Financial mathematics : theory and problems for multi-period models

Cover......Page 1
Title Page......Page 3
Copyright Page
......Page 4
Preface......Page 5
Table of Contents
......Page 7
1 Pricing and hedging......Page 10
1.1.1 Discrete time markets......Page 11
1.1.2 Self-financing and predictable portfolios......Page 12
1.1.3 Relative portfolio......Page 14
1.1.4 Discounted market......Page 15
1.2 Arbitrage and martingale measures......Page 16
1.3.1 Derivative securities......Page 18
1.3.2 Arbitrage pricing......Page 19
1.3.4 Put-Call parity......Page 22
1.4.1 Binomial model......Page 23
1.4.2 Trinomial model......Page 27
1.5 On pricing and hedging in incomplete markets......Page 31
1.6.1 A particular case......Page 32
1.6.2 General case......Page 34
1.7 Solved problems......Page 37
2 Portfolio optimization......Page 70
2.1.1 Strategies with consumption......Page 71
2.1.2 Utility functions......Page 74
2.1.3 Expected utility of terminal wealth......Page 76
2.1.4 Expected utility from intermediate consumption and terminal wealth......Page 79
2.2.1 Complete market: terminal wealth......Page 81
2.2.2 Incomplete market: terminal wealth......Page 87
2.2.3 Complete market: intermediate consumption......Page 90
2.2.4 Complete market: intermediate consumption and terminal wealth......Page 95
2.3.1 Recursive algorithm......Page 97
2.3.2 Proof of Theorem 2.32......Page 101
2.4.1 Terminal utility in the binomial model: MG method......Page 103
2.4.2 Terminal utility in the binomial model: DP method......Page 105
2.4.3 Terminal utility in the completed trinomial model: MG method......Page 108
2.4.4 Terminal utility in the completed trinomial model: DP method......Page 110
2.4.5 Terminal utility in the standard trinomial model: DP method......Page 112
2.4.6 Intermediate consumption in the binomial model: MG method......Page 115
2.4.7 Intermediate consumption in the binomial model: DP method......Page 118
2.4.8 Intermediate consumption in the completed trinomial model: MG method......Page 122
2.4.9 Optimal consumption in the completed trinomial model: DP method......Page 123
2.4.10 Intermediate consumption in the standard trinomial model: DP method......Page 124
2.5 Solved problems......Page 127
3 American options......Page 173
3.1 American derivatives and early exercise strategies......Page 174
3.1.1 Arbitrage pricing......Page 175
3.1.2 Arbitrage price in a complete market......Page 177
3.1.3 Optimal exercise strategies......Page 183
3.1.4 Hedging strategies......Page 186
3.2 American and European options......Page 188
3.3.1 Preliminaries......Page 191
3.3.2 Solved problems......Page 192
4 Interest rates......Page 230
4.1 Bonds and interest rates......Page 231
4.2 Market models for interest rates......Page 234
4.3 Short models......Page 237
4.3.1 Affine models......Page 239
4.3.2 Discrete time Hull-White model......Page 240
4.4 Forward models......Page 244
4.4.1 Binomial forward model......Page 246
4.4.2 Multinomial forward model......Page 248
4.5.1 Caps and Floors......Page 251
4.5.2 Interest Rate Swaps......Page 254
4.5.3 Swaptions and Swap Rate......Page 257
4.6.1 Recalling the basic models......Page 259
4.6.2 Options on T-bonds......Page 262
4.6.3 Caps and Floors......Page 269
4.6.4 Swap Rates and Payer Forward Swaps......Page 278
4.6.5 Swaptions......Page 283
Back matter......Page 293