A course in financial calculus

✍ Scribed by Alison Etheridge

• Publisher: CUP
• Year: 2002
• Tongue: English
• Leaves: 206
• Category: Library

✦ Synopsis

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

Cover......Page 1
Half-title......Page 3
Title......Page 5
Copyright......Page 6
Contents......Page 7
Preface......Page 9
Derivatives......Page 11
The pricing problem......Page 12
Packages......Page 13
The risk-free rate......Page 14
Arbitrage pricing......Page 15
Pricing a European call......Page 16
1.4 A ternary model......Page 18
1.5 A characterisation of no arbitrage......Page 19
A market with N assets......Page 20
Arbitrage pricing......Page 21
1.6 The risk-neutral probability measure......Page 23
Expectation recovered......Page 24
Risk-neutral pricing......Page 25
Complete markets......Page 26
Trading in two different markets......Page 27
Exercises......Page 28
2.1 The multiperiod binary model......Page 31
The cash bond......Page 32
Binomial trees......Page 33
2.2 American options......Page 36
Put on nondividendpaying stock......Page 37
2.3 Discrete parameter martingales and Markov processes......Page 38
Stochastic processes......Page 39
Conditional expectation......Page 40
The martingale property......Page 42
Examples......Page 44
New martingales from old......Page 45
Discrete stochastic integrals......Page 46
The Fundamental Theorem of Asset Pricing......Page 47
Optional stopping......Page 48
Compensation......Page 51
American options and supermartingales......Page 52
2.5 The Binomial Representation Theorem......Page 53
From martingale representation to replicating portfolio......Page 54
2.6 Overture to continuous models......Page 55
Under the martingale measure......Page 56
Exercises......Page 57
A characterisation of simple random walks......Page 61
Definition of Brownian motion......Page 63
Behaviour of Brownian motion......Page 65
A polygonal approximation......Page 66
Convergence to Brownian motion......Page 67
Stopping times......Page 69
The reflection principle......Page 70
Hitting a sloping line......Page 71
3.4 Martingales in continuous time......Page 73
Martingales......Page 74
Brownian hitting time distribution......Page 76
Exercises......Page 77
Summary......Page 81
4.1 Stock prices are not differentiable......Page 82
Bounded variation and arbitrage......Page 83
A differential equation for the stock price......Page 84
Quadratic variation......Page 85
Integrating Brownian motion against itself......Page 86
Defining the integral......Page 87
Integrating simple functions......Page 88
Construction of the Ito integral......Page 91
Other integrators......Page 93
The stochastic chain rule......Page 95
Geometric Brownian motion......Page 97
Ito’s formula for geometric Brownian motion......Page 98
Levy’s characterisation of Brownian motion......Page 100
Stochastic differential equations......Page 101
Solving stochastic differential equations......Page 102
Covariation......Page 103
4.5 The Girsanov Theorem......Page 106
Change of measure in the continuous world......Page 107
4.6 The Brownian Martingale Representation Theorem......Page 110
4.8 The Feynman–Kac representation......Page 112
Solving pde’s probabilistically......Page 113
Kolmogorov equations......Page 114
Exercises......Page 117
5.1 The basic Black–Scholes model......Page 122
Self- financing strategies......Page 123
An equivalent martingale measure......Page 125
The Fundamental Theorem of Asset Pricing......Page 126
5.2 Black–Scholes price and hedge for European options......Page 128
Pricing calls and puts......Page 129
Hedging calls and puts......Page 130
5.3 Foreign exchange......Page 132
Change of numeraire......Page 135
Continuous payments......Page 136
Periodic dividends......Page 139
5.5 Bonds......Page 141
5.6 Market price of risk......Page 142
Martingales and tradables......Page 143
Exercises......Page 144
6.1 European options with discontinuous payoffs......Page 149
Digitals and pin risk......Page 150
6.2 Multistage options......Page 151
General strategy......Page 152
Compound options......Page 153
6.3 Lookbacks and barriers......Page 154
Joint distribution of the stock price and its minimum......Page 155
An expression for the price......Page 157
6.4 Asian options......Page 159
6.5 American options......Page 160
Continuous time......Page 161
An explicit solution......Page 163
Exercises......Page 164
Summary......Page 169
The model......Page 170
Second step to replication......Page 171
The generalised Black–Scholes equation......Page 172
Correlated security prices......Page 173
Multifactor Ito formula......Page 174
Change of measure......Page 176
A martingale measure......Page 177
Replicating the claim......Page 178
The multidimensional Black–Scholes equation......Page 180
Numeraires......Page 181
Quantos......Page 182
Pricing a quanto forward contract......Page 183
A Poisson process of jumps......Page 185
Poisson exponential martingales......Page 187
Change of measure......Page 188
Market price of risk......Page 189
Multiple noises......Page 190
Hedging error......Page 191
Stochastic volatility and implied volatility......Page 193
Exercises......Page 195
Further topics in financial mathematics:......Page 199
Additional references from the text:......Page 200
Martingales and other stochastic processes......Page 201
Miscellaneous......Page 202
Index......Page 203