Finite Difference Methods in Financial Engineering A Partial Differential Equation Approach

by Duffy, Daniel J.

• ISBN13: 9780470858820
• ISBN10: 0470858826
• eBook ISBN(s): 9780470858837
• Edition: 1st
• Format: Hardcover
• Copyright: 2006-05-12
• Publisher: WILEY
• More Book Details
• Purchase Benefits

Summary

The world of quantitative finance (QF) is one of the fastest growing areas of research and its practical applications to derivatives pricing problem. Since the discovery of the famous Black-Scholes equation in the 1970's we have seen a surge in the number of models for a wide range of products such as plain and exotic options, interest rate derivatives, real options and many others. Gone are the days when it was possible to price these derivatives analytically. For most problems we must resort to some kind of approximate method.In this book we employ partial differential equations (PDE) to describe a range of one-factor and multi-factor derivatives products such as plain European and American options, multi-asset options, Asian options, interest rate options and real options. PDE techniques allow us to create a framework for modeling complex and interesting derivatives products. Having defined the PDE problem we then approximate it using the Finite Difference Method (FDM). This method has been used for many application areas such as fluid dynamics, heat transfer, semiconductor simulation and astrophysics, to name just a few. In this book we apply the same techniques to pricing real-life derivative products. We use both traditional (or well-known) methods as well as a number of advanced schemes that are making their way into the QF literature: Crank-Nicolson, exponentially fitted and higher-order schemes for one-factor and multi-factor options Early exercise features and approximation using front-fixing, penalty and variational methods Modelling stochastic volatility models using Splitting methods Critique of ADI and Crank-Nicolson schemes; when they work and when they don't work Modelling jumps using Partial Integro Differential Equations (PIDE) Free and moving boundary value problems in QF Included with the book is a CD containing information on how to set up FDM algorithms, how to map these algorithms to C++ as well as several working programs for one-factor and two-factor models. We also provide source code so that you can customize the applications to suit your own needs.

Author Biography

Daniel Duffy is a numerical analyst who has been working in the IT business since 1979. He has been involved in the analysis, design and implementation of systems using object-oriented, component and (more recently) intelligent agent technologies to large industrial and financial applications. As early as 1993 he was involved in C++ projects for risk management and options applications with a large Dutch bank. His main interest is in finding robust and scalable numerical schemes that approximate the partial differential equations that model financial derivatives products. He has an M.Sc. in the Finite Element Method first-order hyperbolic systems and a Ph.D. in robust finite difference methods for convection-diffusion partial differential equations. Both degrees are from Trinity College, Dublin, Ireland.
Daniel Duffy is founder of Datasim Education and Datasim Component Technology, two companies involved in training, consultancy and software development.

Table of Contents

0 Goals of this Book and Global Overview.

0.1 What is this book?.

0.2 Why has this book been written?.

0.3 For whom is this book intended?.

0.4 Why should I read this book?.

0.5 The structure of this book.

0.6 What this book does not cover.

0.7 Contact, feedback and more information.

PART I THE CONTINUOUS THEORY OF PARTIAL DIFFERENTIAL EQUATIONS.

1 An Introduction to Ordinary Differential Equations.

1.1 Introduction and objectives.

1.2 Two-point boundary value problem.

1.3 Linear boundary value problems.

1.4 Initial value problems.

1.5 Some special cases.

1.6 Summary and conclusions.

2 An Introduction to Partial Differential Equations.

2.1 Introduction and objectives.

2.2 Partial differential equations.

2.3 Specialisations.  

2.4 Parabolic partial differential equations.

2.5 Hyperbolic equations.

2.6 Systems of equations.

2.7 Equations containing integrals.

2.8 Summary and conclusions.

3 Second-Order Parabolic Differential Equations.

3.1 Introduction and objectives.

3.2 Linear parabolic equations.

3.3 The continuous problem.

3.4 The maximum principle for parabolic equations.

3.5 A special case: one-factor generalised Black–Scholes models.

3.6 Fundamental solution and the Green’s function.

3.7 Integral representation of the solution of parabolic PDEs.

3.8 Parabolic equations in one space dimension.

3.9 Summary and conclusions.

4 An Introduction to the Heat Equation in One Dimension.

4.1 Introduction and objectives.

4.2 Motivation and background.

4.3 The heat equation and financial engineering.

4.4 The separation of variables technique.

4.5 Transformation techniques for the heat equation.

4.6 Summary and conclusions.

5 An Introduction to the Method of Characteristics.

5.1 Introduction and objectives.

5.2 First-order hyperbolic equations.

5.3 Second-order hyperbolic equations.

5.4 Applications to financial engineering.

5.5 Systems of equations.

5.6 Propagation of discontinuities.

5.7 Summary and conclusions

PART II FINITE DIFFERENCE METHODS: THE FUNDAMENTALS.

6 AnIntroduction to the Finite Difference Method.

6.1 Introduction and objectives.

6.2 Fundamentals of numerical differentiation.

6.3 Caveat: accuracy and round-off errors.

6.4 Where are divided differences used in instrument pricing?.

6.5 Initial value problems.

6.6 Nonlinear initial value problems.  

6.7 Scalar initial value problems.

6.8 Summary and conclusions.

7 An Introduction to the Method of Lines.

7.1 Introduction and objectives.

7.2 Classifying semi-discretisation methods.

7.3 Semi-discretisation in space using FDM.

7.4 Numerical approximation of first-order systems.

7.5 Summary and conclusions.

8 General Theory of the Finite Difference Method.

8.1 Introduction and objectives.

8.2 Some fundamental concepts.  

8.3 Stability and the Fourier transform.

8.4 The discrete Fourier transform.

8.5 Stability for initial boundary value problems.

8.6 Summary and conclusions.

9 Finite Difference Schemes for First-Order Partial Differential Equations.

9.1 Introduction and objectives.

9.2 Scoping the problem.

9.3 Why first-order equations are different: Essential difficulties.

9.4 A simple explicit scheme.

9.5 Some common schemes for initial value problems.

9.6 Some common schemes for initial boundary value problems.

9.7 Monotone and positive-type schemes.

9.8 Extensions, generalisations and other applications.  

9.9 Summary and conclusions.

10 FDM for the One-Dimensional Convection–Diffusion Equation.

10.1 Introduction and objectives.

10.2 Approximation of derivatives on the boundaries.

10.3 Time-dependent convection–diffusion equations.

10.4 Fully discrete schemes.

10.5 Specifying initial and boundary conditions.

10.6 Semi-discretisation in space.

10.7 Semi-discretisation in time.

10.8 Summary and conclusions.

11 Exponentially Fitted Finite Difference Schemes.

11.1 Introduction and objectives.

11.2 Motivating exponential fitting.

11.3 Exponential fitting and time-dependent convection-diffusion.

11.4 Stability and convergence analysis.

11.5 Approximating the derivative of the solution.

11.6 Special limiting cases.

11.7 Summary and conclusions.

PART III APPLYING FDM TO ONE-FACTOR INSTRUMENT PRICING.

12 Exact Solutions and Explicit Finite Difference Method for One-Factor Models.

12.1 Introduction and objectives.

12.2 Exact solutions and benchmark cases.

12.3 Perturbation analysis and risk engines.

12.4 The trinomial method: Preview.

12.5 Using exponential fitting with explicit time marching.

12.6 Approximating the Greeks.

12.7 Summary and conclusions.

12.8 Appendix: the formula for Vega.

13 An Introduction to the Trinomial Method.

13.1 Introduction and objectives.

13.2 Motivating the trinomial method.

13.3 Trinomial method: Comparisons with other methods.

13.4 The trinomial method for barrier options.

13.5 Summary and conclusions.

14 Exponentially Fitted Difference Schemes for Barrier Options.

14.1 Introduction and objectives.

14.2 What are barrier options?.

14.3 Initial boundary value problems for barrier options.

14.4 Using exponential fitting for barrier options.

14.5 Time-dependent volatility.

14.6 Some other kinds of exotic options.

14.7 Comparisons with exact solutions.

14.8 Other schemes and approximations.

14.9 Extensions to the model.

14.10 Summary and conclusions

15 Advanced Issues in Barrier and Lookback Option Modelling.

15.1 Introduction and objectives.

15.2 Kinds of boundaries and boundary conditions.

15.3 Discrete and continuous monitoring.

15.4 Continuity corrections for discrete barrier options.

15.5 Complex barrier options.

15.6 Summary and conclusions.

16 The Meshless (Meshfree) Method in Financial Engineering.

16.1 Introduction and objectives.

16.2 Motivating the meshless method.

16.3 An introduction to radial basis functions.

16.4 Semi-discretisations and convection–diffusion equations.

16.5 Applications the one-factor Black–Scholes equation.

16.6 Advantages and disadvantages of meshless.

16.7 Summary and conclusions.

17 Extending the Black–Scholes Model: Jump Processes.

17.1 Introduction and objectives.

17.2 Jump–diffusion processes.

17.3 Partial integro-differential equations and financial applications.

17.4 Numerical solution of PIDE: Preliminaries.

17.5 Techniques for the numerical solution of PIDEs.

17.6 Implicit and explicit methods.

17.7 Implicit–explicit Runge–Kutta methods.

17.8 Using operator splitting.

17.9 Splitting and predictor–corrector methods.

17.10 Summary and conclusions.

PART IV FDM FOR MULTIDIMENSIONAL PROBLEMS.

18 Finite Difference Schemes for Multidimensional Problems.

18.1 Introduction and objectives.

18.2 Elliptic equations.

18.3 Diffusion and heat equations.

18.4 Advection equation in two dimensions.

18.5 Convection–diffusion equation.

18.6 Summary and conclusions.

19 An Introduction to Alternating Direction Implicit and Splitting Methods.

19.1 Introduction and objectives.

19.2 What is ADI, really?.

19.3 Improvements on the basic ADI scheme.

19.4 ADI for first-order hyperbolic equations.

19.5 ADI classico and three-dimensional problems.

19.6 The Hopscotch method.

19.7 Boundary conditions.

19.8 Summary and conclusions.

20 Advanced Operator Splitting Methods: Fractional Steps.

20.1 Introduction and objectives.

20.2 Initial examples.

20.3 Problems with mixed derivatives.

20.4 Predictor–corrector methods (approximation correctors).

20.5 Partial integro-differential equations.

20.6 More general results.

20.7 Summary and conclusions.

21 Modern Splitting Methods.

21.1 Introduction and objectives.

21.2 Systems of equations.

21.3 A different kind of splitting: The IMEX schemes.

21.4 Applicability of IMEX schemes to Asian option pricing.

21.5 Summary and conclusions.

PART V APPLYING FDM TO MULTI-FACTOR INSTRUMENT PRICING.

22 Options with Stochastic Volatility: The Heston Model.

22.1 Introduction and objectives.

22.2 An introduction to Ornstein–Uhlenbeck processes.

22.3 Stochastic differential equations and the Heston model.

22.4 Boundary conditions.

22.5 Using finite difference schemes: Prologue.

22.6 A detailed example.

22.7 Summary and conclusions.

23 Finite Difference Methods for Asian Options and Other ‘Mixed’ Problems.

23.1 Introduction and objectives.

23.2 An introduction to Asian options.

23.3 My first PDE formulation.

23.4 Using operator splitting methods.

23.5 Cheyette interest models.

23.6 New developments.

23.7 Summary and conclusions.

24 Multi-Asset Options.

24.1 Introduction and objectives.

24.2 A taxonomy of multi-asset options.

24.3 Common framework for multi-asset options.

24.4 An overview of finite difference schemes for multi-asset problems.

24.5 Numerical solution of elliptic equations.

24.6 Solving multi-asset Black–Scholes equations.

24.7 Special guidelines and caveats.

24.8 Summary and conclusions.

25 Finite Difference Methods for Fixed-Income Problems.

25.1 Introduction and objectives.

25.2 An introduction to interest rate modelling.

25.3 Single-factor models.

25.4 Some specific stochastic models.

25.5 An introduction to multidimensional models.

25.6 The thorny issue of boundary conditions.

25.7 Introduction to approximate methods for interest rate models.

25.8 Summary and conclusions.

PART VI FREE AND MOVING BOUNDARY VALUE PROBLEMS.

26 Background to Free and Moving Boundary Value Problems.

26.1 Introduction and objectives.

26.2 Notation and definitions.

26.3 Some preliminary examples.

26.4 Solutions in financial engineering: A preview.

26.5 Summary and conclusions.

27 Numerical Methods for Free Boundary Value Problems: Front-Fixing Methods.

27.1 Introduction and objectives.

27.2 An introduction to front-fixing methods.

27.3 A crash course on partial derivatives.

27.4 Functions and implicit forms.

27.5 Front fixing for the heat equation.

27.6 Front fixing for general problems.

27.7 Multidimensional problems.

27.8 Front fixing and American options.

27.9 Other finite difference schemes.

27.10 Summary and conclusions.

28 Viscosity Solutions and Penalty Methods for American Option Problems.

28.1 Introduction and objectives.

28.2 Definitions and main results for parabolic problems.  

28.3 An introduction to semi-linear equations and penalty method.

28.4 Implicit, explicit and semi-implicit schemes.

28.5 Multi-asset American options.

28.6 Summary and conclusions.

29 Variational Formulation of American Option Problems.

29.1 Introduction and objectives.

29.2 A short history of variational inequalities.

29.3 A first parabolic variational inequality.

29.4 Functional analysis background.

29.5 Kinds of variational inequalities.  

29.6 Variational inequalities using Rothe’s methods.

29.7 American options and variational inequalities.

29.8 Summary and conclusions.

PART VII DESIGN AND IMPLEMENTATION IN C++.

30 Finding the Appropriate Finite Difference Schemes for your Financial Engineering Problem.

30.1 Introduction and objectives.

30.2 The financial model.

30.3 The viewpoints in the continuous model.

30.4 The viewpoints in the discrete model.

30.5 Auxiliary numerical methods.

30.6 New Developments.

30.7 Summary and conclusions.

31 Design and Implementation of First-Order Problems.

31.1 Introduction and objectives.

31.2 Software requirements.

31.3 Modular decomposition.

31.4 Useful C++ data structures.

31.5 One-factor models.

31.6 Multi-factor models.

31.7 Generalisations and applications to quantitative finance.

31.8 Summary and conclusions.

31.9 Appendix: Useful data structures in C++.

32 Moving to Black–Scholes.

32.1 Introduction and objectives.

32.2 The PDE model.

32.3 The FDM model.

32.4 Algorithms and data structures.

32.5 The C++ model.

32.6 Test case: The two-dimensional heat equation.

32.7 Finite difference solution.

32.8 Moving to software and method implementation.

32.9 Generalisations.  

32.10 Summary and conclusions.

33 C++ Class Hierarchies for One-Factor and Two-Factor Payoffs.

33.1 Introduction and objectives.

33.2 Abstract and concrete payoff classes.

33.3 Using payoff classes.

33.4 Lightweight payoff classes.

33.5 Super-lightweight payoff functions.

33.6 Payoff functions for multi-asset option problems.

33.7 Caveat: non-smooth payoff and convergence degradation

33.8 Summary and conclusions.

Appendices.

A1 An introduction to integral and partial integro-differential equations.

A2 An introduction to the finite element method.

Bibliography.

Index.