Introduction to Mathematical Finance
Discrete Time Models
Stanley R. Pliska

Contents
Preface iii
Acknowledgments viii
1 Single Period Securities Markets 1
1.1 Model Specifications . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Arbitrage and other Economic Considerations . . . . . . . . . . . . . . 4
1.3 Risk Neutral Probability Measures . . . . . . . . . . . . . . . . . . . . 11
1.4 Valuation of Contingent Claims . . . . . . . . . . . . . . . . . . . . . . 16
1.5 Complete and Incomplete Markets . . . . . . . . . . . . . . . . . . . . 21
1.6 Risk and Return . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
2 Single Period Consumption and Investment 32
2.1 Optimal Portfolios and Viability . . . . . . . . . . . . . . . . . . . . . 32
2.2 Risk Neutral Computational Approach . . . . . . . . . . . . . . . . . . 35
2.3 Consumption Investment Problems . . . . . . . . . . . . . . . . . . . . 39
2.4 Mean-Variance Portfolio Analysis . . . . . . . . . . . . . . . . . . . . 45
2.5 Portfolio Management with Short Sales Restrictions and Similar Constraints
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
2.6 Optimal Portfolios in Incomplete Markets . . . . . . . . . . . . . . . . 56
2.7 Equilibrium Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
3 Multiperiod Securities Markets 69
3.1 Model Specifications, Filtrations, and Stochastic Processes . . . . . . . 69
3.1.1 Information Structures . . . . . . . . . . . . . . . . . . . . . . 70
3.1.2 Stochastic Process Models of Security Prices . . . . . . . . . . 73
3.1.3 Trading Strategies . . . . . . . . . . . . . . . . . . . . . . . . 76
3.1.4 Value Processes and Gains Processes . . . . . . . . . . . . . . 77
3.2 Self-Financing Trading Strategies . . . . . . . . . . . . . . . . . . . . 78
3.2.1 Discounted Prices . . . . . . . . . . . . . . . . . . . . . . . . . 79
3.3 Return and Dividend Processes . . . . . . . . . . . . . . . . . . . . . . 80
3.3.1 Returns for Discounted Price Processes . . . . . . . . . . . . . 81
3.3.2 Returns for the Value and Gains Processes . . . . . . . . . . . . 81
3.3.3 Dividend Processes . . . . . . . . . . . . . . . . . . . . . . . . 83
3.4 Conditional Expectation and Martingales . . . . . . . . . . . . . . . . 84
3.5 Economic Considerations . . . . . . . . . . . . . . . . . . . . . . . . . 88
3.6 Markov Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
4 Options, Futures, and Other Derivatives 107
4.1 Contingent Claims . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
4.2 European Options Under the Binomial Model . . . . . . . . . . . . . . 115
4.3 American Options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
4.4 Complete and Incomplete Markets . . . . . . . . . . . . . . . . . . . . 127
4.5 Forward Prices and Cash Stream Valuation . . . . . . . . . . . . . . . . 130
4.6 Futures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
5 Optimal Consumption and Investment Problems 143
5.1 Optimal Portfolios and Dynamic Programming . . . . . . . . . . . . . 143
5.2 Optimal Portfolios and Martingale Methods . . . . . . . . . . . . . . . 149
5.3 Consumption-Investment and Dynamic Programming . . . . . . . . . . 155
5.4 Consumption-Investment and Martingale Methods . . . . . . . . . . . . 160
5.5 Maximum Utility From Consumption and Terminal Wealth . . . . . . . 166
5.6 Optimal Portfolios With Constraints . . . . . . . . . . . . . . . . . . . 170
5.7 Optimal Consumption-Investment With Constraints . . . . . . . . . . . 176
5.8 Portfolio Optimization in Incomplete Markets . . . . . . . . . . . . . . 184
6 Bonds and Interest Rate Derivatives 192
6.1 The Basic Term Structure Model . . . . . . . . . . . . . . . . . . . . . 192
6.2 Lattice, Markov Chain Models . . . . . . . . . . . . . . . . . . . . . . 200
6.3 Yield Curve Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208
6.4 Forward Risk Adjusted Probability Measures . . . . . . . . . . . . . . 213
6.5 Coupon Bonds and Bond Options . . . . . . . . . . . . . . . . . . . . 217
6.6 Swaps and Swaptions . . . . . . . . . . . . . . . . . . . . . . . . . . . 220
6.7 Caps and Floors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225
7 ModelsWith Infinite Sample Spaces 229
7.1 Finite Horizon Models . . . . . . . . . . . . . . . . . . . . . . . . . . 229
7.2 Infinite Horizon Models . . . . . . . . . . . . . . . . . . . . . . . . . . 234
Bibliography 240