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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¡¡L¨¦vy'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