ISBN-10:
3540279237
ISBN-13:
9783540279235
Pub. Date:
10/09/2008
Publisher:
Springer-Verlag New York, LLC
Tools for Computational Finance / Edition 3

Tools for Computational Finance / Edition 3

by Rudiger U. Seydel

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Product Details

ISBN-13: 9783540279235
Publisher: Springer-Verlag New York, LLC
Publication date: 10/09/2008
Series: Universitext Series
Edition description: 3rd ed.
Pages: 299
Product dimensions: 6.10(w) x 9.30(h) x 0.70(d)

Table of Contents

PrefacesV
ContentsXI
NotationXV
Chapter 1Modeling Tools for Financial Options1
1.1Options1
1.2Model of the Financial Market7
1.3Numerical Methods10
1.4The Binomial Method12
1.5Risk-Neutral Valuation21
1.6Stochastic Processes24
1.6.1Wiener Process26
1.6.2Stochastic Integral28
1.7Stochastic Differential Equations31
1.7.1Ito Process31
1.7.2Application to the Stock Market34
1.8Ito Lemma and Implications38
1.9Jump Processes42
Notes and Comments46
Exercises49
Chapter 2Generating Random Numbers with Specified Distributions57
2.1Pseudo-Random Numbers57
2.1.1Linear Congruential Generators58
2.1.2Random Vectors59
2.1.3Fibonacci Generators62
2.2Transformed Random Variables63
2.2.1Inversion64
2.2.2Transformation in IR[superscript 1]66
2.2.3Transformation in IR[superscript n]67
2.3Normally Distributed Random Variables68
2.3.1Method of Box and Muller68
2.3.2Variant of Marsaglia69
2.3.3Correlated Random Variables70
2.4Sequences of Numbers with Low Discrepancy72
2.4.1Monte Carlo Integration72
2.4.2Discrepancy73
2.4.3Examples of Low-Discrepancy Sequences76
Notes and Comments78
Exercises80
Chapter 3Numerical Integration of Stochastic Differential Equations85
3.1Approximation Error86
3.2Stochastic Taylor Expansion89
3.3Examples of Numerical Methods92
3.4Intermediate Values95
3.5Monte Carlo Simulation96
3.5.1The Basic Version96
3.5.2Variance Reduction99
Notes and Comments104
Exercises105
Chapter 4Finite Differences and Standard Options109
4.1Preparations110
4.2Foundations of Finite-Difference Methods112
4.2.1Difference Approximation112
4.2.2The Grid113
4.2.3Explicit Method114
4.2.4Stability116
4.2.5Implicit Method119
4.3Crank-Nicolson Method120
4.4Boundary Conditions123
4.5American Options as Free Boundary-Value Problems126
4.5.1Free Boundary-Value Problems126
4.5.2Black-Scholes Inequality130
4.5.3Obstacle Problems130
4.5.4Linear Complementarity for American Put Options133
4.6Computation of American Options134
4.6.1Discretization with Finite Differences135
4.6.2Iterative Solution136
4.6.3Algorithm for Calculating American Options138
4.7On the Accuracy142
Notes and Comments146
Exercises148
Chapter 5Finite-Element Methods151
5.1Weighted Residuals152
5.1.1The Principle of Weighted Residuals153
5.1.2Examples of Weighting Functions154
5.1.3Examples of Basis Functions155
5.2Galerkin Approach with Hat Functions156
5.2.1Hat Functions157
5.2.2A Simple Application159
5.3Application to Standard Options162
5.4Error Estimates166
5.4.1Classical and Weak Solutions166
5.4.2Approximation on Finite-Dimensional Subspaces168
5.4.3Cea's Lemma170
Notes and Comments172
Exercises173
Chapter 6Pricing of Exotic Options175
6.1Exotic Options176
6.2Asian Options178
6.2.1The Payoff178
6.2.2Modeling in the Black-Scholes Framework180
6.2.3Reduction to a One-Dimensional Equation181
6.2.4Discrete Monitoring183
6.3Numerical Aspects186
6.3.1Convection-Diffusion Problems187
6.3.2Von Neumann Stability Analysis189
6.4Upwind Schemes and Other Methods191
6.4.1Upwind Scheme191
6.4.2Dispersion194
6.5High-Resolution Methods195
6.5.1The Lax-Wendroff Method196
6.5.2Total Variation Diminishing197
6.5.3Numerical Dissipation198
Notes and Comments199
Exercises201
Appendices203
A1Financial Derivatives203
A2Essentials of Stochastics206
A3The Black-Scholes Equation210
A4Numerical Methods214
A5Iterative Methods for Ax = b218
A6Function Spaces220
A7Complementary Formula223
References227
Index235

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