Project Materials

Pricing of Basket Options

TABLE OF CONTENTS

Preliminaries: . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.1.1 -algebra . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.1.2 Probability Space . . . . . . . . . . . . . . . . . . . . . 8
1.1.3 Borel -algebra . . . . . . . . . . . . . . . . . . . . . . 8
1.1.4 A random variable: . . . . . . . . . . . . . . . . . . . . 9
1.1.5 Probability distribution . . . . . . . . . . . . . . . . . 9
1.1.6 Normal distribution . . . . . . . . . . . . . . . . . . . 9
1.1.7 A d-dimensional Normal distribution . . . . . . . . . . 10
1.1.8 Log-normal Distribution . . . . . . . . . . . . . . . . . 11
1.1.9 Mathematical Expectation . . . . . . . . . . . . . . . . 11
1.1.10 Variance and covariance of random variables: . . . . . 12
1.1.11 Characteristic function . . . . . . . . . . . . . . . . . . 12
1.1.12 Stochastic process . . . . . . . . . . . . . . . . . . . . 13
1.1.13 Sample Paths . . . . . . . . . . . . . . . . . . . . . . . 13
1.1.14 Brownian Motion . . . . . . . . . . . . . . . . . . . . . 13
1.1.15 Filtration: . . . . . . . . . . . . . . . . . . . . . . . . . 14
1.1.16 Adaptedness . . . . . . . . . . . . . . . . . . . . . . . 14
1.1.17 Conditional expectation . . . . . . . . . . . . . . . . . 14
1.1.18 Martingale . . . . . . . . . . . . . . . . . . . . . . . . 15
1.1.19 Quadratic variation . . . . . . . . . . . . . . . . . . . 15
1.1.20 Stochastic differential equations . . . . . . . . . . . . . 16
1.1.21 Ito formula and lemma . . . . . . . . . . . . . . . . . . 16
1.1.22 Gamma distribution . . . . . . . . . . . . . . . . . . . 17
1.1.23 Risk-neutral Probabilities . . . . . . . . . . . . . . . . 18
1.2 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2 Literature Review 21
3 Financial Derivatives 24
3.0.1 Forward Contract . . . . . . . . . . . . . . . . . . . . 24
3.0.2 Future Contracts . . . . . . . . . . . . . . . . . . . . . 25
3.0.3 Options . . . . . . . . . . . . . . . . . . . . . . . . . . 25
3.0.4 Hedgers . . . . . . . . . . . . . . . . . . . . . . . . . . 28
3.0.5 Speculators . . . . . . . . . . . . . . . . . . . . . . . . 28
3.0.6 Arbitrageurs . . . . . . . . . . . . . . . . . . . . . . . 29
4 Pricing of Basket option 30
4.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
4.2 Geometric Brownian Motion . . . . . . . . . . . . . . . . . . . 32
4.3 Methods used in pricing Basket options . . . . . . . . . . . . 34
4.3.1 Numerical Methods . . . . . . . . . . . . . . . . . . . 34
4.3.2 Approximation Methods . . . . . . . . . . . . . . . . . 41
5 APPLICATION 48
5.1 Foreign Exchange Market . . . . . . . . . . . . . . . . . . . . 48
5.1.1 Quotation Style . . . . . . . . . . . . . . . . . . . . . . 51
5.2 Foreign Exchange Basket Option . . . . . . . . . . . . . . . . 52
5.2.1 Correlation in foreign exchange . . . . . . . . . . . . . 53
5.3 Numerical Example . . . . . . . . . . . . . . . . . . . . . . . . 54
5.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

CHAPTER ONE

General Introduction

In this chapter we give some definitions in probability theory needed for our thesis and provide some introduction to the work.

1.1 Preliminaries:

We begin by introducing a number of probabilistic concepts.

1.1.1 -algebra

Let

be a non-empty set and B a non-empty collection of subset of, B is called a -algebra if the following properties hold:
i
2 B
ii A 2 ) A0 2
iii fAj : j 2 Jg B )
S
j2J
Aj 2 B for any nite or infinite countable
subset J of N

1.1.2 Probability Space

1. Let

be a nonempty set and B a – algebra of subsets of.

Then
the pair (; B) is called a measurable space and a member of B is called a measurable set.

2. Let (; B) be a measurable space and : B 􀀀! R be a real valued map on . Then is called a probability Measure if the following properties hold: