Mathematical Finance: Core Theory, Problems and Statistical Algorithms

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  • Edition: 1st
  • Format: Nonspecific Binding
  • Copyright: 2007-03-12
  • Publisher: Routledge

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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

Prefacep. ix
Review of probability theoryp. 1
Measure space and probability spacep. 1
Random variablesp. 3
Expectationsp. 4
Equivalent probability measuresp. 7
Conditional probability and expectationp. 7
The [sigma]-algebra generated by a random vectorp. 9
Independencep. 11
Probability distributionsp. 12
Problemsp. 17
Basics of stochastic processesp. 18
Definitions of stochastic processesp. 18
Filtrations, independent processes and martingalesp. 19
Markov timesp. 21
Markov processesp. 22
Problemsp. 23
Discrete time market modelsp. 25
Introduction: basic problems for market modelsp. 25
Discrete time model with free borrowingp. 26
A discrete time bond-stock market modelp. 27
The discounted wealth and stock pricesp. 30
Risk-neutral measurep. 31
Replicating strategiesp. 33
Arbitrage possibilities and arbitrage-free marketp. 34
A case of complete marketp. 35
Cox-Ross-Rubinstein modelp. 36
Option pricingp. 39
Increasing frequency and continuous time limitp. 44
Optimal portfolio selectionp. 47
Possible generalizationsp. 48
Conclusionsp. 49
Problemsp. 49
Basics of Ito calculus and stochastic analysisp. 52
Wiener process (Brownian motion)p. 52
Stochastic integral (Ito integral)p. 54
Ito formulap. 58
Stochastic differential equations (Ito equations)p. 61
Definitionsp. 61
The existence and uniqueness theoremp. 62
Continuous time white noisep. 64
Examples of explicit solutions for Ito equationsp. 64
Diffusion Markov processes and Kolmogorov equationsp. 66
Martingale representation theoremp. 70
Change of measure and the Girsanov theoremp. 72
Problemsp. 76
Continuous time market modelsp. 79
Continuous time model for stock pricep. 79
Continuous time bond-stock market modelp. 81
The discounted wealth and stock pricesp. 82
Risk-neutral measurep. 85
Replicating strategiesp. 88
Arbitrage possibilities and arbitrage-free marketsp. 89
A case of complete marketp. 91
Completeness of the Black-Scholes modelp. 91
Option pricingp. 94
Options and their pricesp. 94
The fair price is arbitrage-freep. 96
Option pricing for a complete marketp. 97
A code for the fair option pricep. 100
Black-Scholes formulap. 100
Dynamic option price processp. 101
Non-uniqueness of the equivalent risk-neutral measurep. 104
Examples of incomplete marketsp. 104
Pricing for an incomplete marketp. 105
A generalization: multistock marketsp. 106
Bond marketsp. 109
Conclusionsp. 112
Problemsp. 112
American options and binomial treesp. 116
The binomial tree for stock pricesp. 116
General descriptionp. 116
Choice of u, d, p for the case of constant r and [sigma]p. 118
Pricing of European options via a binomial treep. 120
American option and non-arbitrage pricesp. 120
Fair price of the American optionp. 124
The basic rule for the American optionp. 126
When American and European options have the same pricep. 131
Stefan problem for the price of American optionsp. 133
Pricing of the American option via a binomial treep. 135
Problemsp. 138
Implied and historical volatilityp. 139
Definitions for historical and implied volatilityp. 139
Calculation of implied volatilityp. 142
A simple market model with volatility smile effectp. 144
Problemsp. 145
Review of statistical estimationp. 146
Some basic facts about discrete time random processesp. 146
Simplest regression and autoregressionp. 148
Least squares (LS) estimationp. 148
The LS estimate of the variance of the error termp. 153
The case of AR(l)p. 154
Maximum likelihoodp. 154
Hypothesis testingp. 155
LS estimate for multiple regressionp. 159
Forecastingp. 161
Heteroscedastic residuals, ARCH and GARCHp. 167
Some tests of heteroscedasticityp. 167
ARCH modelsp. 168
Estimation of parameters for ARCH(1) with the ML methodp. 170
ARCH(q) and GARCH modelsp. 173
Problemsp. 175
Estimation of models for stock pricesp. 176
Review of the continuous time modelp. 176
Examples of special models for stock price evolutionp. 177
Estimation of models with constant volatilityp. 181
Estimation of the log-normal model without mean-revertingp. 181
Estimation of the mean-reverting modelp. 183
Forecast of volatility with ARCH modelsp. 184
Black-Scholes formula and forecast of volatility squarep. 185
Volatility forecast with GARCH and without mean-revertingp. 186
Volatility forecast with GARCH and with mean-revertingp. 188
Problemsp. 189
Legend of notations and abbreviationsp. 191
Selected answers and key figuresp. 192
Bibliographyp. 194
Indexp. 195
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