1 Single period models 1
Summary 1
1.1 Some definitions from finance 1
1.2 Pricing a forward 4
1.3 The one-step binary model 6
1.4 A ternary model 8
1.5 A characterisation of no arbitrage 9
1.6 The risk-neutral probability measure 13
Exercises 18
2 Binomial trees and discrete parameter martingales 21
Summary 21
2.1 The multiperiod binary model 21
2.2 American options 26
2.3 Discrete parameter martingales and Markov processes 28
2.4 Some important martingale theorems 38
2.5 The Binomial Representation Theorem 43
2.6 Overture to continuous models 45
Exercises 47
3 Brownian motion 51
Summary 51
3.1 Definition of the process 51
3.2 Lévy's construction of Brownian motion 56
3.3 The reflection principle and scaling 59
3.4 Martingales in continuous time 63
Exercises 67
4 Stochastic calculus 71
Summary 71
4.1 Stock prices are not differentiable 72
4.2 Stochastic integration 74
4.3 It?'s formula 85
4.4 Integration by parts and a stochastic Fubini Theorem 93
4.5 The Girsanov Theorem 96
4.6 The Brownian Martingale Representation Theorem 100
4.7 Why geometric Brownian motion? 102
4.8 The Feynman-Kac representation 102
Exercises 107
5 The Black-Scholes model 112
Summary 112
5.1 The basic Black-Scholes model 112
5.2 Black-Scholes price and hedge for European options 118
5.3 Foreign exchange 122
5.4 Dividends 126
5.5 Bonds 131
5.6 Market price of risk 132
Exercises 134
6 Oifferent payoffs 139
Summary 139
6.1 European options with discontinuous payoffs 139
6.2 Multistage options 141
6.3 Lookbacks and barriers 144
6.4 Asian options 149
6.5 American options 150
Exercises 154
7 Bigger models 159
Summary 159
7.1 General stock model 160
7.2 Multiple stock models 163
7.3 Asset prices with jumps 175
7.4 Model error 181
Exercises 185
Bibliography 189
Notation 191
Index 193
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