Discrete-time Asset Pricing Models in Applied Stochastic Finance

Title Page......Page 3
Copyright
......Page 4
Contents......Page 5
Preface......Page 10
1.1. Introductory notes......Page 13
1.2. Probability space......Page 14
1.3. Conditional probability and independence......Page 20
1.4. Random variables......Page 24
1.4.1. Discrete random variables......Page 26
1.4.3. Binomial random variables......Page 27
1.4.4. Geometric random variables......Page 28
1.4.5. Poisson random variables......Page 29
1.4.6. Continuous random variables......Page 30
1.4.7. Exponential random variables......Page 32
1.4.9. Gamma random variables......Page 33
1.4.10. Normal random variables......Page 34
1.4.12. Weibull random variables......Page 35
1.5. Expectation and variance of a random variable......Page 36
1.6. Jointly distributed random variables......Page 40
1.6.1. Joint probability distribution of functions of random variables......Page 42
1.7. Moment generating functions......Page 44
1.8. Probability inequalities and limit theorems......Page 49
1.9. Multivariate normal distribution......Page 56
2.1. Introduction......Page 60
2.2. Bonds and basic interest rates......Page 61
2.2.2. Discretely compounded interest rates......Page 62
2.2.3. Continuously compounded interest rate......Page 63
2.2.4. Money-market account......Page 64
2.2.6. Time value of money......Page 66
2.2.7. Coupon-bearing bonds and yield-to-maturity......Page 67
2.3. Forward contracts......Page 69
2.3.1. Arbitrage......Page 70
2.5. Swaps......Page 71
2.6.1. European call option......Page 73
2.6.3. American call option......Page 74
2.6.4. American put option......Page 75
2.6.5. Basic problems and assumptions......Page 76
2.8. Arbitrage relationships between call and put options......Page 78
2.9. Exercises......Page 80
3.1. Introduction......Page 82
3.2. Conditional expectation: the discrete case......Page 83
3.3. Applications of conditional expectations......Page 86
3.3.1. Expectation of the sum of a random number of random variables......Page 87
3.3.2. Expected value of a random number of Bernoulli trials with probability of success being a random variable......Page 88
3.3.3. Number of Bernoulli trials until there are k consecutive successes......Page 89
3.3.4. Conditional variance relationship......Page 90
3.3.5. Variance of the sum of a random number of random variables......Page 91
3.4. Properties of the conditional expectation......Page 92
3.5. Markov chains......Page 96
3.5.1. Probability distribution in the states of a Markov chain......Page 101
3.5.2. Statistical inference in Markov chains......Page 105
3.5.3. The strong Markov property......Page 108
3.5.4. Classification of states of a Markov chain......Page 111
3.5.5. Periodic Markov chains......Page 115
3.5.5.1. Cyclic subclasses......Page 117
3.5.5.2. Algorithm for the cyclic subclasses......Page 120
3.5.6. Classification of states......Page 123
3.5.7. Asymptotic behavior of irreducible homogenous Markov chains......Page 126
3.5.8. The mean time of first entrance in a state of Markov chain......Page 137
3.5.9. The variance of the time of first visit into a state of a Markov chain......Page 140
3.6. Exercises......Page 142
4.1. Introductory notes......Page 148
4.2. Binomial model......Page 149
4.3. Stochastic evolution of the asset prices......Page 152
4.4. Binomial approximation to the lognormal distribution......Page 154
4.5. One-period European call option......Page 156
4.6. Two-period European call option......Page 161
4.7. Multiperiod binomial model......Page 164
4.8. The evolution of the asset prices as a Markov chain......Page 165
4.9. Exercises......Page 169
5.1. Introductory notes......Page 173
5.2. Martingales......Page 174
5.3. Optional sampling theorem......Page 179
5.4. Submartingales, supermartingales and martingales convergence theorem......Page 188
5.5. Martingale transforms......Page 192
5.6. Uniform integrability and Doob’s decomposition......Page 194
5.7. The snell envelope......Page 197
5.8. Exercises......Page 200
6.1. Introductory notes......Page 205
6.2. Equivalent martingale measure and the Randon-Nikodým derivative process......Page 206
6.3. Finite general markets......Page 214
6.3.1. Uniqueness of arbitrage price......Page 220
6.3.2. Equivalent martingale measures......Page 223
6.4. Fundamental theorem of asset pricing......Page 225
6.5. Complete markets and martingale representation......Page 232
6.6. Finding the equivalent martingale measure......Page 238
6.6.1. Exploring the vital equations and conditions......Page 244
6.6.2. Equivalent martingale measures for general finite markets......Page 247
6.7. Exercises......Page 248
7.1. Introductory notes......Page 250
7.2. A three-period American put option......Page 251
7.3. Hedging strategy for an American put option......Page 258
7.4.1.2. Trading strategy for hedging......Page 263
7.5. Optimal time for the holder to exercise......Page 264
7.6. American derivatives in general markets......Page 271
7.7. Extending the concept of self-financing strategies......Page 275
7.8. Exercises......Page 278
8.1. Introductory notes......Page 282
8.2. The zero coupon bonds of all maturities......Page 283
8.3. Arbitrage-free family of bond prices......Page 287
8.4. Interest rate process and the term structure of bond prices......Page 291
8.5. The evolution of the interest rate process......Page 299
8.6. Binomial model with normally distributed spread of interest rates......Page 302
8.7. Binomial model with lognormally distributed spread of interest rates......Page 305
8.8.1. Valuation of the European put call......Page 307
8.8.2. Hedging the European put option......Page 309
8.9. Fixed income derivatives......Page 311
8.9.1. Interest rate swaps......Page 313
8.9.2. Interest rate caps and floors......Page 316
8.10. T-period equivalent forward measure......Page 317
8.11. Futures contracts......Page 326
8.12. Exercises......Page 328
9.1. Introductory notes......Page 331
9.2. Credit ratings and corporate bonds......Page 332
9.3.1. Structural methodologies......Page 334
9.4. Arbitrage pricing of defaultable bonds......Page 335
9.5. Migration process as a Markov chain......Page 338
9.5.1. Change of real-world probability measure to equivalent T*-forward measure......Page 339
9.6. Estimation of the real world transition probabilities......Page 342
9.7. Term structure of credit spread and model calibration......Page 345
9.8. Migration process under the real-world probability measure......Page 349
9.8.1. Stochastic monotonicities in default times......Page 352
9.8.2. Asymptotic behavior......Page 358
9.9. Exercises......Page 360
10.1. Introductory notes......Page 363
10.2.1. Evolution of forward rate process......Page 364
10.2.2. Evolution of the savings account and short-term interest rate process......Page 366
10.2.3. Evolution of the zero-coupon non-defaultable bond process......Page 367
10.2.4. Conditions on the drift and volatility parameters for non-arbitrage......Page 368
10.3. Hedging strategies for zero coupon bonds......Page 370
10.4. Exercises......Page 372
References......Page 373
A.1. Introductory thoughts......Page 383
A.2. Genesis......Page 384
A.3. The decisive steps......Page 386
A.4. A brief glance towards the flow of research paths......Page 395
B.1. Introduction
......Page 398
B.2. The main theorem
......Page 399
Index......Page 401