Derivative securities and difference methods

Library	Item Barcode	Call Number	Material Type	Item Category 1	Status
Searching... PSZ JB	30000010113763	HG6024.A3 Z48 2004	Open Access Book	Book	Searching... Unknown
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This book studies pricing financial derivatives with a partial differential equation approach. The treatment is mathematically rigorous and covers a variety of topics in finance including forward and futures contracts, the Black-Scholes model, European and American type options, free boundary problems, lookback options, interest rate models, interest rate derivatives, swaps, caps, floors, and collars. Each chapter concludes with exercises.

Part I Partial Differential Equations in Finance
1 Introduction	p. 3
1.1 Assets	p. 3
1.2 Derivative Securities	p. 8
1.2.1 Forward and Futures Contracts	p. 9
1.2.2 Options	p. 10
1.2.3 Interest Rate Derivatives	p. 13
1.2.4 Factors Affecting Derivative Prices	p. 13
1.2.5 Functions of Derivative Securities	p. 14
Problems	p. 15
2 Basic Options	p. 17
2.1 Asset Price Model and Ito's Lemma	p. 17
2.1.1 A Model for Asset Prices	p. 17
2.1.2 Itô's Lemma	p. 20
2.1.3 Expectation and Variance of Lognormal Random Variables	p. 22
2.2 Derivation of the Black-Scholes Equation	p. 25
2.2.1 Arbitrage Arguments	p. 25
2.2.2 The Black-Scholes Equation	p. 26
2.2.3 Final Conditions for the Black-Scholes Equation	p. 29
2.2.4 Hedging and Greeks	p. 30
2.3 Two Transformations on the Black-Scholes Equation	p. 32
2.3.1 Converting the Black-Scholes Equation into a Heat Equation	p. 32
2.3.2 Transforming the Black-Scholes Equation into an Equation Defined on a Finite Domain	p. 35
2.4 Solutions of European Options	p. 39
2.4.1 The Solutions of Parabolic Equations	p. 39
2.4.2 Solutions of the Black-Scholes Equation	p. 42
2.4.3 Prices of Forward Contracts and Delivery Prices	p. 44
2.4.4 Derivation of the Black-Scholes Formulae	p. 44
2.4.5 Put-Call Parity Relation	p. 48
2.4.6 An Explanation in Terms of Probability	p. 49
2.5 American Option Problems as LC Problems	p. 51
2.5.1 Constraints on American Options	p. 51
2.5.2 Formulation of the Linear Complementarity Problem in (S,t)- Plane	p. 52
2.5.3 Formulation of the Linear Complementarity Problem in (x. \bar {{\tau}} )-Plane	p. 58
2.5.4 Formulation of the Linear Complementarity Problem on a Finite Domain	p. 60
2.5.5 More General Form of the Linear Complementarity Problems	p. 61
2.6 American Option Problems as FBPs	p. 62
2.6.1 Free Boundaries	p. 62
2.6.2 Free-Boundary Problems	p. 67
2.6.3 Put-Call Symmetry Relations	p. 70
2.7 Equations for Some Greeks	p. 75
2.8 Perpetual Options	p. 77
2.9 General Equations for Derivatives	p. 78
2.9.1 Models for Random Variables	p. 79
2.9.2 Generalization of Itô's Lemma	p. 81
2.9.3 Derivation of Equations for Financial Derivatives	p. 82
2.9.4 Three Types of State Variables	p. 84
2.9.5 Uniqueness of Solutions	p. 85
2.10 Jump Conditions	p. 90
2.10.1 Hyperbolic Equations with a Dirac Delta Function	p. 90
2.10.2 Jump Conditions for Options with Discrete Dividends and Discrete Sampling	p. 91
2.11 More Arbitrage Theory	p. 92
2.11.1 Three Conclusions and Some Portfolios	p. 92
2.11.2 Bounds of Option Prices	p. 94
2.11.3 Relations Between Call and Put Prices	p. 97
Problems	p. 100
3 Exotic Options	p. 113
3.1 Introduction	p. 113
3.2 Barrier Options	p. 114
3.2.1 Knock-out and Knock-in Options	p. 114
3.2.2 Closed-Form Solutions of Some European Barrier Options	p. 115
3.2.3 Formulation of American Barrier Options	p. 121
3.2.4 Parisian Options	p. 122
3.3 Asian Options	p. 126
3.3.1 Average Strike, Average Price and Double Average Options	p. 126
3.3.2 Continuously and Discretely Sampled Arithmetic Averages	p. 127
3.3.3 Derivation of Equations	p. 129
3.3.4 Reducing to One-Dimensional Problems	p. 130
3.3.5 Jump Conditions	p. 133
3.3.6 American Asian Options	p. 133
3.3.7 Some Examples	p. 136
3.4 Lookback Options	p. 138
3.4.1 Equations for Lookback Options	p. 138
3.4.2 Reducing to One-Dimensional Problems	p. 143
3.4.3 Closed-Form Solutions for European Lookback Options	p. 145
3.4.4 American Options Formulated as Free-Boundary Problems	p. 152
3.4.5 A Closed-Form Solution for a Perpetual American Lookback Option	p. 154
3.4.6 Lookback-Asian Options	p. 156
3.4.7 Some Examples	p. 159
3.5 Multi-Asset Options	p. 161
3.5.1 Equations for Multi-Asset Options and Green's Formula	p. 161
3.5.2 Exchange Options	p. 166
3.5.3 Options on the Extremum of Several Assets	p. 168
3.5.4 Formulation of Multi-Asset Option Problems on a Finite Domain	p. 178
3.6 Some Other Exotic Options	p. 182
3.6.1 Binary Options	p. 183
3.6.2 Forward Start Options (Delayed Strike Options)	p. 183
3.6.3 Compound Options	p. 184
3.6.4 Chooser Options	p. 190
Problems	p. 191
4 Interest Rate Derivative Securities	p. 205
4.1 Introduction	p. 205
4.2 Bonds	p. 208
4.2.1 Bond Values for Deterministic Spot Rates	p. 208
4.2.2 Bond Equations for Random Spot Rates	p. 210
4.3 Some Explicit Solutions of Bond Equations	p. 211
4.3.1 Analytic Solutions for the Vasicek and Cox-Ingersoll-Ross Models	p. 212
4.3.2 Explicit Solutions for the Ho-Lee and Hull-White Models	p. 218
4.4 Inverse Problem on the Market Price of Risk	p. 220
4.5 Application of Bond Equations	p. 223
4.5.1 Bond Options and Bond Futures Contract Options	p. 223
4.5.2 Interest Rate Swaps and Swaptions	p. 226
4.5.3 Interest Rate Caps, Floors and Collars	p. 233
4.6 Multi-Factor Interest Rate Models	p. 235
4.6.1 Brief Description of Several Multi-Factor Interest Rate Models	p. 235
4.6.2 Reducing the Randomness of a Zero-Coupon Bond Curve to That of a Few Zero-Coupon Bonds	p. 237
4.6.3 A Three-Factor Interest Rate Model and the Equation for Interest Rate Derivatives	p. 241
4.7 Two-Factor Convertible Bonds	p. 248
Problems	p. 254
Part II Numerical Methods for Derivative Securities
5 Basic Numerical Methods	p. 267
5.1 Approximations	p. 267
5.1.1 Interpolation	p. 267
5.1.2 Approximation of Partial Derivatives	p. 271
5.1.3 Approximate Integration	p. 274
5.1.4 Least Squares Approximation	p. 276
5.2 Solution of Systems and Eigenvalue Problems	p. 277
5.2.1 LU Decomposition of Linear Systems	p. 277
5.2.2 Iteration Methods for Linear Systems	p. 280
5.2.3 Iteration Methods for Nonlinear Systems	p. 282
5.2.4 Obtaining Eigenvalues and Eigenvectors	p. 287
5.3 Finite-Difference Methods	p. 292
5.4 Stability and Convergence Analysis	p. 302
5.4.1 Stability	p. 302
5.4.2 Convergence	p. 311
5.5 Extrapolation of Numerical Solutions	p. 314
5.6 Determination of Parameters in Models	p. 318
5.6.1 Constant Variances and Covariances	p. 319
5.6.2 Variable Parameters	p. 321
Problems	p. 323
Projects	p. 328
6 Initial-Boundary Value and LC Problems	p. 331
6.1 Explicit Methods _	p. 331
6.1.1 Pricing European Options by Using \bar {{V}} , ¿, ¿ or u, x, \bar {{\tau}} Variables	p. 331
6.1.2 Projected Methods for LC Problems	p. 335
6.1.3 Binomial and Trinomial Methods	p. 338
6.1.4 Relations Between the Lattice Methods and the Explicit Finite-Difference Methods	p. 346
6.1.5 Examples of Unstable Schemes	p. 349
6.2 Implicit Methods	p. 349
6.2.1 Pricing European Options by Using \bar {{V}} , ¿, ¿ Variables	p. 349
6.2.2 European Options with Discrete Dividends and Asian and Lookback Options with Discrete Sampling	p. 351
6.2.3 Projected Direct Methods for the LC Problem	p. 354
6.2.4 Projected Iteration Methods for the LC Problem	p. 357
6.2.5 Comparison with Explicit Methods	p. 359
6.3 Singularity-Separating Method	p. 360
6.3.1 Barrier Options	p. 361
6.3.2 European Vanilla Options with Variable Volatilities	p. 365
6.3.3 Bermudan Options	p. 370
6.3.4 European Parisian Options	p. 376
6.3.5 European Average Price Options	p. 378
6.3.6 European Two-Factor Options	p. 381
6.3.7 Two-Factor Convertible Bonds with D 0 = 0	p. 391
6.4 Pseudo-Spectral Methods	p. 391
Problems	p. 396
Projects	p. 401
7 Free-Boundary Problems	p. 405
7.1 SSM for Free-Boundary Problems	p. 406
7.1.1 One-Dimensional Cases	p. 406
7.1.2 Two-Dimensional Cases	p. 411
7.2 Implicit Finite-Difference Methods	p. 422
7.2.1 Solution of One-Dimensional Problems	p. 422
7.2.2 Solution of Greeks	p. 427
7.2.3 Numerical Results of Vanilla Options and Comparison	p. 429
7.2.4 Solution and Numerical Results of Exotic Options	p. 437
7.2.5 Solution of Two-Dimensional Problems	p. 445
7.2.6 Numerical Results of Two-Factor Options	p. 449
7.3 Pseudo-Spectral Methods	p. 456
7.3.1 The Description of the Pseudo-Spectral Methods for Two-Factor Convertible Bonds	p. 456
7.3.2 Numerical Results of Two-Factor Convertible Bonds	p. 462
Problems	p. 467
Projects	p. 471
8 Interest Rate Modelling	p. 473
8.1 Inverse Problems	p. 473
8.1.1 Another Formulation of the Inverse Problem	p. 473
8.1.2 Numerical Methods for the Inverse Problem	p. 478
8.1.3 Numerical Results on Market Prices of Risk	p. 481
8.2 Numerical Results of One-Factor Models	p. 484
8.3 Pricing Derivatives with Multi-Factor Models	p. 490
8.3.1 Determining Models from the Market Data	p. 490
8.3.2 Numerical Methods and Results	p. 494
Problems	p. 498
Projects	p. 500
References	p. 503
Index	p. 509

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