Discrete-time Asset Pricing Models in Applied Stochastic Finance: Vassiliou/Discrete-time Asset Pricing Models in Applied Stochastic Finance

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. Content: Chapter 1 Probability and Random Variables (pages 1–48): P?C.G. Vassiliou Chapter 2 An Introduction to Financial Instruments and Derivatives (pages 49–70): P?C.G. Vassiliou Chapter 3 Conditional Expectation and Markov Chains (pages 71–136): P?C.G. Vassiliou Chapter 4 The No?Arbitrage Binomial Pricing Model (pages 137–162): P?C.G. Vassiliou Chapter 5 Martingales (pages 163–194): P?C.G. Vassiliou Chapter 6 Equivalent Martingale Measures, No?Arbitrage and Complete Markets (pages 195–240): P?C.G. Vassiliou Chapter 7 American Derivative Securities (pages 241–272): P?C.G. Vassiliou Chapter 8 Fixed?Income Markets and Interest Rates (pages 273–322): P?C.G. Vassiliou Chapter 9 Credit Risk (pages 323–354): P?C.G. Vassiliou Chapter 10 The Heath?Jarrow?Morton Model (pages 355–364): P?C.G. Vassiliou 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 Title Page 3 Copyright 4 Contents 5 Preface 10 Chapter 1. Probability and Random Variables 13 1.1. Introductory notes 13 1.2. Probability space 14 1.3. Conditional probability and independence 20 1.4. Random variables 24 1.4.1. Discrete random variables 26 1.4.2. Bernoulli random variables 27 1.4.3. Binomial random variables 27 1.4.4. Geometric random variables 28 1.4.5. Poisson random variables 29 1.4.6. Continuous random variables 30 1.4.7. Exponential random variables 32 1.4.8. Uniform random variables 33 1.4.9. Gamma random variables 33 1.4.10. Normal random variables 34 1.4.11. Lognormal random variables 35 1.4.12. Weibull random variables 35 1.5. Expectation and variance of a random variable 36 1.6. Jointly distributed random variables 40 1.6.1. Joint probability distribution of functions of random variables 42 1.7. Moment generating functions 44 1.8. Probability inequalities and limit theorems 49 1.9. Multivariate normal distribution 56 Chapter 2. An Introduction to Financial Instruments and Derivatives 60 2.1. Introduction 60 2.2. Bonds and basic interest rates 61 2.2.1. Simple interest rates 62 2.2.2. Discretely compounded interest rates 62 2.2.3. Continuously compounded interest rate 63 2.2.4. Money-market account 64 2.2.5. Basic interest rates 66 2.2.5.1. Treasury rate 66 2.2.5.2. LIBOR rates 66 2.2.6. Time value of money 66 2.2.7. Coupon-bearing bonds and yield-to-maturity 67 2.3. Forward contracts 69 2.3.1. Arbitrage 70 2.4. Futures contracts 71 2.5. Swaps 71 2.6. Options 73 2.6.1. European call option 73 2.6.2. European put option 74 2.6.3. American call option 74 2.6.4. American put option 75 2.6.5. Basic problems and assumptions 76 2.7. Types of market participants 78 2.7.1. Hedgers 78 2.7.2. Speculators 78 2.7.3. Arbitrageurs 78 2.8. Arbitrage relationships between call and put options 78 2.9. Exercises 80 Chapter 3. Conditional Expectation and Markov Chains 82 3.1. Introduction 82 3.2. Conditional expectation: the discrete case 83 3.3. Applications of conditional expectations 86 3.3.1. Expectation of the sum of a random number of random variables 87 3.3.2. Expected value of a random number of Bernoulli trials with probability of success being a random variable 88 3.3.3. Number of Bernoulli trials until there are k consecutive successes 89 3.3.4. Conditional variance relationship 90 3.3.5. Variance of the sum of a random number of random variables 91 3.4. Properties of the conditional expectation 92 3.5. Markov chains 96 3.5.1. Probability distribution in the states of a Markov chain 101 3.5.2. Statistical inference in Markov chains 105 3.5.3. The strong Markov property 108 3.5.4. Classification of states of a Markov chain 111 3.5.5. Periodic Markov chains 115 3.5.5.1. Cyclic subclasses 117 3.5.5.2. Algorithm for the cyclic subclasses 120 3.5.6. Classification of states 123 3.5.7. Asymptotic behavior of irreducible homogenous Markov chains 126 3.5.8. The mean time of first entrance in a state of Markov chain 137 3.5.9. The variance of the time of first visit into a state of a Markov chain 140 3.6. Exercises 142 Chapter 4. The No- Arbitrage Binomial Pricing Model 148 4.1. Introductory notes 148 4.2. Binomial model 149 4.3. Stochastic evolution of the asset prices 152 4.4. Binomial approximation to the lognormal distribution 154 4.5. One-period European call option 156 4.6. Two-period European call option 161 4.7. Multiperiod binomial model 164 4.8. The evolution of the asset prices as a Markov chain 165 4.9. Exercises 169 Chapter 5. Martingales 173 5.1. Introductory notes 173 5.2. Martingales 174 5.3. Optional sampling theorem 179 5.4. Submartingales, supermartingales and martingales convergence theorem 188 5.5. Martingale transforms 192 5.6. Uniform integrability and Doob’s decomposition 194 5.7. The snell envelope 197 5.8. Exercises 200 Chapter 6. Equivalent Martingale Measures, No-Arbitrage and Complete Markets 205 6.1. Introductory notes 205 6.2. Equivalent martingale measure and the Randon-Nikodým derivative process 206 6.3. Finite general markets 214 6.3.1. Uniqueness of arbitrage price 220 6.3.2. Equivalent martingale measures 223 6.4. Fundamental theorem of asset pricing 225 6.5. Complete markets and martingale representation 232 6.6. Finding the equivalent martingale measure 238 6.6.1. Exploring the vital equations and conditions 244 6.6.2. Equivalent martingale measures for general finite markets 247 6.7. Exercises 248 Chapter 7. American Derivative Securities 250 7.1. Introductory notes 250 7.2. A three-period American put option 251 7.3. Hedging strategy for an American put option 258 7.4. The algorithm of the American put option 263 7.4.1. Algorithm of the American put option 263 7.4.1.1. Pricing of the American put option 263 7.4.1.2. Trading strategy for hedging 263 7.5. Optimal time for the holder to exercise 264 7.6. American derivatives in general markets 271 7.7. Extending the concept of self-financing strategies 275 7.8. Exercises 278 Chapter 8. Fixed-Income Markets and Interest Rates 282 8.1. Introductory notes 282 8.2. The zero coupon bonds of all maturities 283 8.3. Arbitrage-free family of bond prices 287 8.4. Interest rate process and the term structure of bond prices 291 8.5. The evolution of the interest rate process 299 8.6. Binomial model with normally distributed spread of interest rates 302 8.7. Binomial model with lognormally distributed spread of interest rates 305 8.8. Option arbitrage pricing on zero coupon bonds 307 8.8.1. Valuation of the European put call 307 8.8.2. Hedging the European put option 309 8.9. Fixed income derivatives 311 8.9.1. Interest rate swaps 313 8.9.2. Interest rate caps and floors 316 8.10. T-period equivalent forward measure 317 8.11. Futures contracts 326 8.12. Exercises 328 Chapter 9. Credit Risk 331 9.1. Introductory notes 331 9.2. Credit ratings and corporate bonds 332 9.3. Credit risk methodologies 334 9.3.1. Structural methodologies 334 9.3.2. Reduced-form methodologies 335 9.4. Arbitrage pricing of defaultable bonds 335 9.5. Migration process as a Markov chain 338 9.5.1. Change of real-world probability measure to equivalent T*-forward measure 339 9.6. Estimation of the real world transition probabilities 342 9.7. Term structure of credit spread and model calibration 345 9.8. Migration process under the real-world probability measure 349 9.8.1. Stochastic monotonicities in default times 352 9.8.2. Asymptotic behavior 358 9.9. Exercises 360 Chapter 10. The Heath-Jarrow-Morton Model 363 10.1. Introductory notes 363 10.2. Heath-Jarrow-Morton model 364 10.2.1. Evolution of forward rate process 364 10.2.2. Evolution of the savings account and short-term interest rate process 366 10.2.3. Evolution of the zero-coupon non-defaultable bond process 367 10.2.4. Conditions on the drift and volatility parameters for non-arbitrage 368 10.3. Hedging strategies for zero coupon bonds 370 10.4. Exercises 372 References 373 Appendix A: The Evolution of Stochastic Mathematics that Changed the Financial World 383 A.1. Introductory thoughts 383 A.2. Genesis 384 A.3. The decisive steps 386 A.4. A brief glance towards the flow of research paths 395 B. Appendix B 398 B.1. Introduction 398 B.2. The main theorem 399 Index 401

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.