Discrete-time Asset Pricing Models in Applied Stochastic Finance

Transcription

1 Discrete-time Asset Pricing Models in Applied Stochastic Finance P.C.G. Vassiliou ) WILEY

2 Table of Contents Preface xi Chapter ^Probability and Random Variables Introductory notes Probability space Conditional probability and independence Random variables Discrete random variables Bernoulli random variables Binomial random variables Geometric random variables Poisson random variables Continuous random variables Exponential random variables Uniform random variables Gamma random variables Normal random variables ll.Lognormal random variables Weibull random variables Expectation and variance of a random variable Jointly distributed random variables Joint probability distribution of functions of random variables Moment generating functions Probability inequalities and limit theorems Multivariate normal distribution 44 Chapter 2. An Introduction to Financial Instruments and Derivatives Introduction Bonds and basic interest rates 50

3 vi Applied Stochastic Finance Simple interest rates Discretely compounded interest rates Continuously compounded interest rate Money-market account Basic interest rates Treasury rate LIBOR rates Time value of money Coupon-bearing bonds and yield-to-maturity Forward contracts Arbitrage Futures contracts Swaps Options European call option European put option American call option American put option Basic problems and assumptions Types of market participants Hedgers Speculators Arbitrageurs Arbitrage relationships between call and put options Exercises 69 Chapter 3. Conditional Expectation and Markov Chains Introduction Conditional expectation: the discrete case Applications of conditional expectations Expectation of the sum of a random number of random variables Expected value of a random number of Bernoulli trials with probability of success being a random variable Number of Bernoulli trials until there are k consecutive successes Conditional variance relationship Variance of the sum of a random number of random variables Properties of the conditional expectation Markov chains Probability distribution in the states of a Markov chain Statistical inference in Markov chains The strong Markov property Classification of states of a Markov chain Periodic Markov chains 104

4 Table of Contents vii Cyclic subclasses Algorithm for the cyclic subclasses Classification of states Asymptotic behavior of irreducible homogenous Markov chains The mean time of first entrance in a state of Markov chain The variance of the time of first visit into a state of a Markov chain Exercises 131 Chapter 4. The No-Arbitrage Binomial Pricing Model Introductory notes Binomial model Stochastic evolution of the asset prices Binomial approximation to the lognormal distribution One-period European call option Two-period European call option Multiperiod binomial model The evolution of the asset prices as a Markov chain Exercises 158 Chapter 5. Martingales Introductory notes Martingales Optional sampling theorem Submartingales, supermartingales and martingales convergence theorem Martingale transforms Uniform integrability and Doob's decomposition Doob decomposition The snell envelope Exercises 190 Chapter 6. Equivalent Martingale Measures, No-Arbitrage and Complete Markets Introductory notes Equivalent martingale measure and the Randon-Nikodym derivative process Finite general markets Uniqueness of arbitrage price Equivalent martingale measures Fundamental theorem of asset pricing Complete markets and martingale representation 222

5 viii Applied Stochastic Finance 6.6. Finding the equivalent martingale measure Exploring the vital equations and conditions Equivalent martingale measures for general finite markets Exercises 238 Chapter 7. American Derivative Securities Introductory notes A three-period American put option Hedging strategy for an American put option The algorithm of the American put option Algorithm of the American put option Pricing of the American put option Trading strategy for hedging Optimal time for the holder to exercise American derivatives in general markets Extending the concept of self-financing strategies Exercises 269 Chapter 8. Fixed-Income Markets and Interest Rates Introductory notes The zero coupon bonds of all maturities Arbitrage-free family of bond prices Interest rate process and the term structure of bond prices The evolution of the interest rate process Binomial model with normally distributed spread of interest rates Binomial model with lognormally distributed spread of interest rates Option arbitrage pricing on zero coupon bonds Valuation of the European put call Hedging the European put option Fixed income derivatives Interest rate swaps Interest rate caps and floors T-period equivalent forward measure Futures contracts Exercises 319 Chapter 9. Credit Risk Introductory notes Credit ratings and corporate bonds Credit risk methodologies Structural methodologies Reduced-form methodologies 327

6 Table of Contents ix 9.4. Arbitrage pricing of defaultable bonds Migration process as a Markov chain Change of real-world probability measure to equivalent T* -forward measure Estimation of the real world transition probabilities Term structure of credit spread and model calibration Migration process under the real-world probability measure Stochastic monotonicities in default times Asymptotic behavior.\ Exercises 352 Chapter 10. The Heath-Jarrow-Morton Model Introductory notes Heath-Jarrow-Morton model ^2.1. Evolution of forward rate process Evolution of the savings account and short-term interest rate process Evolution of the zero-coupon non-defaultable bond process Conditions on the drift and volatility parameters for non-arbitrage Hedging strategies for zero coupon bonds Exercises 364 References 365 Appendices 374 A. Appendix A 375 A.I. 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.I. Introduction 391 B.2. The main theorem 392 Index 395