Mathematical Finance Book

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