Finite Difference Methods in Financial Engineering

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0 Goals of this Book and Global Overview 1 0.1 What is this book? 1 0.2 Why has this book been written? 2 0.3 For whom is this book intended? 2 0.4 Why should I read this book? 2 0.5 The structure of this book 3 0.6 What this book does not cover 4 0.7 Contact, feedback and more information 4 PART I THE CONTINUOUS THEORY OF PARTIAL DIFFERENTIAL EQUATIONS 5 1 An Introduction to Ordinary Differential Equations 7 1.1 Introduction and objectives 7 1.2 Two-point boundary value problem 8 1.3 Linear boundary value problems 9 1.4 Initial value problems 10 1.5 Some special cases 10 1.6 Summary and conclusions 11 2 An Introduction to Partial Differential Equations 13 2.1 Introduction and objectives 13 2.2 Partial differential equations 13 2.3 Specialisations 15 2.4 Parabolic partial differential equations 18 2.5 Hyperbolic equations 20 2.6 Systems of equations 22 2.7 Equations containing integrals 23 2.8 Summary and conclusions 24 3 Second-Order Parabolic Differential Equations 25 3.1 Introduction and objectives 25 3.2 Linear parabolic equations 25 3.3 The continuous problem 26 3.4 The maximum principle for parabolic equations 28 3.5 A special case: one-factor generalised Black?Scholes models 29 3.6 Fundamental solution and the Green?s function 30 3.7 Integral representation of the solution of parabolic PDEs 31 3.8 Parabolic equations in one space dimension 33 3.9 Summary and conclusions 35 4 An Introduction to the Heat Equation in One Dimension 37 4.1 Introduction and objectives 37 4.2 Motivation and background 38 4.3 The heat equation and financial engineering 39 4.4 The separation of variables technique 40 4.5 Transformation techniques for the heat equation 44 4.6 Summary and conclusions 46 5 An Introduction to the Method of Characteristics 47 5.1 Introduction and objectives 47 5.2 First-order hyperbolic equations 47 5.3 Second-order hyperbolic equations 50 5.4 Applications to financial engineering 53 5.5 Systems of equations 55 5.6 Propagation of discontinuities 57 5.7 Summary and conclusions 59 PART II FINITE DIFFERENCE METHODS: THE FUNDAMENTALS 61 6 An Introduction to the Finite Difference Method 63 6.1 Introduction and objectives 63 6.2 Fundamentals of numerical differentiation 63 6.3 Caveat: accuracy and round-off errors 65 6.4 Where are divided differences used in instrument pricing? 67 6.5 Initial value problems 67 6.6 Nonlinear initial value problems 72 6.7 Scalar initial value problems 75 6.8 Summary and conclusions 76 7 An Introduction to the Method of Lines 79 7.1 Introduction and objectives 79 7.2 Classifying semi-discretisation methods 79 7.3 Semi-discretisation in space using FDM 80 7.4 Numerical approximation of first-order systems 85 7.5 Summary and conclusions 89 8 General Theory of the Finite Difference Method 91 8.1 Introduction and objectives 91 8.2 Some fundamental concepts 91 8.3 Stability and the Fourier transform 94 8.4 The discrete Fourier transform 96 8.5 Stability for initial boundary value problems 99 8.6 Summary and conclusions 101 9 Finite Difference Schemes for First-Order Partial Differential Equations 103 9.1 Introduction and objectives 103 9.2 Scoping the problem 103 9.3 Why first-order equations are different: Essential difficulties 105 9.4 A simple explicit scheme 106 9.5 Some common schemes for initial value problems 108 9.6 Some common schemes for initial boundary value problems 110 9.7 Monotone and positive-type schemes 110 9.8 Extensions, generalisations and other applications 111 9.9 Summary and conclusions 115 10 FDM for the One-Dimensional Convection?Diffusion Equation 117 10.1 Introduction and objectives 117 10.2 Approximation of derivatives on the boundaries 118 10.3 Time-dependent convection?diffusion equations 120 10.4 Fully discrete schemes 120 10.5 Specifying initial and boundary conditions 121 10.6 Semi-discretisation in space 121 10.7 Semi-discretisation in time 122 10.8 Summary and conclusions 122 11 Exponentially Fitted Finite Difference Schemes 123 11.1 Introduction and objectives 123 11.2 Motivating exponential fitting 123 11.3 Exponential fitting and time-dependent convection-diffusion 128 11.4 Stability and convergence analysis 129 11.5 Approximating the derivative of the solution 131 11.6 Special limiting cases 132 11.7 Summary and conclusions 132 PART III APPLYING FDM TO ONE-FACTOR INSTRUMENT PRICING 135 12 Exact Solutions and Explicit Finite Difference Method for One-Factor Models 137 12.1 Introduction and objectives 137 12.2 Exact solutions and benchmark cases 137 12.3 Perturbation analysis and risk engines 139 12.4 The trinomial method: Preview 139 12.5 Using exponential fitting with explicit time marching 142 12.6 Approximating the Greeks 142 12.7 Summary and conclusions 144 12.8 Appendix: the formula for Vega 144 13 An Introduction to the Trinomial Method 147 13.1 Introduction and objectives 147 13.2 Motivating the trinomial method 147 13.3 Trinomial method: Comparisons with other methods 149 13.4 The trinomial method for barrier options 151 13.5 Summary and conclusions 152 14 Exponentially Fitted Difference Schemes for Barrier Options 153 14.1 Introduction and objectives 153 14.2 What are barrier options? 153 14.3 Initial boundary value problems for barrier options 154 14.4 Using exponential fitting for barrier options 154 14.5 Time-dependent volatility 156 14.6 Some other kinds of exotic options 157 14.7 Comparisons with exact solutions 159 14.8 Other schemes and approximations 162 14.9 Extensions to the model 162 14.10 Summary and conclusions 163 15 Advanced Issues in Barrier and Lookback Option Modelling 165 15.1 Introduction and objectives 165 15.2 Kinds of boundaries and boundary conditions 165 15.3 Discrete and continuous monitoring 168 15.4 Continuity corrections for discrete barrier options 171 15.5 Complex barrier options 171 15.6 Summary and conclusions 173 16 The Meshless (Meshfree) Method in Financial Engineering 175 16.1 Introduction and objectives 175 16.2 Motivating the meshless method 175 16.3 An introduction to radial basis functions 177 16.4 Semi-discretisations and convection?diffusion equations 177 16.5 Applications of the one-factor Black?Scholes equation 179 16.6 Advantages and disadvantages of meshless 180 16.7 Summary and conclusions 181 17 Extending the Black?Scholes Model: Jump Processes 183 17.1 Introduction and objectives 183 17.2 Jump?diffusion processes 183 17.2.1 Convolution transformations 185 17.3 Partial integro-differential equations and financial applications 186 17.4 Numerical solution of PIDE: Preliminaries 187 17.5 Techniques for the numerical solution of PIDEs 188 17.6 Implicit and explicit methods 188 17.7 Implicit?explicit Runge?Kutta methods 189 17.8 Using operator splitting 189 17.9 Splitting and predictor?corrector methods 190 17.10 Summary and conclusions 191 PART IV FDM FOR MULTIDIMENSIONAL PROBLEMS 193 18 Finite Difference Schemes for Multidimensional Problems 195 18.1 Introduction and objectives 195 18.2 Elliptic equations 195 18.3 Diffusion and heat equations 202 18.4 Advection equation in two dimensions 205 18.5 Convection?diffusion equation 207 18.6 Summary and conclusions 208 19 An Introduction to Alternating Direction Implicit and Splitting Methods 209 19.1 Introduction and objectives 209 19.2 What is ADI, really? 210 19.3 Improvements on the basic ADI scheme 212 19.4 ADI for first-order hyperbolic equations 215 19.5 ADI classico and three-dimensional problems 217 19.6 The Hopscotch method 218 19.7 Boundary conditions 219 19.8 Summary and conclusions 221 20 Advanced Operator Splitting Methods: Fractional Steps 223 20.1 Introduction and objectives 223 20.2 Initial examples 223 20.3 Problems with mixed derivatives 224 20.4 Predictor?corrector methods (approximation correctors) 226 20.5 Partial integro-differential equations 227 20.6 More general results 228 20.7 Summary and conclusions 228 21 Modern Splitting Methods 229 21.1 Introduction and objectives 229 21.2 Systems of equations 229 21.3 A different kind of splitting: The IMEX schemes 232 21.4 Applicability of IMEX schemes to Asian option pricing 234 21.5 Summary and conclusions 235 PART V APPLYING FDM TO MULTI-FACTOR INSTRUMENT PRICING 237 22 Options with Stochastic Volatility: The Heston Model 239 22.1 Introduction and objectives 239 22.2 An introduction to Ornstein?Uhlenbeck processes 239 22.3 Stochastic differential equations and the Heston model 240 22.4 Boundary conditions 241 22.5 Using finite difference schemes: Prologue 243 22.6 A detailed example 243 22.7 Summary and conclusions 246 23 Finite Difference Methods for Asian Options and Other ?Mixed? Problems 249 23.1 Introduction and objectives 249 23.2 An introduction to Asian options 249 23.3 My first PDE formulation 250 23.4 Using operator splitting methods 251 23.5 Cheyette interest models 253 23.6 New developments 254 23.7 Summary and conclusions 255 24 Multi-Asset Options 257 24.1 Introduction and objectives 257 24.2 A taxonomy of multi-asset options 257 24.3 Common framework for multi-asset options 265 24.4 An overview of finite difference schemes for multi-asset problems 266 24.5 Numerical solution of elliptic equations 267 24.6 Solving multi-asset Black?Scholes equations 269 24.7 Special guidelines and caveats 270 24.8 Summary and conclusions 271 25 Finite Difference Methods for Fixed-Income Problems 273 25.1 Introduction and objectives 273 25.2 An introduction to interest rate modelling 273 25.3 Single-factor models 274 25.4 Some specific stochastic models 276 25.5 An introduction to multidimensional models 278 25.6 The thorny issue of boundary conditions 280 25.7 Introduction to approximate methods for interest rate models 282 25.8 Summary and conclusions 283 PART VI FREE AND MOVING BOUNDARY VALUE PROBLEMS 285 26 Background to Free and Moving Boundary Value Problems 287 26.1 Introduction and objectives 287 26.2 Notation and definitions 287 26.3 Some preliminary examples 288 26.4 Solutions in financial engineering: A preview 293 26.5 Summary and conclusions 294 27 Numerical Methods for Free Boundary Value Problems: Front-Fixing Methods 295 27.1 Introduction and objectives 295 27.2 An introduction to front-fixing methods 295 27.3 A crash course on partial derivatives 295 27.4 Functions and implicit forms 297 27.5 Front fixing for the heat equation 299 27.6 Front fixing for general problems 300 27.7 Multidimensional problems 300 27.8 Front fixing and American options 303 27.9 Other finite difference schemes 305 27.10 Summary and conclusions 306 28 Viscosity Solutions and Penalty Methods for American Option Problems 307 28.1 Introduction and objectives 307 28.2 Definitions and main results for parabolic problems 307 28.3 An introduction to semi-linear equations and penalty method 310 28.4 Implicit, explicit and semi-implicit schemes 311 28.5 Multi-asset American options 312 28.6 Summary and conclusions 314 29 Variational Formulation of American Option Problems 315 29.1 Introduction and objectives 315 29.2 A short history of variational inequalities 316 29.3 A first parabolic variational inequality 316 29.4 Functional analysis background 318 29.5 Kinds of variational inequalities 319 29.6 Variational inequalities using Rothe?s methods 323 29.7 American options and variational inequalities 324 29.8 Summary and conclusions 324 PART VII DESIGN AND IMPLEMENTATION IN C++ 325 30 Finding the Appropriate Finite Difference Schemes for your Financial Engineering Problem 327 30.1 Introduction and objectives 327 30.2 The financial model 328 30.3 The viewpoints in the continuous model 328 30.4 The viewpoints in the discrete model 332 30.5 Auxiliary numerical methods 335 30.6 New Developments 336 30.7 Summary and conclusions 336 31 Design and Implementation of First-Order Problems 337 31.1 Introduction and objectives 337 31.2 Software requirements 337 31.3 Modular decomposition 338 31.4 Useful C++ data structures 339 31.5 One-factor models 339 31.6 Multi-factor models 343 31.7 Generalisations and applications to quantitative finance 346 31.8 Summary and conclusions 347 31.9 Appendix: Useful data structures in C++ 348 32 Moving to Black?Scholes 353 32.1 Introduction and objectives 353 32.2 The PDE model 354 32.3 The FDM model 355 32.4 Algorithms and data structures 355 32.5 The C++ model 356 32.6 Test case: The two-dimensional heat equation 357 32.7 Finite difference solution 357 32.8 Moving to software and method implementation 358 32.9 Generalisations 361 32.10 Summary and conclusions 362 33 C++ Class Hierarchies for One-Factor and Two-Factor Payoffs 363 33.1 Introduction and objectives 363 33.2 Abstract and concrete payoff classes 364 33.3 Using payoff classes 367 33.4 Lightweight payoff classes 368 33.5 Super-lightweight payoff functions 369 33.6 Payoff functions for multi-asset option problems 371 33.7 Caveat: non-smooth payoff and convergence degradation 373 33.8 Summary and conclusions 374 Appendices 375 A1 An introduction to integral and partial integro-differential equations 375 A2 An introduction to the finite element method 393 Bibliography 409 Index 417

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.

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