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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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