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Finite Difference Methods in Financial Engineering [2006]

1
Goals of this Book and Global Overview
5
An Introduction to Ordinary Differential Equations
13
An Introduction to Partial Differential Equations
25
Second‐Order Parabolic Differential Equations
38
An Introduction to the Heat Equation in One Dimension
47
An Introduction to the Method of Characteristics
61
An Introduction to the Finite Difference Method
79
An Introduction to the Method of Lines
91
General Theory of the Finite Difference Method
103
Finite Difference Schemes for First‐Order Partial Differential Equations
117
FDM for the One‐Dimensional Convection–Diffusion Equation
123
Exponentially Fitted Finite Difference Schemes
135
Exact Solutions and Explicit Finite Difference Method for One‐Factor Models
147
An Introduction to the Trinomial Method
153
Exponentially Fitted Difference Schemes for Barrier Options
165
Advanced Issues in Barrier and Lookback Option Modelling
175
The Meshless (Meshfree) Method in Financial Engineering
183
Extending the Black–Scholes Model: Jump Processes
193
Finite Difference Schemes for Multidimensional Problems
209
An Introduction to Alternating Direction Implicit and Splitting Methods
223
Advanced Operator Splitting Methods: Fractional Steps
229
Modern Splitting Methods
237
Options with Stochastic Volatility: The Heston Model
249
Finite Difference Methods for Asian Options and Other ‘Mixed’ Problems
257
Multi‐Asset Options
273
Finite Difference Methods for Fixed‐Income Problems
285
Background to Free and Moving Boundary Value Problems
295
Numerical Methods for Free Boundary Value Problems: Front‐Fixing Methods
307
Viscosity Solutions and Penalty Methods for American Option Problems
315
Variational Formulation of American Option Problems
325
Finding the Appropriate Finite Difference Schemes for your Financial Engineering Problem
337
Design and Implementation of First‐Order Problems
353
Moving to Black–Scholes
363
C++ Class Hierarchies for One‐Factor and Two‐Factor Payoffs
375
Appendix 1: An introduction to Integral and Partial Integro‐Differential Equations
393
Appendix 2: An Introduction to the Finite Element Method
409
Bibliography
417
Index
i
Front Matter
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