Mathematical models of financial derivatives

Cover......Page 1
Springer Finance......Page 2
Mathematical Models of Financial Derivatives, Second Edition......Page 4
ISBN 978-3-540-42288-4......Page 5
Objectives and Audience......Page 7
Guide to the Chapters......Page 8
Final Words on the Book Cover Design......Page 10
Contents......Page 11
1 Introduction to Derivative Instruments......Page 16
Financial Options and Their Trading Strategies......Page 17
Trading Strategies Involving Options......Page 20
Rational Boundaries for Option Values......Page 25
Effects of Dividend Payments......Page 31
Put-Call Parity Relations......Page 33
Foreign Currency Options......Page 34
Values and Prices of Forward Contracts......Page 36
Relation between Forward and Futures Prices......Page 39
Swap Contracts......Page 40
Interest Rate Swaps......Page 41
Currency Swaps......Page 43
Problems......Page 44
2 Financial Economics and Stochastic Calculus......Page 50
Single Period Securities Models......Page 51
Dominant Trading Strategies and Linear Pricing Measures......Page 52
Arbitrage Opportunities and Risk Neutral Probability Measures......Page 58
Valuation of Contingent Claims......Page 63
Principles of Binomial Option Pricing Model......Page 67
Filtrations, Martingales and Multiperiod Models......Page 70
Information Structures and Filtrations......Page 71
Conditional Expectations and Martingales......Page 73
Stopping Times and Stopped Processes......Page 77
Multiperiod Securities Models......Page 79
Multiperiod Binomial Models......Page 84
Asset Price Dynamics and Stochastic Processes......Page 87
Random Walk Models......Page 88
Brownian Processes......Page 91
Stochastic Integrals......Page 94
Ito\'s Lemma and Stochastic Differentials......Page 97
Ito\'s Processes and Feynman-Kac Representation Formula......Page 100
Change of Measure: Radon-Nikodym Derivative and Girsanov\'s Theorem......Page 102
Problems......Page 104
3 Option Pricing Models: Black-Scholes-Merton Formulation......Page 114
Riskless Hedging Principle......Page 116
Dynamic Replication Strategy......Page 119
Risk Neutrality Argument......Page 121
Martingale Pricing Theory......Page 123
Equivalent Martingale Measure and Risk Neutral Valuation......Page 124
Black-Scholes Model Revisited......Page 127
Black-Scholes Pricing Formulas and Their Properties......Page 129
Pricing Formulas for European Options......Page 130
Comparative Statics......Page 136
Options on a Dividend-Paying Asset......Page 142
Futures Options......Page 147
Chooser Options......Page 150
Compound Options......Page 151
Merton\'s Model of Risky Debts......Page 154
Exchange Options......Page 157
Equity Options with Exchange Rate Risk Exposure......Page 159
Beyond the Black-Scholes Pricing Framework......Page 162
Transaction Costs Models......Page 164
Jump-Diffusion Models......Page 166
Implied and Local Volatilities......Page 168
Stochastic Volatility Models......Page 174
Problems......Page 179
4 Path Dependent Options......Page 196
Barrier Options......Page 197
European Down-and-Out Call Options......Page 198
Transition Density Function and First Passage Time Density......Page 203
Options with Double Barriers......Page 210
Lookback Options......Page 216
European Fixed Strike Lookback Options......Page 218
European Floating Strike Lookback Options......Page 220
More Exotic Forms of European Lookback Options......Page 222
Differential Equation Formulation......Page 224
Discretely Monitored Lookback Options......Page 226
Asian Options......Page 227
Partial Differential Equation Formulation......Page 228
Continuously Monitored Geometric Averaging Options......Page 229
Continuously Monitored Arithmetic Averaging Options......Page 232
Put-Call Parity and Fixed-Floating Symmetry Relations......Page 234
Fixed Strike Options with Discrete Geometric Averaging......Page 237
Fixed Strike Options with Discrete Arithmetic Averaging......Page 240
Problems......Page 245
5 American Options......Page 265
American Options on an Asset Paying Dividend Yield......Page 267
Smooth Pasting Condition......Page 269
Optimal Exercise Boundary for an American Call......Page 270
Put-Call Symmetry Relations......Page 274
American Call Options on an Asset Paying Single Dividend......Page 277
One-Dividend and Multidividend American Put Options......Page 281
Linear Complementarity Formulation......Page 284
Optimal Stopping Problem......Page 286
Integral Representation of the Early Exercise Premium......Page 288
American Barrier Options......Page 292
American Lookback Options......Page 294
Analytic Approximation Methods......Page 296
Compound Option Approximation Method......Page 297
Numerical Solution of the Integral Equation......Page 298
Quadratic Approximation Method......Page 301
Options with Voluntary Reset Rights......Page 303
Valuation of the Shout Floor......Page 304
Reset-Strike Put Options......Page 306
Problems......Page 311
6 Numerical Schemes for Pricing Options......Page 326
Binomial Model Revisited......Page 328
Continuous Limits of the Binomial Model......Page 329
Discrete Dividend Models......Page 333
Early Exercise Feature and Callable Feature......Page 335
Trinomial Schemes......Page 336
Forward Shooting Grid Methods......Page 340
Finite Difference Algorithms......Page 345
Construction of Explicit Schemes......Page 346
Implicit Schemes and Their Implementation Issues......Page 350
Front Fixing Method and Point Relaxation Technique......Page 353
Truncation Errors and Order of Convergence......Page 357
Numerical Stability and Oscillation Phenomena......Page 359
Numerical Approximation of Auxiliary Conditions......Page 362
Monte Carlo Simulation......Page 365
Variance Reduction Techniques......Page 368
Low Discrepancy Sequences......Page 371
Valuation of American Options......Page 372
Problems......Page 382
7 Interest Rate Models and Bond Pricing......Page 393
Bond Prices and Interest Rates......Page 394
Bond Prices and Yield Curves......Page 395
Forward Rate Agreement, Bond Forward and Vanilla Swap......Page 396
Forward Rates and Short Rates......Page 399
Bond Prices under Deterministic Interest Rates......Page 401
One-Factor Short Rate Models......Page 402
Short Rate Models and Bond Prices......Page 403
Vasicek Mean Reversion Model......Page 408
Cox-Ingersoll-Ross Square Root Diffusion Model......Page 409
Generalized One-Factor Short Rate Models......Page 411
Calibration to Current Term Structures of Bond Prices......Page 412
Multifactor Interest Rate Models......Page 415
Short Rate/Long Rate Models......Page 416
Stochastic Volatility Models......Page 419
Affine Term Structure Models......Page 420
Heath-Jarrow-Morton Framework......Page 423
Forward Rate Drift Condition......Page 425
Short Rate Processes and Their Markovian Characterization......Page 426
Forward LIBOR Processes under Gaussian HJM Framework......Page 430
Problems......Page 432
8 Interest Rate Derivatives: Bond Options, LIBOR and Swap Products......Page 453
Forward Measure......Page 455
Pricing of Equity Options under Stochastic Interest Rates......Page 458
Futures Process and Futures-Forward Price Spread......Page 460
Options on Discount Bonds and Coupon-Bearing Bonds......Page 462
Range Notes......Page 469
Caps and LIBOR Market Models......Page 472
Pricing of Caps under Gaussian HJM Framework......Page 473
Black Formulas and LIBOR Market Models......Page 474
Swap Products and Swaptions......Page 480
Forward Swap Rates and Swap Measure......Page 481
Approximate Pricing of Swaption under Lognormal LIBOR Market Model......Page 485
Cross-Currency Swaps......Page 489
Problems......Page 497
References......Page 518
Author Index......Page 528
Subject Index......Page 532