Applied Stochastic Finance: Number 1: Discrete-time Asset Pricing Models, Vol. 1 Book

Preface xi

Chapter 1 Probability and Random Variables 1

1.1 Introductory notes 1

1.2 Probability space 2

1.3 Conditional probability and independence 8

1.4 Random variables 12

1.4.1 Discrete random variables 14

1.4.2 Bernoulli random variables 15

1.4.3 Binomial random variables 15

1.4.4 Geometric random variables 16

1.4.5 Poisson random variables 17

1.4.6 Continuous random variables 18

1.4.7 Exponential random variables 20

1.4.8 Uniform random variables 21

1.4.9 Gamma random variables 21

1.4.10 Normal random variables 22

1.4.11 Lognormal random variables 23

1.4.12 Weibull random variables 23

1.5 Expectation and variance of a random variable 24

1.6 Jointly distributed random variables 28

1.6.1 Joint probability distribution of functions of random variables 30

1.7 Moment generating functions 32

1.8 Probability inequalities and limit theorems 37

1.9 Multivariate normal distribution 44

Chapter 2 An Introduction to Financial Instruments and Derivatives 49

2.1 Introduction 49

2.2 Bonds and basic interest rates 50

2.2.1 Simple interest rates 51

2.2.2 Discretely compounded interest rates 51

2.2.3 Continuously compounded interest rate 52

2.2.4 Money-market account 53

2.2.5 Basic interest rates 55

2.2.5.1 Treasury rate 55

2.2.5.2 LIBOR rates 55

2.2.6 Time value of money 55

2.2.7 Coupon-bearing bonds and yield-to-maturity 56

2.3 Forward contracts 58

2.3.1 Arbitrage 59

2.4 Futures contracts 60

2.5 Swaps 60

2.6 Options 62

2.6.1 European call option 62

2.6.2 European put option 63

2.6.3 American call option 63

2.6.4 American put option 64

2.6.5 Basic problems and assumptions 65

2.7 Types of market participants 67

2.7.1 Hedgers 67

2.7.2 Speculators 67

2.7.3 Arbitrageurs 67

2.8 Arbitrage relationships between call and put options 67

2.9 Exercises 69

Chapter 3 Conditional Expectation and Markov Chains 71

3.1 Introduction 71

3.2 Conditional expectation: the discrete case 72

3.3 Applications of conditional expectations 75

3.3.1 Expectation of the sum of a random number of random variables 76

3.3.2 Expected value of a random number of Bernoulli trials with probability of success being a random variable 77

3.3.3 Number of Bernoulli trials until there are k consecutive successes 78

3.3.4 Conditional variance relationship 79

3.3.5 Variance of the sum of a random number of random variables 80

3.4 Properties of the conditional expectation 81

3.5 Markov chains 85

3.5.1 Probability distribution in the states of a Markov chain 90

3.5.2 Statistical inference in Markov chains 94

3.5.3 The strong Markov property 97

3.5.4 Classification of states of a Markov chain 100

3.5.5 Periodic Markov chains 104

3.5.5.1 Cyclic subclasses 106

3.5.5.2 Algorithm for the cyclic subclasses 109

3.5.6 Classification of states 112

3.5.7 Asymptotic behavior of irreducible homogenous Markov chains 115

3.5.8 The mean time of first entrance in a state of Markov chain 126

3.5.9 The variance of the time of first visit into a state of a Markov chain 129

3.6 Exercises 131

Chapter 4 The No-Arbitrage Binomial Pricing Model 137

4.1 Introductory notes 137

4.2 Binomial model 138

4.3 Stochastic evolution of the asset prices 141

4.4 Binomial approximation to the lognormal distribution 143

4.5 One-period European call option 145

4.6 Two-period European call option 150

4.7 Multiperiod binomial model 153

4.8 The evolution of the asset prices as a Markov chain 154

4.9 Exercises 158

Chapter 5 Martingales 163

5.1 Introductory notes 163

5.2 Martingales 164

5.3 Optional sampling theorem 169

5.4 Submartingales, supermartingales and martingales convergence theorem 178

5.5 Martingale transforms 182

5.6 Uniform integrability and Doob's decomposition 184

5.6.1 Doob decomposition 186

5.7 The snell envelope 187

5.8 Exercises 190

Chapter 6 Equivalent Martingale Measures, No-Arbitrage and Complete Markets 195

6.1 Introductory notes 195

6.2 Equivalent martingale measure and the Randon-Nikodým derivative process 196

6.3 Finite general markets 204

6.3.1 Uniqueness of arbitrage price 210

6.3.2 Equivalent martingale measures 213

6.4 Fundamental theorem of asset pricing 215

6.5 Complete markets and martingale representation 222

6.6 Finding the equivalent martingale measure 228

6.6.1 Exploring the vital equations and conditions 234

6.6.2 Equivalent martingale measures for general finite markets 237

6.7 Exercises 238

Chapter 7 American Derivative Securities 241

7.1 Introductory notes 241

7.2 A three-period American put option 242

7.3 Hedging strategy for an American put option 249

7.4 The algorithm of the American put option 254

7.4.1 Algorithm of the American put option 254

7.4.1.1 Pricing of the American put option 254

7.4.1.2 Trading strategy for hedging 254

7.5 Optimal time for the holder to exercise 255

7.6 American derivatives in general markets 262

7.7 Extending the concept of self-financing strategies 266

7.8 Exercises 269

Chapter 8 Fixed-Income Markets and Interest Rates 273

8.1 Introductory notes 273

8.2 The zero coupon bonds of all maturities 274

8.3 Arbitrage-free family of bond prices 278

8.4 Interest rate process and the term structure of bond prices 282

8.5 The evolution of the interest rate process 290

8.6 Binomial model with normally distributed spread of interest rates 293

8.7 Binomial model with lognormally distributed spread of interest rates 296

8.8 Option arbitrage pricing on zero coupon bonds 298

8.8.1 Valuation of the European put call 298

8.8.2 Hedging the European put option 300

8.9 Fixed income derivatives 302

8.9.1 Interest rate swaps 304

8.9.2 Interest rate caps and floors 307

8.10 T-period equivalent forward measure 308

8.11 Futures contracts 317

8.12 Exercises 319

Chapter 9 Credit Risk 323

9.1 Introductory notes 323

9.2 Credit ratings and corporate bonds 324

9.3 Credit risk methodologies 326

9.3.1 Structural methodologies 326

9.3.2 Reduced-form methodologies 327

9.4 Arbitrage pricing of defaultable bonds 327

9.5 Migration process as a Markov chain 330

9.5.1 Change of real-world probability measure to equivalent T^*-forward measure 331

9.6 Estimation of the real world transition probabilities 334

9.7 Term structure of credit spread and model calibration 337

9.8 Migration process under the real-world probability measure 341

9.8.1 Stochastic monotonicities in default times 344

9.8.2 Asymptotic behavior 350

9.9 Exercises 352

Chapter 10 The Health-Jarrow-Morton Model 355

10.1 Introductory notes 355

10.2 Heath-Jarrow-Morton model 356

10.2.1 Evolution of forward rate process 356

10.2.2 Evolution of the savings account and short-term interest rate process 358

10.2.3 Evolution of the zero-coupon non-defaultable bond process 359

10.2.4 Conditions on the drift and volatility parameters for non-arbitrage 360

10.3 Hedging strategies for zero coupon bonds 362

10.4 Exercises 364

References 365

Appendices 374

A Appendix A 375

A.1 Introductory thoughts 375

A.2 Genesis 376

A.3 The decisive steps 378

A.4 A brief glance towards the flow of research paths 387

B Appendix B 391

B.1 Introduction 391

B.2 The main theorem 392

Index 395