Pricing of Compound Options

Pricing of Compound Options

TABLE OF CONTENTS

INTRODUCTION AND PRELIMINARIES 6
1.1 PRELIMINARIES . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.1.1 -ALGEBRA: . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.1.2 BOREL -ALGEBRA: . . . . . . . . . . . . . . . . . . . . . . 6
1.1.3 PROBABILITY SPACE: . . . . . . . . . . . . . . . . . . . . 7
1.1.4 MEASURABLE MAP: . . . . . . . . . . . . . . . . . . . . . 7
1.1.5 RANDOM VARIABLES/VECTORS: . . . . . . . . . . . . . . . . 7
1.1.6 PROBABILITY DISTRIBUTION: . . . . . . . . . . . . . . . . . 7
1.1.7 MATHEMATICAL EXPECTATION: . . . . . . . . . . . . . . . 8
1.1.8 VARIANCE AND COVARIANCE OF RANDOM VARIABLES: . . . . . 8
1.1.9 STOCHASTIC PROCESS: . . . . . . . . . . . . . . . . . . . . 8
1.1.10 BROWNIAN MOTION: . . . . . . . . . . . . . . . . . . . . . 8
1.1.11 FILTRATIONS AND FILTERED PROBABILITY SPACE: . . . . . . . 9
1.1.12 ADAPTEDNESS: . . . . . . . . . . . . . . . . . . . . . . . 10
1.1.13 CONDITIONAL EXPECTATION: . . . . . . . . . . . . . . . . . 10
1.1.14 MARTINGALES: . . . . . . . . . . . . . . . . . . . . . . . . 10
1.1.15 ITO CALCULUS: . . . . . . . . . . . . . . . . . . . . . . . . 10
1.1.16 QUADRATIC VARIATION: . . . . . . . . . . . . . . . . . . . 11
1.1.17 STOCHASTIC DIERENTIAL EQUATIONS: . . . . . . . . . . . . 11
1.1.18 ITO FORMULA AND LEMMA: . . . . . . . . . . . . . . . . . 11
1.1.19 RISK-NEUTRAL PROBABILITIES: . . . . . . . . . . . . . . . . 12
1.1.20 LOG-NORMAL DISTRIBUTION: . . . . . . . . . . . . . . . . . 13
1.1.21 BIVARIATE NORMAL DENSITY FUNCTION: . . . . . . . . . . . 13
1.1.22 CUMULATIVE BIVARIATE NORMAL DISTRIBUTION FUNCTION: . 13
1.1.23 MARKOV PROCESS: . . . . . . . . . . . . . . . . . . . . . . 13
1.1.24 BACKWARD KOLMOGOROV EQUATION: . . . . . . . . . . . . 14
1.1.25 FORKKER-PLANCK EQUATION: . . . . . . . . . . . . . . . . 14
1.1.26 DIUSION PROCESS: . . . . . . . . . . . . . . . . . . . . . 14
1.2 INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2 LITERATURE REVIEW 17
3 FINANCIAL DERIVATIVES AND COMPOUND OPTIONS 20
3.1 FINANCIAL DERIVATIVES . . . . . . . . . . . . . . . . . . 20
3.2 CATEGORIES OF DERIVATIVES . . . . . . . . . . . . . . . 21
3.2.1 FORWARDS . . . . . . . . . . . . . . . . . . . . . . . . . . 21
3.2.2 FUTURES . . . . . . . . . . . . . . . . . . . . . . . . . . 21
3.2.3 SWAPS . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
3.2.4 OPTIONS . . . . . . . . . . . . . . . . . . . . . . . . . . 22
3.2.5 FINANCIAL MARKETS . . . . . . . . . . . . . . . . 25
3.2.6 TYPES OF TRADERS . . . . . . . . . . . . . . . . . 25
3.2.7 EXOTIC OPTIONS . . . . . . . . . . . . . . . . . . . 27
3.2.8 SIMULTANEOUS AND SEQUENTIAL COMPOUND OPTIONS . . . 33
4 PRICING COMPOUND OPTIONS 34
4.1 FACTORS AFFECTING OPTION PRICES . . . . . . . . . . 34
4.1.1 EXERCISE PRICE OF THE OPTION . . . . . . . . . . . . . . . 34
4.1.2 CURRENT VALUE OF THE UNDERLYING ASSET . . . . . . . . . 34
4.1.3 TIME TO EXPIRATION ON THE OPTION . . . . . . . . . . . . 35
4.1.4 VARIANCE IN VALUE OF UNDERLYING ASSET . . . . . . . . . . 35
4.1.5 RISK FREE INTEREST RATE . . . . . . . . . . . . . . . . . . 35
4.2 BLACK-SCHOLES-MERTON MODEL . . . . . . . . . . . . . 35
4.2.1 BLACK-SCHOLES OPTION PRICING . . . . . . . . . . . . . . 35
4.2.2 THE GENERALISED BLACK-SCHOLES-MERTON OPTION PRICING
FORMULA . . . . . . . . . . . . . . . . . . . . . . . . . . 45
4.2.3 COMPOUND OPTIONS . . . . . . . . . . . . . . . . . . . . 46
4.2.4 PUT-CALL PARITY COMPOUND OPTIONS . . . . . . . . . . . 48
4.3 BINOMIAL LATTICE MODEL . . . . . . . . . . . . . . . . . 49
4.3.1 COMPOUND OPTION MODEL IN A TWO PERIOD BINOMIAL TREE 49
4.3.2 FOUR-PERIOD BINOMIAL LATTICE MODEL . . . . . . . . . . . 53
4.4 THE FORWARD VALUATION OF COMPOUND OPTIONS 57
5 APPLICATIONS 65
5.1 BLACK-SCHOLES-MERTON MODEL . . . . . . . . . . . . . . . . . . . 65
5.2 BINOMIAL LATTICE MODEL . . . . . . . . . . . . . . . . . . . . . . 70

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CHAPTER ONE

INTRODUCTION AND PRELIMINARIES

1.1 Preliminaries

1.1.1 -algebra:

Let

be a non empty set, and a non empty collection of subsets of.

Then is called a -algebra if the following properties hold:
(i)
2
(ii) If A 2 , then A0 2
(iii) If fAj : j 2 Jg , then
[
j2J
Aj 2
for any nite or infinite countable subset of N.

1.1.2 Borel -algebra:

Let X be a non empty set and a topology on X i.e. is the collection of subsets of X. Then ( ) is called the Borel -algebra of the topological space (X; )

1.1.3 Probability Space:

Let

be a non-empty set and be a -algebra of subsets of. Then the pair (,) is called a measurable space, and a member of is called a measurable set. Let (,) be a measurable space and be a real-valued map on . Then is called a probability measure on (,) if the following properties hold:
I (A) 0; 8A 2

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