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Discrete-time Asset Pricing Models in Applied Stochastic Finance

ISBN: 978-1-84821-158-2
416 pages
March 2010, Wiley-ISTE
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Stochastic finance and financial engineering have been rapidly expanding fields of science over the past four decades, mainly due to the success of sophisticated quantitative methodologies in helping professionals manage financial risks. In recent years, we have witnessed a tremendous acceleration in research efforts aimed at better comprehending, modeling and hedging this kind of risk.

These two volumes aim to provide a foundation course on applied stochastic finance. They are designed for three groups of readers: firstly, students of various backgrounds seeking a core knowledge on the subject of stochastic finance; secondly financial analysts and practitioners in the investment, banking and insurance industries; and finally other professionals who are interested in learning advanced mathematical and stochastic methods, which are basic knowledge in many areas, through finance.

Volume 1 starts with the introduction of the basic financial instruments and the fundamental principles of financial modeling and arbitrage valuation of derivatives. Next, we use the discrete-time binomial model to introduce all relevant concepts. The mathematical simplicity of the binomial model also provides us with the opportunity to introduce and discuss in depth concepts such as conditional expectations and martingales in discrete time. However, we do not expand beyond the needs of the stochastic finance framework. Numerous examples, each highlighted and isolated from the text for easy reference and identification, are included.

The book concludes with the use of the binomial model to introduce interest rate models and the use of the Markov chain model to introduce credit risk. This volume is designed in such a way that, among other uses, makes it useful as an undergraduate course.

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Table of Contents

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

1.5. Expectation and variance of a random variable 24

1.6. Jointly distributed random variables 28

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

2.3. Forward contracts 58

2.4. Futures contracts 60

2.5.Swaps 60

2.6.Options 62

2.7. Types of market participants 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

3.4. Properties of the conditional expectation 81

3.5. Markov chains 85

3.6. Exercises 131

4.1. Introductory notes 137

Chapter 4. The No-Arbitrage Binomial Pricing Model 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

5.7.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

6.4. Fundamental theorem of asset pricing 215

6.5.Completemarkets andmartingale representation 222

6.6. Finding the equivalent martingale measure 228

6.7.Exercises 238

Chapter 7. American Derivative Securities 241

7.1. Introductory notes 241

7.2.A three-periodAmerican put option 242

7.3. Hedging strategy for an American put option 249

7.4.The algorithm of the American put option 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

8.9. Fixed income derivatives 302

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

9.4. Arbitrage pricing of defaultable bonds 327

9.5. Migration process as a Markov chain 330

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

9.9.Exercises 352

Chapter 10. The Heath-Jarrow-Morton Model 355

10.1. Introductory notes 355

10.2. Heath-Jarrow-Morton model 356

10.3. Hedging strategies for zero coupon bonds 362

10.4.Exercises 364

References 365

Appendices 374

Index 395

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