Tools for Computational Finance

by Rüdiger U. Seydel

Overview

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.

Product Details

ISBN-13:	9781447173380
Publisher:	Springer-Verlag New York, LLC
Publication date:	08/17/2017
Series:	Universitext
Sold by:	Barnes & Noble
Format:	eBook
File size:	8 MB

About the Author

Rüdiger U. Seydel is professor emeritus of numerical analysis. He is the former head of a research group on computational finance at the University of Cologne. He also worked in bifurcation and dynamical systems.

Table of Contents

	Prefaces	V
	Contents	XI
	Notation	XV
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
	Notes and Comments	46
	Exercises	49
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
	Notes and Comments	78
	Exercises	80
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
	Notes and Comments	104
	Exercises	105
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
	Notes and Comments	146
	Exercises	148
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
	Notes and Comments	172
	Exercises	173
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
	Notes and Comments	199
	Exercises	201
	Appendices	203
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
	References	227
	Index	235