Tools for computational finance / Rüdiger Seydel.

Format:

• Book

Author/Creator:

• Seydel, R. (Rüdiger), 1947-

Series:

• Universitext

Language:

English

Subjects (All):

• Finance--Mathematical models.

Physical Description:

xvi, 240 pages : illustrations ; 24 cm.

Edition:

Second edition.

Place of Publication:

Berlin ; New York : Springer, [2004]

Summary:

For the second edition the author has enlarged the section on Monte Carlo Simulation, has added more figures, more exercises, more references, more material in the appendices and has written a new section on jump processes.

Contents:

• Chapter 1 Modeling Tools for Financial Options 1
• 1.1 Options 1
• 1.2 Model of the Financial Market 7
• 1.3 Numerical Methods 10
• 1.4 The Binomial Method 12
• 1.5 Risk-Neutral Valuation 21
• 1.6 Stochastic Processes 24
• 1.6.1 Wiener Process 26
• 1.6.2 Stochastic Integral 28
• 1.7 Stochastic Differential Equations 31
• 1.7.1 Ito Process 31
• 1.7.2 Application to the Stock Market 34
• 1.8 Ito Lemma and Implications 38
• 1.9 Jump Processes 42
• Chapter 2 Generating Random Numbers with Specified Distributions 57
• 2.1 Pseudo-Random Numbers 57
• 2.1.1 Linear Congruential Generators 58
• 2.1.2 Random Vectors 59
• 2.1.3 Fibonacci Generators 62
• 2.2 Transformed Random Variables 63
• 2.2.1 Inversion 64
• 2.2.2 Transformation in IR[superscript 1] 66
• 2.2.3 Transformation in IR[superscript n] 67
• 2.3 Normally Distributed Random Variables 68
• 2.3.1 Method of Box and Muller 68
• 2.3.2 Variant of Marsaglia 69
• 2.3.3 Correlated Random Variables 70
• 2.4 Sequences of Numbers with Low Discrepancy 72
• 2.4.1 Monte Carlo Integration 72
• 2.4.2 Discrepancy 73
• 2.4.3 Examples of Low-Discrepancy Sequences 76
• Chapter 3 Numerical Integration of Stochastic Differential Equations 85
• 3.1 Approximation Error 86
• 3.2 Stochastic Taylor Expansion 89
• 3.3 Examples of Numerical Methods 92
• 3.4 Intermediate Values 95
• 3.5 Monte Carlo Simulation 96
• 3.5.1 The Basic Version 96
• 3.5.2 Variance Reduction 99
• Chapter 4 Finite Differences and Standard Options 109
• 4.1 Preparations 110
• 4.2 Foundations of Finite-Difference Methods 112
• 4.2.1 Difference Approximation 112
• 4.2.2 The Grid 113
• 4.2.3 Explicit Method 114
• 4.2.4 Stability 116
• 4.2.5 Implicit Method 119
• 4.3 Crank-Nicolson Method 120
• 4.4 Boundary Conditions 123
• 4.5 American Options as Free Boundary-Value Problems 126
• 4.5.1 Free Boundary-Value Problems 126
• 4.5.2 Black-Scholes Inequality 130
• 4.5.3 Obstacle Problems 130
• 4.5.4 Linear Complementarity for American Put Options 133
• 4.6 Computation of American Options 134
• 4.6.1 Discretization with Finite Differences 135
• 4.6.2 Iterative Solution 136
• 4.6.3 Algorithm for Calculating American Options 138
• 4.7 On the Accuracy 142
• Chapter 5 Finite-Element Methods 151
• 5.1 Weighted Residuals 152
• 5.1.1 The Principle of Weighted Residuals 153
• 5.1.2 Examples of Weighting Functions 154
• 5.1.3 Examples of Basis Functions 155
• 5.2 Galerkin Approach with Hat Functions 156
• 5.2.1 Hat Functions 157
• 5.2.2 A Simple Application 159
• 5.3 Application to Standard Options 162
• 5.4 Error Estimates 166
• 5.4.1 Classical and Weak Solutions 166
• 5.4.2 Approximation on Finite-Dimensional Subspaces 168
• 5.4.3 Cea's Lemma 170
• Chapter 6 Pricing of Exotic Options 175
• 6.1 Exotic Options 176
• 6.2 Asian Options 178
• 6.2.1 The Payoff 178
• 6.2.2 Modeling in the Black-Scholes Framework 180
• 6.2.3 Reduction to a One-Dimensional Equation 181
• 6.2.4 Discrete Monitoring 183
• 6.3 Numerical Aspects 186
• 6.3.1 Convection-Diffusion Problems 187
• 6.3.2 Von Neumann Stability Analysis 189
• 6.4 Upwind Schemes and Other Methods 191
• 6.4.1 Upwind Scheme 191
• 6.4.2 Dispersion 194
• 6.5 High-Resolution Methods 195
• 6.5.1 The Lax-Wendroff Method 196
• 6.5.2 Total Variation Diminishing 197
• 6.5.3 Numerical Dissipation 198
• A1 Financial Derivatives 203
• A2 Essentials of Stochastics 206
• A3 The Black-Scholes Equation 210
• A4 Numerical Methods 214
• A5 Iterative Methods for Ax = b 218
• A6 Function Spaces 220
• A7 Complementary Formula 223.

Notes:

Includes bibliographical references (pages [227]-234) and index.

ISBN:

3540406042

OCLC:

53291172