[...]... implicit (ADI) methods are popular in the financial engineering literature; second, we discuss operator splitting methods (or the method of fractional steps) that have their origins in the former Soviet Union Finally, we discuss some modern developments in this area 4 Finite Difference Methods in Financial Engineering Part V applies the results and schemes from Part IV to approximating some multi-factor... 10 Finite Difference Methods in Financial Engineering 1.4 INITIAL VALUE PROBLEMS In the previous section we examined a differential equation on a bounded interval In this case we assumed that the solution was defined in this interval and that certain boundary conditions were defined at the interval’s end-points We now consider a different problem where we wish to find the solution on a semi -in nite interval,... b(S) = r S c(S) = −r f (S) = 0 (1.4) In the following chapters our intention is to solve problems of the form (1.1) and we then apply our results to the specialised equations in quantitative finance 8 Finite Difference Methods in Financial Engineering 1.2 TWO-POINT BOUNDARY VALUE PROBLEM Let us examine a general second-order ordinary differential equation given in the form u = f (x; u, u ) (1.5) where... VII 325 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 30.3.1 Payoff functions 30.3.2 Boundary conditions 30.3.3 Transformations 327 327 328 328 329 330 331 xiv Contents 30.4 The viewpoints in the discrete model 30.4.1 Functional... the parts in the book discuss the following topics: I II III IV V VI VII The Continuous Theory of Partial Differential Equations Finite Difference Methods: the Fundamentals Applying FDM to One-Factor Instrument Pricing FDM for Multidimensional Problems Applying FDM to Multi-Factor Instrument Pricing Free and Moving Boundary Value Problems Design and Implementation in C++ Part I presents an introduction... 135 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 12.4.1 Stability of the trinomial method 12.5 Using exponential fitting with explicit time marching 12.6 Approximating the... this knowledge in a meaningful way This book is also a good companion to my other book, Financial Instrument Pricing in C++ (Duffy, 2004) The algorithms in the present book can be mapped to C++, the de-facto object-oriented language for financial engineering applications In short, it is hoped that this book will help you to master all the details needed for a good understanding of FDM in your daily work... diffusion 26.4 Solutions in financial engineering: A preview 26.4.1 What kinds of early exercise features? 26.4.2 What kinds of numerical techniques? 26.5 Summary and conclusions 287 287 287 288 288 289 290 290 291 293 293 293 294 294 Contents xiii 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... BOOK The book has been partitioned into seven parts, each of which deals with one specific topic in detail Furthermore, each part contains material that is required by its successor In general, we interleave the parts by first discussing the theory (for example, basic finite difference schemes) in a given part and then applying this theory to a problem in financial engineering This ‘separation of concerns’... An overview of finite difference schemes for multi-asset problems Numerical solution of elliptic equations Solving multi-asset Black–Scholes equations Special guidelines and caveats Summary and conclusions 264 265 265 265 266 267 269 270 271 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 . 0 Finite Difference Methods in Financial Engineering i 0470858826FM JWBK073-Duffy February 9, 2006 20:25 Char Count= 0 For other titles in the Wiley Finance. AND IMPLEMENTATION IN C++ 325 30 Finding the Appropriate Finite Difference Schemes for your Financial Engineering Problem 327 30.1 Introduction and objectives