Table Of Contents:
Preface xiii

PART I PRELIMINARIES 1(138)

Introduction and Overview 3(18)

A Tour of Derivatives and Markets 4(10)

Forward Contracts 4(2)

Futures 6(2)

``Vanilla'' Options 8(4)

Other Derivative Products 12(2)

An Overview of Derivatives Pricing 14(7)

Replication: Static and Dynamic 15(1)

Approaches to Valuation when Replication is Possible 16(3)

Markets: Complete and Otherwise 19(1)

Derivatives Pricing in Incomplete Markets 20(1)

Mathematical Preparation 21(72)

Analytical Tools 22(25)

Order Notation 22(1)

Series Expansions and Finite Sums 23(2)

Measures 25(2)

Measurable Functions 27(1)

Variation and Absolute Continuity of Functions 28(1)

Integration 29(12)

Change of Measure: Radon-Nikodym Theorem 41(1)

Special Functions and Integral Transforms 42(5)

Probability 47(46)

Probability Spaces 47(2)

Random Variables and Their Distributions 49(6)

Mathematical Expectation 55(11)

Radon-Nikodym for Probability Measures 66(2)

Conditional Probability and Expectation 68(5)

Stochastic Convergence 73(4)

Models for Distributions 77(6)

Introduction to Stochastic Processes 83(10)

Tools for Continuous-Time Models 93(46)

Wiener Processes 93(4)

Definition and Background 93(1)

Essential Properties 94(3)

Ito Integrals and Processes 97(13)

A Motivating Example 97(2)

Integrals with Respect to Brownian Motions 99(5)

Ito Processes 104(6)

Ito's Formula 110(8)

The Result, and Some Intuition 110(1)

Outline of Proof 111(2)

Functions of Time and an Ito Process 113(1)

Illustrations 113(2)

Functions of Higher-Dimensional Processes 115(2)

Self-Financing Portfolios in Continuous Time 117(1)

Tools for Martingale Pricing 118(7)

Girsanov's Theorem and Changes of Measure 118(4)

Representation of Martingales 122(1)

Numeraires, Changes of Numeraire, and Changes of Measure 123(2)

Tools for Discontinuous Processes 125(14)

J Processes 125(5)

More General Processes 130(9)

PART II PRICING THEORY 139(386)

Dynamics-Free Pricing 141(33)

Bond Prices and Interest Rates 143(7)

Spot Bond Prices and Rates 144(3)

Forward Bond Prices and Rates 147(2)

Uncertainty in Future Bond Prices and Rates 149(1)

Forwards and Futures 150(9)

Forward Prices and Values of Forward Contracts 151(2)

Determining Futures Prices 153(3)

Illustrations and Caveats 156(2)

A Preview of Martingale Pricing 158(1)

Options 159(15)

Payoff Distributions for European Options 160(1)

Put-Call Parity 161(4)

Bounds on Option Prices 165(4)

How Prices Vary with T, X, and St 169(5)

Pricing under Bernoulli Dynamics 174(73)

The Structure of Bernoulli Dynamics 176(3)

Replication and Binomial Pricing 179(6)

Interpreting the Binomial Solution 185(11)

The P.D.E. Interpretation 186(1)

The Risk-Neutral or Martingale Interpretation 186(10)

Specific Applications 196(24)

European Stock Options 196(3)

Futures and Futures Options 199(3)

American-Style Derivatives 202(7)

Derivatives on Assets That Pay Dividends 209(11)

Implementing the Binomial Method 220(19)

Modeling the Dynamics 220(10)

Efficient Calculation 230(9)

Inferring Trees from Prices of Traded Options 239(8)

Assessing the Implicit Risk-Neutral Distribution of S$T 240(3)

Building the Tree 243(3)

Appraisal 246(1)

Black-Scholes Dynamics 247(45)

The Structure of Black-Scholes Dynamics 248(2)

Approaches to Arbitrage-Free Pricing 250(11)

The Differential-Equation Approach 251(3)

The Equivalent-Martingale Approach 254(7)

Applications 261(12)

Forward Contracts 261(1)

European Options on Primary Assets 262(8)

Extensions of the Black-Scholes Theory 270(3)

Properties of Black-Scholes Formulas 273(19)

Symmetry and Put-Call Parity 274(1)

Extreme Values and Comparative Statics 275(4)

Implicit Volatility 279(2)

Delta Hedging and Synthetic Options 281(3)

Instantaneous Risks and Expected Returns of European Options 284(3)

Holding-Period Returns for European Options 287(5)

American Options and ``Exotics'' 292(75)

American Options 292(19)

Calls on Stocks Paying Lump-Sum Dividends 293(3)

Options on Assets Paying Continuous Dividends 296(12)

Indefinitely Lived American Options 308(3)

Compound and Extendable Options 311(16)

Options on Options 311(10)

Options with Extra Lives 321(6)

Other Path-Independent Claims 327(12)

Digital Options 327(1)

Threshold Options 328(1)

``As-You-Like-It'' or ``Chooser'' Options 329(1)

Forward-Start Options 330(1)

Options on the Max or Min 331(5)

Quantos 336(3)

Path-Dependent Options 339(28)

Extrema of Brownian Paths 339(5)

Lookback Options 344(4)

Barrier Options 348(5)

Ladder Options 353(3)

Asian Options 356(11)

Models with Uncertain Volatility 367(34)

Empirical Motivation 367(3)

Brownian Motion Does Not Fit Underlying Prices 367(2)

Black-Scholes No Longer Fits Option Prices 369(1)

Price-Dependent Volatility 370(12)

Qualitative Features of Derivatives Prices 371(2)

Two Specific Models 373(6)

Numerical Methods 379(1)

Limitations of Price-Dependent Volatility 379(1)

Incorporating Dependence on Past Prices 380(2)

Stochastic-Volatility Models 382(13)

Nonuniqueness of Arbitrage-Free Prices 383(5)

Specific S.V. Models 388(7)

Computational Issues 395(6)

Inverting C.f.s 396(1)

Two One-Step Approaches 397(4)

Discontinuous Processes 401(56)

Derivatives with Random Payoff Times 402(7)

Derivatives on Mixed Jump/Diffusions 409(9)

Jumps Plus Constant-Volatility Diffusions 409(2)

Nonuniqueness of the Martingale Measure 411(2)

European Options under Jump Dynamics 413(2)

Properties of Jump-Dynamics Option Prices 415(2)

Options Subject to Early Exercise 417(1)

Jumps Plus Stochastic Volatility 418(8)

The S.V.-Jump Model 419(3)

Further Variations 422(4)

Pure-Jump Models 426(20)

The Variance-Gamma Model 426(6)

The Hyperbolic Model 432(4)

A Levy Process with Finite Levy Measure 436(2)

Modeling Prices as Branching Processes 438(7)

Assessing the Pure-Jump Models 445(1)

A Markov-Switching Model 446(11)

Interest-Rate Dynamics 457(68)

Preliminaries 457(5)

A Summary of Basic Concepts 457(1)

Spot and Forward Measures 458(3)

A Preview of Things to Come 461(1)

Spot-Rate Models 462(10)

Bond Prices under Vasicek 463(5)

Bond Prices under Cox, Ingersoll, Ross 468(4)

A Forward-Rate Model 472(26)

The One-Factor HJM Model 474(4)

Allowing Additional Risk Sources 478(2)

Implementation and Applications 480(18)

The LIBOR Market Model 498(14)

Deriving Black's Formulas 500(4)

Applying the Model 504(8)

Modeling Default Risk 512(13)

Endogenous Risk: The Black-Scholes-Merton Model 514(4)

Exogenous Default Risk 518(7)

PART III COMPUTATIONAL METHODS 525(80)

Simulation 527(50)

Generating Pseudorandom Deviates 529(8)

Uniform Deviates 529(3)

Deviates from Other Distributions 532(5)

Variance-Reduction Techniques 537(10)

Stratified Sampling 537(3)

Importance Sampling 540(1)

Antithetic Variates 541(3)

Control Variates 544(2)

Richardson Extrapolation 546(1)

Applications 547(30)

``Basket'' Options 547(4)

European Options under Stochastic Volatility 551(2)

Lookback Options under Stochastic Volatility 553(1)

American-Style Derivatives 554(23)

Solving P.D.E.s Numerically 577(18)

Setting Up for Solution 579(3)

Approximating the Derivatives 579(1)

Constructing a Discrete Time/Price Grid 580(1)

Specifying Boundary Conditions 581(1)

Obtaining a Solution 582(9)

The Explicit Method 582(3)

A First-Order Implicit Method 585(4)

Crank-Nicolson's Second-Order Implicit Method 589(2)

Comparison of Methods 591(1)

Extensions 591(4)

More General P.D.E.s 591(3)

Allowing for Lump-Sum Dividends 594(1)

Programs 595(10)

Generate and Test Random Deviates 596(3)

Generating Uniform Deviates 596(1)

Generating Poisson Deviates 596(1)

Generating Normal Deviates 596(1)

Testing for Randomness 597(1)

Testing for Uniformity 597(1)

Anderson-Darling Test for Normality 598(1)

ICF Test for Normality 598(1)

General Computation 599(2)

Standard Normal CDF 599(1)

Expectation of Function of Normal Variate 599(1)

Standard Inversion of Characteristic Function 600(1)

Inversion of Characteristic Function by FFT 600(1)

Discrete-Time Pricing 601(1)

Binomial Pricing 601(1)

Solving PDEs under Geometric Brownian Motion 602(1)

Crank-Nicolson Solution of General PDE 602(1)

Continuous-Time Pricing 602(3)

Shell for Black-Scholes with Input/Output 602(1)

Basic Black-Scholes Routine 603(1)

Pricing under the C.E.V. Model 603(1)

Pricing a Compound Option 603(1)

Pricing an Extendable Option 604(1)

Pricing under Jump Dynamics 604(1)
Bibliography 605(12)
Subject Index 617