PART I THE CONTINUOUS THEORY OF PARTIAL DIFFERENTIAL EQUATIONS.

1 An Introduction to Ordinary Differential Equations.

2 An Introduction to Partial Differential Equations.

3 Second-Order Parabolic Differential Equations.

4 An Introduction to the Heat Equation in One Dimension.

5 An Introduction to the Method of Characteristics.

PART II FINITE DIFFERENCE METHODS: THE FUNDAMENTALS.

6 AnIntroduction to the Finite Difference Method.

7 An Introduction to the Method of Lines.

8 General Theory of the Finite Difference Method.

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

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

11 Exponentially Fitted Finite Difference Schemes.

PART III APPLYING FDM TO ONE-FACTOR INSTRUMENT PRICING.

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

13 An Introduction to the Trinomial Method.

14 Exponentially Fitted Difference Schemes for Barrier Options.

15 Advanced Issues in Barrier and Lookback Option Modelling.

16 The Meshless (Meshfree) Method in Financial Engineering.

17 Extending the Black–Scholes Model: Jump Processes.

PART IV FDM FOR MULTIDIMENSIONAL PROBLEMS.

18 Finite Difference Schemes for Multidimensional Problems.

19 An Introduction to Alternating Direction Implicit and Splitting Methods.

20 Advanced Operator Splitting Methods: Fractional Steps. 

21 Modern Splitting Methods.

PART V APPLYING FDM TO MULTI-FACTOR INSTRUMENT PRICING.

22 Options with Stochastic Volatility: The Heston Model.

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

24 Multi-Asset Options.

25 Finite Difference Methods for Fixed-Income Problems.

PART VI FREE AND MOVING BOUNDARY VALUE PROBLEMS.

26 Background to Free and Moving Boundary Value Problems.

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

28 Viscosity Solutions and Penalty Methods for American Option Problems.

29 Variational Formulation of American Option Problems.

PART VII DESIGN AND IMPLEMENTATION IN C++.

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

31 Design and Implementation of First-Order Problems.

32 Moving to Black–Scholes.

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

Appendices.

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

A2 An introduction to the finite element method.

Bibliography.

Index.