sigma }\sqrt{\mathit{\theta }}\mathit{g}\mathrm{-}{\mathit{\sigma }}^{\mathrm{2}}\frac{\mathit{\theta }}{\mathrm{2}}}\mathrm{-}\mathit{K}{\mathit{e}}^{\mathrm{-}\mathit{r}\mathit{\theta }}\mathrm{)}{\mathit{I}}_{\mathrm{\{}\mathit{g}\mathrm{+}{\mathit{d}}_{\mathrm{2}}\mathrm{\ge }\mathrm{0}\mathrm{\}}\mathrm{]}}$

$\mathrm{=}{{\displaystyle \int }}_{\mathrm{-}{\mathit{d}}_{\mathrm{2}}}^{\mathrm{\infty }}\mathrm{(}\mathit{x}{\mathit{e}}^{\mathit{\sigma }\sqrt{\mathit{\theta }}\mathit{g}\mathrm{-}{\mathit{\sigma }}^{\mathrm{2}}\frac{\mathit{\theta }}{\mathrm{2}}}\mathrm{-}\mathit{K}{\mathit{e}}^{\mathrm{-}\mathit{r}\mathit{\theta }}\mathrm{)}\frac{{\mathit{e}}^{\mathrm{-}\frac{{\mathit{y}}^{\mathrm{2}}}{\mathrm{2}}}}{\sqrt{\mathrm{2}\mathit{\pi }}}\mathit{d}\mathit{y}$

$\mathrm{=}{{\displaystyle \int }}_{\mathrm{-}\mathrm{\infty }}^{{\mathit{d}}_{\mathrm{2}}}\mathrm{(}\mathit{x}{\mathit{e}}^{\mathrm{-}\mathit{\sigma }\sqrt{\mathit{\theta }}\mathit{g}\mathrm{-}{\mathit{\sigma }}^{\mathrm{2}}\frac{\mathit{\theta }}{\mathrm{2}}}\mathrm{-}\mathit{K}{\mathit{e}}^{\mathrm{-}\mathit{r}\mathit{\theta }}\mathrm{)}\frac{{\mathit{e}}^{\mathrm{-}\frac{{\mathit{y}}^{\mathrm{2}}}{\mathrm{2}}}}{\sqrt{\mathrm{2}\mathit{\pi }}}\mathit{d}\mathit{y}$

$\mathrm{=}{{\displaystyle \int }}_{\mathrm{-}\mathrm{\infty }}^{{\mathit{d}}_{\mathrm{2}}}\mathrm{(}\mathit{x}{\mathit{e}}^{\mathrm{-}\mathit{\sigma }\sqrt{\mathit{\theta }}\mathit{g}\mathrm{-}{\mathit{\sigma }}^{\mathrm{2}}\frac{\mathit{\theta }}{\mathrm{2}}}\mathrm{-}{{\displaystyle \int }}_{\mathrm{-}\mathrm{\infty }}^{{\mathit{d}}_{\mathrm{2}}}\mathit{K}{\mathit{e}}^{\mathrm{-}\mathit{r}\mathit{\theta }}\mathrm{)}\frac{{\mathit{e}}^{\mathrm{-}\frac{{\mathit{y}}^{\mathrm{2}}}{\mathrm{2}}}}{\sqrt{\mathrm{2}\mathit{\pi }}}\mathit{d}\mathit{y}$.

Now, using the change of variable $\mathit{z}$ $\mathrm{=}\mathit{y}\mathrm{+}\mathit{\sigma }\sqrt{\mathit{\theta }}$ we get

$\mathit{F}\mathrm{(}\mathit{t}\mathrm{,}\mathrm{}\mathit{x}\mathrm{)}\mathrm{=}\mathit{x}\mathit{N}\mathrm{\left(}{\mathit{d}}_{\mathrm{1}}\mathrm{\right)}\mathrm{-}\mathit{K}{\mathit{e}}^{\mathrm{-}\mathit{r}\mathit{\theta }}\mathit{N}\mathrm{\left(}{\mathit{d}}_{\mathrm{2}}\mathrm{\right)}$,

where

$\mathit{N}\mathrm{\left(}\mathit{d}\mathrm{\right)}\mathrm{=}\frac{\mathrm{1}}{\sqrt{\mathrm{2}\mathit{\pi }}}{{\displaystyle \int }}_{\mathrm{-}\mathrm{\infty }}^{\mathit{d}}{\mathit{e}}^{\mathrm{-}\frac{{\mathit{x}}^{\mathrm{2}}}{\mathrm{2}}}\mathit{d}\mathit{x}$.

Similar calculations show that the price of the put option is

$\mathit{F}\mathrm{(}\mathit{t}\mathrm{,}\mathrm{}\mathit{x}\mathrm{)}\mathrm{=}\mathit{K}{\mathit{e}}^{\mathrm{-}\mathit{r}\mathit{\theta }}\mathit{N}\mathrm{(}\mathrm{-}{\mathit{d}}_{\mathrm{2}}\mathrm{)}\mathrm{-}\mathit{x}\mathit{N}$