Pricing Derivative Securities

✍ Scribed by Thomas W. Epps

• Publisher: World Scientific Publishing Company
• Year: 2007
• Tongue: English
• Leaves: 644
• Edition: 2
• Category: Library

✦ Synopsis

This book presents techniques for valuing derivative securities at a level suitable for practitioners, students in doctoral programs in economics and finance, and those in masters-level programs in financial mathematics and computational finance. It provides the necessary mathematical tools from analysis, probability theory, the theory of stochastic processes, and stochastic calculus, making extensive use of examples. It also covers pricing theory, with emphasis on martingale methods. The chapters are organized around the assumptions made about the dynamics of underlying price processes. Readers begin with simple, discrete-time models that require little mathematical sophistication, proceed to the basic Black Scholes theory, and then advance to continuous-time models with multiple risk sources. The second edition takes account of the major developments in the field since 2000. New topics include the use of simulation to price American-style derivatives, a new one-step approach to pricing options by inverting characteristic functions, and models that allow jumps in volatility and Markov-driven changes in regime. The new chapter on interest-rate derivatives includes extensive coverage of the LIBOR market model and an introduction to the modeling of credit risk. As a supplement to the text, the book contains an accompanying CD-ROM with user-friendly FORTRAN, C++, and VBA program components.

✦ Table of Contents

Contents......Page 6
Preface......Page 14
PART I PRELIMINARIES......Page 18
1 Introduction and Overview......Page 20
1.1.1 Forward Contracts......Page 21
1.1.2 Futures......Page 23
1.1.3 “Vanilla” Options......Page 25
1.1.4 Other Derivative Products......Page 29
1.2 An Overview of Derivatives Pricing......Page 31
1.2.1 Replication: Static and Dynamic......Page 32
1.2.2 Approaches to Valuation when Replication is Possible......Page 33
1.2.3 Markets: Complete and Otherwise......Page 36
1.2.4 Derivatives Pricing in Incomplete Markets......Page 37
2 Mathematical Preparation......Page 38
2.1.1 Order Notation......Page 39
2.1.2 Series Expansions and Finite Sums......Page 40
2.1.3 Measures......Page 42
2.1.4 Measurable Functions......Page 44
2.1.5 Variation and Absolute Continuity of Functions......Page 45
2.1.6 Integration......Page 46
2.1.7 Change of Measure: Radon-Nikodym Theorem......Page 58
2.1.8 Special Functions and Integral Transforms......Page 59
2.2.1 Probability Spaces......Page 64
2.2.2 Random Variables and Their Distributions......Page 66
2.2.3 Mathematical Expectation......Page 72
2.2.4 Radon-Nikodym for Probability Measures......Page 83
2.2.5 Conditional Probability and Expectation......Page 85
2.2.6 Stochastic Convergence......Page 90
2.2.7 Models for Distributions......Page 94
2.2.8 Introduction to Stochastic Processes......Page 100
3.1.1 Definition and Background......Page 110
3.1.2 Essential Properties......Page 111
3.2.1 A Motivating Example......Page 114
3.2.2 Integrals with Respect to Brownian Motions......Page 116
3.2.3 Ito Processes......Page 121
3.3.1 The Result, and Some Intuition......Page 127
3.3.2 Outline of Proof......Page 128
3.3.4 Illustrations......Page 130
3.3.5 Functions of Higher-Dimensional Processes......Page 132
3.3.6 Self-Financing Portfolios in Continuous Time......Page 134
3.4.1 Girsanov’s Theorem and Changes of Measure......Page 135
3.4.2 Representation of Martingales......Page 139
3.4.3 Numeraires, Changes of Numeraire, and Changes of Measure......Page 140
3.5.1 J Processes......Page 142
3.5.2 More General Processes......Page 147
PART II PRICING THEORY......Page 156
4 Dynamics-Free Pricing......Page 158
4.1 Bond Prices and Interest Rates......Page 160
4.1.1 Spot Bond Prices and Rates......Page 161
4.1.2 Forward Bond Prices and Rates......Page 164
4.1.3 Uncertainty in Future Bond Prices and Rates......Page 166
4.2 Forwards and Futures......Page 167
4.2.1 Forward Prices and Values of Forward Contracts......Page 168
4.2.2 Determining Futures Prices......Page 170
4.2.3 Illustrations and Caveats......Page 173
4.2.4 A Preview of Martingale Pricing......Page 175
4.3 Options......Page 176
4.3.1 Payoff Distributions for European Options......Page 177
4.3.2 Put-Call Parity......Page 178
4.3.3 Bounds on Option Prices......Page 182
4.3.4 How Prices Vary with T , X, and St......Page 186
5 Pricing under Bernoulli Dynamics......Page 191
5.1 The Structure of Bernoulli Dynamics......Page 193
5.2 Replication and Binomial Pricing......Page 196
5.3 Interpreting the Binomial Solution......Page 202
5.3.2 The Risk-Neutral or Martingale Interpretation......Page 203
5.4.1 European Stock Options......Page 213
5.4.2 Futures and Futures Options......Page 216
5.4.3 American-Style Derivatives......Page 219
5.4.4 Derivatives on Assets That Pay Dividends......Page 226
5.5.1 Modeling the Dynamics......Page 237
5.5.2 Efficient Calculation......Page 247
5.6 Inferring Trees from Prices of Traded Options......Page 256
5.6.1 Assessing the Implicit Risk-Neutral Distribution of ST......Page 257
5.6.2 Building the Tree......Page 260
5.6.3 Appraisal......Page 263
6 Black-Scholes Dynamics......Page 264
6.1 The Structure of Black-Scholes Dynamics......Page 265
6.2 Approaches to Arbitrage-Free Pricing......Page 267
6.2.1 The Differential-Equation Approach......Page 268
6.2.2 The Equivalent-Martingale Approach......Page 271
6.3.1 Forward Contracts......Page 278
6.3.2 European Options on Primary Assets......Page 279
6.3.3 Extensions of the Black-Scholes Theory......Page 287
6.4 Properties of Black-Scholes Formulas......Page 290
6.4.1 Symmetry and Put-Call Parity......Page 291
6.4.2 Extreme Values and Comparative Statics......Page 292
6.4.3 Implicit Volatility......Page 296
6.4.4 Delta Hedging and Synthetic Options......Page 298
6.4.5 Instantaneous Risks and Expected Returns of European Options......Page 301
6.4.6 Holding-Period Returns for European Options......Page 304
7.1 American Options......Page 309
7.1.1 Calls on Stocks Paying Lump-Sum Dividends......Page 310
7.1.2 Options on Assets Paying Continuous Dividends......Page 313
7.1.3 Indefinitely Lived American Options......Page 325
7.2.1 Options on Options......Page 328
7.2.2 Options with Extra Lives......Page 338
7.3.1 Digital Options......Page 344
7.3.2 Threshold Options......Page 345
7.3.3 “As-You-Like-It” or “Chooser” Options......Page 346
7.3.4 Forward-Start Options......Page 347
7.3.5 Options on the Max or Min......Page 348
7.3.6 Quantos......Page 353
7.4.1 Extrema of Brownian Paths......Page 356
7.4.2 Lookback Options......Page 361
7.4.3 Barrier Options......Page 365
7.4.4 Ladder Options......Page 370
7.4.5 Asian Options......Page 373
8.1.1 Brownian Motion Does Not Fit Underlying Prices......Page 384
8.1.2 Black-Scholes No Longer Fits Option Prices......Page 386
8.2 Price-Dependent Volatility......Page 387
8.2.1 Qualitative Features of Derivatives Prices......Page 388
8.2.2 Two Specific Models......Page 390
8.2.4 Limitations of Price-Dependent Volatility......Page 396
8.2.5 Incorporating Dependence on Past Prices......Page 397
8.3 Stochastic-Volatility Models......Page 399
8.3.1 Nonuniqueness of Arbitrage-Free Prices......Page 400
8.3.2 Specific S.V. Models......Page 405
8.4 Computational Issues......Page 412
8.4.1 Inverting C.f.s......Page 413
8.4.2 Two One-Step Approaches......Page 414
9 Discontinuous Processes......Page 418
9.1 Derivatives with Random Payoff Times......Page 419
9.2.1 Jumps Plus Constant-Volatility Diffusions......Page 426
9.2.2 Nonuniqueness of the Martingale Measure......Page 428
9.2.3 European Options under Jump Dynamics......Page 430
9.2.4 Properties of Jump-Dynamics Option Prices......Page 432
9.2.5 Options Subject to Early Exercise......Page 434
9.3 Jumps Plus Stochastic Volatility......Page 435
9.3.1 The S.V.-Jump Model......Page 436
9.3.2 Further Variations......Page 439
9.4.1 The Variance-Gamma Model......Page 443
9.4.2 The Hyperbolic Model......Page 449
9.4.3 A Levy Process with Finite Levy Measure......Page 453
9.4.4 Modeling Prices as Branching Processes......Page 455
9.4.5 Assessing the Pure-Jump Models......Page 462
9.5 A Markov-Switching Model......Page 463
10.1.1 A Summary of Basic Concepts......Page 474
10.1.2 Spot and Forward Measures......Page 475
10.1.3 A Preview of Things to Come......Page 478
10.2 Spot-Rate Models......Page 479
10.2.1 Bond Prices under Vasicek......Page 480
10.2.2 Bond Prices under Cox, Ingersoll, Ross......Page 485
10.3 A Forward-Rate Model......Page 489
10.3.1 The One-Factor HJM Model......Page 491
10.3.2 Allowing Additional Risk Sources......Page 495
10.3.3 Implementation and Applications......Page 497
10.4 The LIBOR Market Model......Page 515
10.4.1 Deriving Black’s Formulas......Page 517
10.4.2 Applying the Model......Page 521
10.5 Modeling Default Risk......Page 529
10.5.1 Endogenous Risk: The Black-Scholes-Merton Model......Page 531
10.5.2 Exogenous Default Risk......Page 535
PART III COMPUTATIONAL METHODS......Page 542
11 Simulation......Page 544
11.1.1 Uniform Deviates......Page 546
11.1.2 Deviates from Other Distributions......Page 549
11.2.1 Stratified Sampling......Page 554
11.2.2 Importance Sampling......Page 557
11.2.3 Antithetic Variates......Page 558
11.2.4 Control Variates......Page 561
11.2.5 Richardson Extrapolation......Page 563
11.3.1 “Basket” Options......Page 564
11.3.2 European Options under Stochastic Volatility......Page 568
11.3.3 Lookback Options under Stochastic Volatility......Page 570
11.3.4 American-Style Derivatives......Page 571
12 Solving P.D.E.s Numerically......Page 594
12.1.1 Approximating the Derivatives......Page 596
12.1.2 Constructing a Discrete Time/Price Grid......Page 597
12.1.3 Specifying Boundary Conditions......Page 598
12.2.1 The Explicit Method......Page 599
12.2.2 A First-Order Implicit Method......Page 602
12.2.3 Crank-Nicolson’s Second-Order Implicit Method......Page 606
12.3.1 More General P.D.E.s......Page 608
12.3.2 Allowing for Lump-Sum Dividends......Page 611
13 Programs......Page 612
13.1.3 Generating Normal Deviates......Page 613
13.1.5 Testing for Uniformity......Page 614
13.1.7 ICF Test for Normality......Page 615
13.2.2 Expectation of Function of Normal Variate......Page 616
13.2.4 Inversion of Characteristic Function by FFT......Page 617
13.3.1 Binomial Pricing......Page 618
13.4.1 Shell for Black-Scholes with Input/Output......Page 619
13.4.4 Pricing a Compound Option......Page 620
13.4.6 Pricing under Jump Dynamics......Page 621
Bibliography......Page 622
Subject Index......Page 634