Goals of this Book and Global Overview / 0: |
What is this book? / 0.1: |
Why has this book been written? / 0.2: |
For whom is this book intended? / 0.3: |
Why should I read this book? / 0.4: |
The structure of this book / 0.5: |
What this book does not cover / 0.6: |
Contact, feedback and more information / 0.7: |
The Continuous Theory of Partial Differential Equations / Part I: |
An Introduction to Ordinary Differential Equations / 1: |
Introduction and objectives / 1.1: |
Two-point boundary value problem / 1.2: |
Linear boundary value problems / 1.3: |
Initial value problems / 1.4: |
Some special cases / 1.5: |
Summary and conclusions / 1.6: |
An Introduction to Partial Differential Equations / 2: |
Partial differential equations / 2.1: |
Specialisations / 2.3: |
Parabolic partial differential equations / 2.4: |
Hyperbolic equations / 2.5: |
Systems of equations / 2.6: |
Equations containing integrals / 2.7: |
Second-Order Parabolic Differential Equations / 2.8: |
Linear parabolic equations / 3.1: |
The continuous problem / 3.3: |
The maximum principle for parabolic equations / 3.4: |
A special case: one-factor generalised Black-Scholes models / 3.5: |
Fundamental solution and the Green's function / 3.6: |
Integral representation of the solution of parabolic PDEs / 3.7: |
Parabolic equations in one space dimension / 3.8: |
An Introduction to the Heat Equation in One Dimension / 3.9: |
Motivation and background / 4.1: |
The heat equation and financial engineering / 4.3: |
The separation of variables technique / 4.4: |
Transformation techniques for the heat equation / 4.5: |
An Introduction to the Method of Characteristics / 4.6: |
First-order hyperbolic equations / 5.1: |
Second-order hyperbolic equations / 5.3: |
Applications to financial engineering / 5.4: |
Propagation of discontinuities / 5.5: |
Finite Difference Methods: The Fundamentals / 5.7: |
An Introduction to the Finite Difference Method / 6: |
Fundamentals of numerical differentiation / 6.1: |
Caveat: accuracy and round-off errors / 6.3: |
Where are divided differences used in instrument pricing? / 6.4: |
Nonlinear initial value problems / 6.5: |
Scalar initial value problems / 6.7: |
An Introduction to the Method of Lines / 6.8: |
Classifying semi-discretisation methods / 7.1: |
Semi-discretisation in space using FDM / 7.3: |
Numerical approximation of first-order systems / 7.4: |
General Theory of the Finite Difference Method / 7.5: |
Some fundamental concepts / 8.1: |
Stability and the Fourier transform / 8.3: |
The discrete Fourier transform / 8.4: |
Stability for initial boundary value problems / 8.5: |
Finite Difference Schemes for First-Order Partial Differential Equations / 8.6: |
Scoping the problem / 9.1: |
Why first-order equations are different: Essential difficulties / 9.3: |
A simple explicit scheme / 9.4: |
Some common schemes for initial value problems / 9.5: |
Some common schemes for initial boundary value problems / 9.6: |
Monotone and positive-type schemes / 9.7: |
Extensions, generalisations and other applications / 9.8: |
FDM for the One-Dimensional Convection-Diffusion Equation / 9.9: |
Approximation of derivatives on the boundaries / 10.1: |
Time-dependent convection-diffusion equations / 10.3: |
Fully discrete schemes / 10.4: |
Specifying initial and boundary conditions / 10.5: |
Semi-discretisation in space / 10.6: |
Semi-discretisation in time / 10.7: |
Exponentially Fitted Finite Difference Schemes / 10.8: |
Motivating exponential fitting / 11.1: |
Exponential fitting and time-dependent convection-diffusion / 11.3: |
Stability and convergence analysis / 11.4: |
Approximating the derivative of the solution / 11.5: |
Special limiting cases / 11.6: |
Applying FDM to One-Factor Instrument Pricing / 11.7: |
Exact Solutions and Explicit Finite Difference Method for One-Factor Models / 12: |
Exact solutions and benchmark cases / 12.1: |
Perturbation analysis and risk engines / 12.3: |
The trinomial method: Preview / 12.4: |
Using exponential fitting with explicit time marching / 12.5: |
Approximating the Greeks / 12.6: |
Appendix: the formula for Vega / 12.7: |
An Introduction to the Trinomial Method / 13: |
Motivating the trinomial method / 13.1: |
Trinomial method: Comparisons with other methods / 13.3: |
The trinomial method for barrier options / 13.4: |
Exponentially Fitted Difference Schemes for Barrier Options / 13.5: |
What are barrier options? / 14.1: |
Initial boundary value problems for barrier options / 14.3: |
Using exponential fitting for barrier options / 14.4: |
Time-dependent volatility / 14.5: |
Some other kinds of exotic options / 14.6: |
Comparisons with exact solutions / 14.7: |
Other schemes and approximations / 14.8: |
Extensions to the model / 14.9: |
Advanced Issues in Barrier and Lookback Option Modelling / 14.10: |
Kinds of boundaries and boundary conditions / 15.1: |
Discrete and continuous monitoring / 15.3: |
Continuity corrections for discrete barrier options / 15.4: |
Complex barrier options / 15.5: |
The Meshless (Meshfree) Method in Financial Engineering / 15.6: |
Motivating the meshless method / 16.1: |
An introduction to radial basis functions / 16.3: |
Semi-discretisations and convection-diffusion equations / 16.4: |
Applications of the one-factor Black-Scholes equation / 16.5: |
Advantages and disadvantages of meshless / 16.6: |
Extending the Black-Scholes Model: Jump Processes / 16.7: |
Jump-diffusion processes / 17.1: |
Partial integro-differential equations and financial applications / 17.3: |
Numerical solution of PIDE: Preliminaries / 17.4: |
Techniques for the numerical solution of PIDEs / 17.5: |
Implicit and explicit methods / 17.6: |
Implicit-explicit Runge-Kutta methods / 17.7: |
Using operator splitting / 17.8: |
Splitting and predictor-corrector methods / 17.9: |
FDM for Multidimensional Problems / 17.10: |
Finite Difference Schemes for Multidimensional Problems / 18: |
Elliptic equations / 18.1: |
Diffusion and heat equations / 18.3: |
Advection equation in two dimensions / 18.4: |
Convection-diffusion equation / 18.5: |
An Introduction to Alternating Direction Implicit and Splitting Methods / 18.6: |
What is ADI, really? / 19.1: |
Improvements on the basic ADI scheme / 19.3: |
ADI for first-order hyperbolic equations / 19.4: |
ADI classico and three-dimensional problems / 19.5: |
The Hopscotch method / 19.6: |
Boundary conditions / 19.7: |
Advanced Operator Splitting Methods: Fractional Steps / 19.8: |
Initial examples / 20.1: |
Problems with mixed derivatives / 20.3: |
Predictor-corrector methods (approximation correctors) / 20.4: |
Partial integro-differential equations / 20.5: |
More general results / 20.6: |
Modern Splitting Methods / 20.7: |
A different kind of splitting: The IMEX schemes / 21.1: |
Applicability of IMEX schemes to Asian option pricing / 21.4: |
Applying FDM to Multi-Factor Instrument Pricing / 21.5: |
Options with Stochastic Volatility: The Heston Model / 22: |
An introduction to Ornstein-Uhlenbeck processes / 22.1: |
Stochastic differential equations and the Heston model / 22.3: |
Using finite difference schemes: Prologue / 22.4: |
A detailed example / 22.6: |
Finite Difference Methods for Asian Options and Other 'Mixed' Problems / 22.7: |
An introduction to Asian options / 23.1: |
My first PDE formulation / 23.3: |
Using operator splitting methods / 23.4: |
Cheyette interest models / 23.5: |
New developments / 23.6: |
Multi-Asset Options / 23.7: |
A taxonomy of multi-asset options / 24.1: |
Common framework for multi-asset options / 24.3: |
An overview of finite difference schemes for multi-asset problems / 24.4: |
Numerical solution of elliptic equations / 24.5: |
Solving multi-asset Black-Scholes equations / 24.6: |
Special guidelines and caveats / 24.7: |
Finite Difference Methods for Fixed-Income Problems / 24.8: |
An introduction to interest rate modelling / 25.1: |
Single-factor models / 25.3: |
Some specific stochastic models / 25.4: |
An introduction to multidimensional models / 25.5: |
The thorny issue of boundary conditions / 25.6: |
Introduction to approximate methods for interest rate models / 25.7: |
Free and Moving Boundary Value Problems / 25.8: |
Background to Free and Moving Boundary Value Problems / 26: |
Notation and definitions / 26.1: |
Some preliminary examples / 26.3: |
Solutions in financial engineering: A preview / 26.4: |
Numerical Methods for Free Boundary Value Problems: Front-Fixing Methods / 26.5: |
An introduction to front-fixing methods / 27.1: |
A crash course on partial derivatives / 27.3: |
Functions and implicit forms / 27.4: |
Front fixing for the heat equation / 27.5: |
Front fixing for general problems / 27.6: |
Multidimensional problems / 27.7: |
Front fixing and American options / 27.8: |
Other finite difference schemes / 27.9: |
Viscosity Solutions and Penalty Methods for American Option Problems / 27.10: |
Definitions and main results for parabolic problems / 28.1: |
An introduction to semi-linear equations and penalty method / 28.3: |
Implicit, explicit and semi-implicit schemes / 28.4: |
Multi-asset American options / 28.5: |
Variational Formulation of American Option Problems / 28.6: |
A short history of variational inequalities / 29.1: |
A first parabolic variational inequality / 29.3: |
Functional analysis background / 29.4: |
Kinds of variational inequalities / 29.5: |
Variational inequalities using Rothe's methods / 29.6: |
American options and variational inequalities / 29.7: |
Design and Implementation In C++ / 29.8: |
Finding the Appropriate Finite Difference Schemes for your Financial Engineering Problem / 30: |
The financial model / 30.1: |
The viewpoints in the continuous model / 30.3: |
The viewpoints in the discrete model / 30.4: |
Auxiliary numerical methods / 30.5: |
New Developments / 30.6: |
Design and Implementation of First-Order Problems / 30.7: |
Software requirements / 31.1: |
Modular decomposition / 31.3: |
Useful C++ data structures / 31.4: |
One-factor models / 31.5: |
Multi-factor models / 31.6: |
Generalisations and applications to quantitative finance / 31.7: |
Appendix: Useful data structures in C++ / 31.8: |
Moving to Black-Scholes / 32: |
The PDE model / 32.1: |
The FDM model / 32.3: |
Algorithms and data structures / 32.4: |
The C++ model / 32.5: |
Test case: The two-dimensional heat equation / 32.6: |
Finite difference solution / 32.7: |
Moving to software and method implementation / 32.8: |
Generalisations / 32.9: |
C++ Class Hierarchies for One-Factor and Two-Factor Payoffs / 32.10: |
Abstract and concrete payoff classes / 33.1: |
Using payoff classes / 33.3: |
Lightweight payoff classes / 33.4: |
Super-lightweight payoff functions / 33.5: |
Payoff functions for multi-asset option problems / 33.6: |
Caveat: non-smooth payoff and convergence degradation / 33.7: |
Appendices / 33.8: |
An introduction to integral and partial integro-differential equations / A1: |
An introduction to the finite element method / A2: |
Bibliography |
Index |