Notes on Stochastic Finance

Introduction......Page 9
Definitions and Formalism......Page 17
Portfolio Allocation and Short-Selling......Page 18
Arbitrage......Page 19
Risk-Neutral Measures......Page 21
Hedging of Contingent Claims......Page 22
Example......Page 24
Exercises......Page 30
Stochastic Processes......Page 33
Portfolio Strategies......Page 34
Contingent Claims......Page 37
Martingales and Conditional Expectation......Page 39
Risk-Neutral Probability Measures......Page 42
Market Completeness......Page 43
The Cox-Ross-Rubinstein (CRR) Market Model......Page 44
Exercises......Page 46
Pricing of Contingent Claims......Page 47
Hedging of Contingent Claims - Backward Induction......Page 51
Pricing of Vanilla Options in the CRR Model......Page 52
Hedging of Vanilla Options in the CRR model......Page 55
Hedging of Exotic Options in the CRR Model......Page 58
Convergence of the CRR Model......Page 62
Exercises......Page 66
Brownian Motion......Page 69
Wiener Stochastic Integral......Page 73
Itô Stochastic Integral......Page 78
Deterministic Calculus......Page 82
Stochastic Calculus......Page 83
Geometric Brownian Motion......Page 86
Stochastic Differential Equations......Page 89
Exercises......Page 90
Self-Financing Portfolio Strategies......Page 95
Arbitrage and Risk-Neutral Measures......Page 99
Completeness......Page 100
The Black-Scholes PDE......Page 101
The Heat Equation......Page 110
Solution of the Black-Scholes PDE......Page 112
Exercises......Page 114
Martingale Property of the Itô Integral......Page 117
Risk-neutral Measures......Page 119
Pricing by the Martingale Method......Page 123
Hedging Strategies......Page 126
Pricing and Hedging of Some Exotic Options......Page 131
Exercises......Page 138
Historical Volatility......Page 147
Implied Volatility......Page 148
The Black-Scholes Formula vs Market Data......Page 149
Local Volatility......Page 154
Stopping Times and Martingales......Page 157
Perpetual American Options......Page 167
Finite Expiration American Options......Page 178
Exercises......Page 186
Notion of Numéraire......Page 191
Change of Numéraire......Page 193
Foreign Exchange......Page 197
Pricing under Change of Numéraire......Page 199
Pricing of Exchange Options......Page 202
Self-Financing Hedging by Change of Numéraire......Page 204
Exercises......Page 206
Short Term Models......Page 211
Zero-Coupon Bonds......Page 213
Forward Rates......Page 217
The HJM Model......Page 219
Forward Vasicek Rates......Page 223
Modeling Issues......Page 229
The LIBOR Model......Page 237
Exercises......Page 239
Forward Measures and Tenor Structure......Page 245
Caplets and Caps......Page 248
Forward Swap Measures......Page 250
Swaption Pricing on the LIBOR......Page 252
Exercises......Page 255
Exponential Distribution......Page 263
Survival Probabilities......Page 264
Stochastic Default......Page 265
Defaultable Bonds......Page 267
Credit Default Swaps......Page 268
Exercises......Page 270
The Poisson Process......Page 273
Compound Poisson Processes......Page 278
Stochastic Integrals with Jumps......Page 280
Itô Formula with Jumps......Page 281
Stochastic Differential Equations with Jumps......Page 283
Girsanov Theorem for Jump Processes......Page 287
Exercises......Page 291
Risk-Neutral Measures......Page 293
Pricing in Jump Models......Page 294
Black-Scholes PDE with Jumps......Page 295
Exponential models......Page 297
Self-Financing Hedging with Jumps......Page 298
Exercises......Page 301
The heat equation......Page 303
The Black-Scholes PDE......Page 305
Euler discretization......Page 309
Milshtein discretization......Page 310
Probability Spaces......Page 313
Events......Page 314
Probability Measures......Page 316
Conditional Probabilities and Independence......Page 317
Random Variables......Page 318
Probability Distributions......Page 320
Expectation of a Random Variable......Page 325
Characteristic Functions and Laplace Transforms......Page 329
Exercise Solutions......Page 335
References......Page 387
Index......Page 391