Mathematical Finance: Core Theory, Problems and Statistical Algorithms

by Dokuchaev; Nikolai

• ISBN13: 9780415414470
• ISBN10: 0415414474
• Edition: 1st
• Format: Hardcover
• Copyright: 2007-04-25
• Publisher: Routledge

Summary

Written in a rigorous yet logical and easy to use style, spanning a range of disciplines, including business, mathematics, finance and economics, this comprehensive textbook offers a systematic, self-sufficient yet concise presentation of the main topics and related parts of Stochastic Analysis and statistical finance that are covered in the majority of university programmes. Providing all explanations of basic concepts and results with proofs and numerous examples and problems, it includes: an introduction to probability theory a detailed study of discrete and continuous time market models a comprehensive review of Ito calculus and statistical methods as a basis for statistical estimation of models for pricing a detailed discussion of options and their pricing, including American options in continuous time setting. An excellent introduction to the topic, this textbook is an essential resource for all students on undergraduate andpostgraduate courses and advanced degree programs in econometrics, finance, applied mathematics and mathematical modelling as well as academics and practitioners.

Author Biography

Nikolai Dokuchaev is Associate Professor in the Department of Mathematics, Trent University, Ontario, Canada.

Table of Contents

Preface	p. ix
Review of probability theory	p. 1
Measure space and probability space	p. 1
Random variables	p. 3
Expectations	p. 4
Equivalent probability measures	p. 7
Conditional probability and expectation	p. 7
The [sigma]-algebra generated by a random vector	p. 9
Independence	p. 11
Probability distributions	p. 12
Problems	p. 17
Basics of stochastic processes	p. 18
Definitions of stochastic processes	p. 18
Filtrations, independent processes and martingales	p. 19
Markov times	p. 21
Markov processes	p. 22
Problems	p. 23
Discrete time market models	p. 25
Introduction: basic problems for market models	p. 25
Discrete time model with free borrowing	p. 26
A discrete time bond-stock market model	p. 27
The discounted wealth and stock prices	p. 30
Risk-neutral measure	p. 31
Replicating strategies	p. 33
Arbitrage possibilities and arbitrage-free market	p. 34
A case of complete market	p. 35
Cox-Ross-Rubinstein model	p. 36
Option pricing	p. 39
Increasing frequency and continuous time limit	p. 44
Optimal portfolio selection	p. 47
Possible generalizations	p. 48
Conclusions	p. 49
Problems	p. 49
Basics of Ito calculus and stochastic analysis	p. 52
Wiener process (Brownian motion)	p. 52
Stochastic integral (Ito integral)	p. 54
Ito formula	p. 58
Stochastic differential equations (Ito equations)	p. 61
Definitions	p. 61
The existence and uniqueness theorem	p. 62
Continuous time white noise	p. 64
Examples of explicit solutions for Ito equations	p. 64
Diffusion Markov processes and Kolmogorov equations	p. 66
Martingale representation theorem	p. 70
Change of measure and the Girsanov theorem	p. 72
Problems	p. 76
Continuous time market models	p. 79
Continuous time model for stock price	p. 79
Continuous time bond-stock market model	p. 81
The discounted wealth and stock prices	p. 82
Risk-neutral measure	p. 85
Replicating strategies	p. 88
Arbitrage possibilities and arbitrage-free markets	p. 89
A case of complete market	p. 91
Completeness of the Black-Scholes model	p. 91
Option pricing	p. 94
Options and their prices	p. 94
The fair price is arbitrage-free	p. 96
Option pricing for a complete market	p. 97
A code for the fair option price	p. 100
Black-Scholes formula	p. 100
Dynamic option price process	p. 101
Non-uniqueness of the equivalent risk-neutral measure	p. 104
Examples of incomplete markets	p. 104
Pricing for an incomplete market	p. 105
A generalization: multistock markets	p. 106
Bond markets	p. 109
Conclusions	p. 112
Problems	p. 112
American options and binomial trees	p. 116
The binomial tree for stock prices	p. 116
General description	p. 116
Choice of u, d, p for the case of constant r and [sigma]	p. 118
Pricing of European options via a binomial tree	p. 120
American option and non-arbitrage prices	p. 120
Fair price of the American option	p. 124
The basic rule for the American option	p. 126
When American and European options have the same price	p. 131
Stefan problem for the price of American options	p. 133
Pricing of the American option via a binomial tree	p. 135
Problems	p. 138
Implied and historical volatility	p. 139
Definitions for historical and implied volatility	p. 139
Calculation of implied volatility	p. 142
A simple market model with volatility smile effect	p. 144
Problems	p. 145
Review of statistical estimation	p. 146
Some basic facts about discrete time random processes	p. 146
Simplest regression and autoregression	p. 148
Least squares (LS) estimation	p. 148
The LS estimate of the variance of the error term	p. 153
The case of AR(l)	p. 154
Maximum likelihood	p. 154
Hypothesis testing	p. 155
LS estimate for multiple regression	p. 159
Forecasting	p. 161
Heteroscedastic residuals, ARCH and GARCH	p. 167
Some tests of heteroscedasticity	p. 167
ARCH models	p. 168
Estimation of parameters for ARCH(1) with the ML method	p. 170
ARCH(q) and GARCH models	p. 173
Problems	p. 175
Estimation of models for stock prices	p. 176
Review of the continuous time model	p. 176
Examples of special models for stock price evolution	p. 177
Estimation of models with constant volatility	p. 181
Estimation of the log-normal model without mean-reverting	p. 181
Estimation of the mean-reverting model	p. 183
Forecast of volatility with ARCH models	p. 184
Black-Scholes formula and forecast of volatility square	p. 185
Volatility forecast with GARCH and without mean-reverting	p. 186
Volatility forecast with GARCH and with mean-reverting	p. 188
Problems	p. 189
Legend of notations and abbreviations	p. 191
Selected answers and key figures	p. 192
Bibliography	p. 194
Index	p. 195
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