A Course in Financial Calculus

by Alison Etheridge

• ISBN13: 9780521813853
• ISBN10: 0521813859
• Format: Hardcover
• Copyright: 2002-09-16
• Publisher: Cambridge University Press

Summary

This text is designed for first courses in financial calculus aimed at students with a good background in mathematics. Key concepts such as martingales and change of measure are introduced in the discrete time framework, allowing an accessible account of Brownian motion and stochastic calculus. The Black-Scholes pricing formula is first derived in the simplest financial context. Subsequent chapters are devoted to increasing the financial sophistication of the models and instruments. The final chapter introduces more advanced topics including stock price models with jumps, and stochastic volatility. A large number of exercises and examples illustrate how the methods and concepts can be applied to realistic financial questions.

Table of Contents

Preface	vii
Single period models	1	(20)
Summary	1	(1)
Some definitions from finance	1	(3)
Pricing a forward	4	(2)
The one-step binary model	6	(2)
A ternary model	8	(1)
A characterisation of no arbitrage	9	(4)
The risk-neutral probability measure	13	(8)
Exercises	18	(3)
Binomial trees and discrete parameter martingales	21	(30)
Summary	21	(1)
The multiperiod binary model	21	(5)
American options	26	(2)
Discrete parameter martingales and Markov processes	28	(10)
Some important martingale theorems	38	(5)
The Binomial Representation Theorem	43	(2)
Overture to continuous models	45	(6)
Exercises	47	(4)
Brownian motion	51	(20)
Summary	51	(1)
Definition of the process	51	(5)
Levy's construction of Brownian motion	56	(3)
The reflection principle and scaling	59	(4)
Martingales in continuous time	63	(8)
Exercises	67	(4)
Stochastic calculus	71	(41)
Summary	71	(1)
Stock prices are not differentiable	72	(2)
Stochastic integration	74	(11)
Ito's formula	85	(8)
Integration by parts and a stochastic Fubini Theorem	93	(3)
The Girsanov Theorem	96	(4)
The Brownian Martingale Representation Theorem	100	(2)
Why geometric Brownian motion?	102	(1)
The Feynman--Kac representation	102	(10)
Exercises	107	(5)
The Black-Scholes model	112	(27)
Summary	112	(1)
The basic Black-Scholes model	112	(6)
Black--Scholes price and hedge for European options	118	(4)
Foreign exchange	122	(4)
Dividends	126	(5)
Bonds	131	(1)
Market price of risk	132	(7)
Exercises	134	(5)
Different payoffs	139	(20)
Summary	139	(1)
European options with discontinuous payoffs	139	(2)
Multistage options	141	(3)
Lookbacks and barriers	144	(5)
Asian options	149	(1)
American options	150	(9)
Exercises	154	(5)
Bigger models	159	(30)
Summary	159	(1)
General stock model	160	(3)
Multiple stock models	163	(12)
Asset prices with jumps	175	(6)
Model error	181	(8)
Exercises	185	(4)
Bibliography	189	(2)
Notation	191	(2)
Index	193