金融数学教程(英文版)

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