1 Single period models

Summary

1.1 Some definitions from finance

1.2 Pricing aforward

1.3 The one-step binary model

1.4 A ternary model

1.5 A characterisation of no arbitrage

1.6 The risk-neutral probability measure

Exercises

2 Binomial trees and discrete parameter martingales

Summary

2.1 The multiperiod binary model

2.2 American options

2.3 Discrete parameter martingales and Markov processes

2.4 Some important martingale theorems

2.5 The Binomial Representation Theorem

2.6 Overture to continuous models

Exercises

3 Brownian motion

Summary

3.1 Definition of the process

3.2 Lévy's construction of Brownian motion

3.3 The reflection principle and scaling

3.4 Martingales in continuous time

Exercises

4 Stochastic calculus

Summary

4.1 Stock prices are not differentiable

4.2 Stochastic integration

4.3 It？'s formula

4.4 Integration by parts and a stochastic Fubini Theorem

4.5 The Girsanov Theorem

4.6 The Brownian Martingale Representation Theorem

4.7 Why geometric Brownian motion?

4.8 The Feynman-Kac representation

Exercises

5 The Black-Scholes model

Summary

5.1 The basic Black-Scholes model

5.2 Black-Scholes price and hedge for European options

5.3 Foreign exchange

5.4 Dividends

5.5 Bonds

5.6 Market price of risk

Exercises

6 Different payoffs

Summary

6.1 European options with discontinuous payoffs

6.2 Multistage options

6.3 Lookbacks and barriers

6.4 Asian options

6.5 American options

Exercises

7 Bigger models

Summary

7.1 General stock model

7.2 Multiple stock models

7.3 Asset prices with jumps

7.4 Model error

Exercises

Bibliography

Notation

Index