<p></p><p><br/></p><p><font color="#0909f7" size="4">Finite Difference Methods In Financial Engineering: A Partial Differential Equation Approach</font></p><p><font color="#0909f7" size="4">金融工程的有限差方法：偏微分方程研究(英文原版)</font><br/>&nbsp;</p><div class="goods-title"><a href="javascript:;"><img alt="Finite Difference Methods In Financial Engineering: A Partial Differential Equation Approach财政工程的有限差方法：偏微分方程研究(英文原版进口)" src="http://shop.biox.cn/images/200708/1187317101433877926.jpg" width="347" height="281" style="WIDTH: 347px; HEIGHT: 281px;"/></a></div><div class="goods-title">详细介绍</div><div class="article-content"><p>Author: Daniel J. Duffy <br/>Format: Hardcover, 423 pages <br/>Publication Date: May 2006 <br/>Publisher: John Wiley &amp; Sons Inc <br/>Dimensions: 10"H x 7"W x 1.25"D; 2.25 lbs. <br/>ISBN-10: 0470858826 <br/>ISBN-13: 9780470858820</p><p>【内容简介】 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. <br/>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: <br/><br/>Crank-Nicolson, exponentially fitted and higher-order schemes for one-factor and multi-factor options <br/>Early exercise features and approximation using front-fixing, penalty and variational methods <br/>Modelling stochastic volatility models using Splitting methods <br/>Critique of ADI and Crank-Nicolson schemes; when they work and when they don't work <br/>Modelling jumps using Partial Integro Differential Equations (PIDE) <br/>Free and moving boundary value problems in QF <br/>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. </p><p><a href="javascript:;"></a>&nbsp;0 Goals of this Book and Global Overview 1<br/>0.1 What is this book? 1<br/>0.2 Why has this book been written? 2<br/>0.3 For whom is this book intended? 2<br/>0.4 Why should I read this book? 2<br/>0.5 The structure of this book 3<br/>0.6 What this book does not cover 4<br/>0.7 Contact, feedback and more information 4<br/>PART I THE CONTINUOUS THEORY OF PARTIAL<br/>DIFFERENTIAL EQUATIONS 5<br/>1 An Introduction to Ordinary Differential Equations 7<br/>1.1 Introduction and objectives 7<br/>1.2 Two-point boundary value problem 8<br/>1.2.1 Special kinds of boundary condition 8<br/>1.3 Linear boundary value problems 9<br/>1.4 Initial value problems 10<br/>1.5 Some special cases 10<br/>1.6 Summary and conclusions 11<br/>2 An Introduction to Partial Differential Equations 13<br/>2.1 Introduction and objectives 13<br/>2.2 Partial differential equations 13<br/>2.3 Specialisations 15<br/>2.3.1 Elliptic equations 15<br/>2.3.2 Free boundary value problems 17<br/>2.4 Parabolic partial differential equations 18<br/>2.4.1 Special cases 20<br/>2.5 Hyperbolic equations 20<br/>2.5.1 Second-order equations 20<br/>2.5.2 First-order equations 21</p><p>2.6 Systems of equations 22<br/>2.6.1 Parabolic systems 22<br/>2.6.2 First-order hyperbolic systems 22<br/>2.7 Equations containing integrals 23<br/>2.8 Summary and conclusions 24<br/>3 Second-Order Parabolic Differential Equations 25<br/>3.1 Introduction and objectives 25<br/>3.2 Linear parabolic equations 25<br/>3.3 The continuous problem 26<br/>3.4 The maximum principle for parabolic equations 28<br/>3.5 A special case: one-factor generalised Black–Scholes models 29<br/>3.6 Fundamental solution and the Green’s function 30<br/>3.7 Integral representation of the solution of parabolic PDEs 31<br/>3.8 Parabolic equations in one space dimension 33<br/>3.9 Summary and conclusions 35<br/>4 An Introduction to the Heat Equation in One Dimension 37<br/>4.1 Introduction and objectives 37<br/>4.2 Motivation and background 38<br/>4.3 The heat equation and financial engineering 39<br/>4.4 The separation of variables technique 40<br/>4.4.1 Heat flow in a road with ends held at constant temperature 42<br/>4.4.2 Heat flow in a rod whose ends are at a specified<br/>variable temperature 42<br/>4.4.3 Heat flow in an infinite rod 43<br/>4.4.4 Eigenfunction expansions 43<br/>4.5 Transformation techniques for the heat equation 44<br/>4.5.1 Laplace transform 45<br/>4.5.2 Fourier transform for the heat equation 45<br/>4.6 Summary and conclusions 46<br/>5 An Introduction to the Method of Characteristics 47<br/>5.1 Introduction and objectives 47<br/>5.2 First-order hyperbolic equations 47<br/>5.2.1 An example 48<br/>5.3 Second-order hyperbolic equations 50<br/>5.3.1 Numerical integration along the characteristic lines 50<br/>5.4 Applications to financial engineering 53<br/>5.4.1 Generalisations 55<br/>5.5 Systems of equations 55<br/>5.5.1 An example 57<br/>5.6 Propagation of discontinuities 57<br/>5.6.1 Other problems 58<br/>5.7 Summary and conclusions 59</p><p>6 An Introduction to the Finite Difference Method 63<br/>6.1 Introduction and objectives 63<br/>6.2 Fundamentals of numerical differentiation 63<br/>6.3 Caveat: accuracy and round-off errors 65<br/>6.4 Where are divided differences used in instrument pricing? 67<br/>6.5 Initial value problems 67<br/>6.5.1 Pad&acute;e matrix approximations 68<br/>6.5.2 Extrapolation 71<br/>6.6 Nonlinear initial value problems 72<br/>6.6.1 Predictor–corrector methods 73<br/>6.6.2 Runge–Kutta methods 74<br/>6.7 Scalar initial value problems 75<br/>6.7.1 Exponentially fitted schemes 76<br/>6.8 Summary and conclusions 76<br/>7 An Introduction to the Method of Lines 79<br/>7.1 Introduction and objectives 79<br/>7.2 Classifying semi-discretisation methods 79<br/>7.3 Semi-discretisation in space using FDM 80<br/>7.3.1 A test case 80<br/>7.3.2 Toeplitz matrices 82<br/>7.3.3 Semi-discretisation for convection-diffusion problems 82<br/>7.3.4 Essentially positive matrices 84<br/>7.4 Numerical approximation of first-order systems 85<br/>7.4.1 Fully discrete schemes 86<br/>7.4.2 Semi-linear problems 87<br/>7.5 Summary and conclusions 89<br/>8 General Theory of the Finite Difference Method 91<br/>8.1 Introduction and objectives 91<br/>8.2 Some fundamental concepts 91<br/>8.2.1 Consistency 93<br/>8.2.2 Stability 93<br/>8.2.3 Convergence 94<br/>8.3 Stability and the Fourier transform 94<br/>8.4 The discrete Fourier transform 96<br/>8.4.1 Some other examples 98<br/>8.5 Stability for initial boundary value problems 99<br/>8.5.1 Gerschgorin’s circle theorem 100<br/>8.6 Summary and conclusions 101<br/>9 Finite Difference Schemes for First-Order Partial Differential Equations 103<br/>9.1 Introduction and objectives 103<br/>9.2 Scoping the problem 103</p><p>9.3 Why first-order equations are different: Essential difficulties 105<br/>9.3.1 Discontinuous initial conditions 106<br/>9.4 A simple explicit scheme 106<br/>9.5 Some common schemes for initial value problems 108<br/>9.5.1 Some other schemes 110<br/>9.6 Some common schemes for initial boundary value problems 110<br/>9.7 Monotone and positive-type schemes 110<br/>9.8 Extensions, generalisations and other applications 111<br/>9.8.1 General linear problems 112<br/>9.8.2 Systems of equations 112<br/>9.8.3 Nonlinear problems 114<br/>9.8.4 Several independent variables 114<br/>9.9 Summary and conclusions 115<br/>10 FDM for the One-Dimensional Convection–Diffusion Equation 117<br/>10.1 Introduction and objectives 117<br/>10.2 Approximation of derivatives on the boundaries 118<br/>10.3 Time-dependent convection–diffusion equations 120<br/>10.4 Fully discrete schemes 120<br/>10.5 Specifying initial and boundary conditions 121<br/>10.6 Semi-discretisation in space 121<br/>10.7 Semi-discretisation in time 122<br/>10.8 Summary and conclusions 122<br/>11 Exponentially Fitted Finite Difference Schemes 123<br/>11.1 Introduction and objectives 123<br/>11.2 Motivating exponential fitting 123<br/>11.2.1 ‘Continuous’ exponential approximation 124<br/>11.2.2 ‘Discrete’ exponential approximation 125<br/>11.2.3 Where is exponential fitting being used? 128<br/>11.3 Exponential fitting and time-dependent convection-diffusion 128<br/>11.4 Stability and convergence analysis 129<br/>11.5 Approximating the derivative of the solution 131<br/>11.6 Special limiting cases 132<br/>11.7 Summary and conclusions 132<br/>PART III APPLYING FDM TO ONE-FACTOR INSTRUMENT PRICING 135<br/>12 Exact Solutions and Explicit Finite Difference Method<br/>for One-Factor Models 137<br/>12.1 Introduction and objectives 137<br/>12.2 Exact solutions and benchmark cases 137<br/>12.3 Perturbation analysis and risk engines 139<br/>12.4 The trinomial method: Preview 139<br/>12.4.1 Stability of the trinomial method 141<br/>12.5 Using exponential fitting with explicit time marching 142<br/>12.6 Approximating the Greeks 142</p><p>12.7 Summary and conclusions 144<br/>12.8 Appendix: the formula for Vega 144<br/>13 An Introduction to the Trinomial Method 147<br/>13.1 Introduction and objectives 147<br/>13.2 Motivating the trinomial method 147<br/>13.3 Trinomial method: Comparisons with other methods 149<br/>13.3.1 A general formulation 150<br/>13.4 The trinomial method for barrier options 151<br/>13.5 Summary and conclusions 152<br/>14 Exponentially Fitted Difference Schemes for Barrier Options 153<br/>14.1 Introduction and objectives 153<br/>14.2 What are barrier options? 153<br/>14.3 Initial boundary value problems for barrier options 154<br/>14.4 Using exponential fitting for barrier options 154<br/>14.4.1 Double barrier call options 156<br/>14.4.2 Single barrier call options 156<br/>14.5 Time-dependent volatility 156<br/>14.6 Some other kinds of exotic options 157<br/>14.6.1 Plain vanilla power call options 158<br/>14.6.2 Capped power call options 158<br/>14.7 Comparisons with exact solutions 159<br/>14.8 Other schemes and approximations 162<br/>14.9 Extensions to the model 162<br/>14.10 Summary and conclusions 163<br/>15 Advanced Issues in Barrier and Lookback Option Modelling 165<br/>15.1 Introduction and objectives 165<br/>15.2 Kinds of boundaries and boundary conditions 165<br/>15.3 Discrete and continuous monitoring 168<br/>15.3.1 What is discrete monitoring? 168<br/>15.3.2 Finite difference schemes and jumps in time 169<br/>15.3.3 Lookback options and jumps 170<br/>15.4 Continuity corrections for discrete barrier options 171<br/>15.5 Complex barrier options 171<br/>15.6 Summary and conclusions 173<br/>16 The Meshless (Meshfree) Method in Financial Engineering 175<br/>16.1 Introduction and objectives 175<br/>16.2 Motivating the meshless method 175<br/>16.3 An introduction to radial basis functions 177<br/>16.4 Semi-discretisations and convection–diffusion equations 177<br/>16.5 Applications of the one-factor Black–Scholes equation 179<br/>16.6 Advantages and disadvantages of meshless 180<br/>16.7 Summary and conclusions 181</p><p>17 Extending the Black–Scholes Model: Jump Processes 183<br/>17.1 Introduction and objectives 183<br/>17.2 Jump–diffusion processes 183<br/>17.2.1 Convolution transformations 185<br/>17.3 Partial integro-differential equations and financial applications 186<br/>17.4 Numerical solution of PIDE: Preliminaries 187<br/>17.5 Techniques for the numerical solution of PIDEs 188<br/>17.6 Implicit and explicit methods 188<br/>17.7 Implicit–explicit Runge–Kutta methods 189<br/>17.8 Using operator splitting 189<br/>17.9 Splitting and predictor–corrector methods 190<br/>17.10 Summary and conclusions 191<br/>PART IV FDM FOR MULTIDIMENSIONAL PROBLEMS 193<br/>18 Finite Difference Schemes for Multidimensional Problems 195<br/>18.1 Introduction and objectives 195<br/>18.2 Elliptic equations 195<br/>18.2.1 A self-adjoint elliptic operator 198<br/>18.2.2 Solving the matrix systems 199<br/>18.2.3 Exact solutions to elliptic problems 200<br/>18.3 Diffusion and heat equations 202<br/>18.3.1 Exact solutions to the heat equation 204<br/>18.4 Advection equation in two dimensions 205<br/>18.4.1 Initial boundary value problems 207<br/>18.5 Convection–diffusion equation 207<br/>18.6 Summary and conclusions 208<br/>19 An Introduction to Alternating Direction Implicit and Splitting Methods 209<br/>19.1 Introduction and objectives 209<br/>19.2 What is ADI, really? 210<br/>19.3 Improvements on the basic ADI scheme 212<br/>19.3.1 The D’Yakonov scheme 212<br/>19.3.2 Approximate factorization of operators 213<br/>19.3.3 ADI classico for two-factor models 215<br/>19.4 ADI for first-order hyperbolic equations 215<br/>19.5 ADI classico and three-dimensional problems 217<br/>19.6 The Hopscotch method 218<br/>19.7 Boundary conditions 219<br/>19.8 Summary and conclusions 221<br/>20 Advanced Operator Splitting Methods: Fractional Steps 223<br/>20.1 Introduction and objectives 223<br/>20.2 Initial examples 223<br/>20.3 Problems with mixed derivatives 224<br/>20.4 Predictor–corrector methods (approximation correctors) 226<br/>20.5 Partial integro-differential equations 227</p><p>20.6 More general results 228<br/>20.7 Summary and conclusions 228<br/>21 Modern Splitting Methods 229<br/>21.1 Introduction and objectives 229<br/>21.2 Systems of equations 229<br/>21.2.1 ADI and splitting for parabolic systems 230<br/>21.2.2 Compound and chooser options 231<br/>21.2.3 Leveraged knock-in options 232<br/>21.3 A different kind of splitting: The IMEX schemes 232<br/>21.4 Applicability of IMEX schemes to Asian option pricing 234<br/>21.5 Summary and conclusions 235<br/>PART V APPLYING FDM TO MULTI-FACTOR INSTRUMENT PRICING 237<br/>22 Options with Stochastic Volatility: The Heston Model 239<br/>22.1 Introduction and objectives 239<br/>22.2 An introduction to Ornstein–Uhlenbeck processes 239<br/>22.3 Stochastic differential equations and the Heston model 240<br/>22.4 Boundary conditions 241<br/>22.4.1 Standard european call option 242<br/>22.4.2 European put options 242<br/>22.4.3 Other kinds of boundary conditions 242<br/>22.5 Using finite difference schemes: Prologue 243<br/>22.6 A detailed example 243<br/>22.7 Summary and conclusions 246<br/>23 Finite Difference Methods for Asian Options and Other ‘Mixed’ Problems 249<br/>23.1 Introduction and objectives 249<br/>23.2 An introduction to Asian options 249<br/>23.3 My first PDE formulation 250<br/>23.4 Using operator splitting methods 251<br/>23.4.1 For sake of completeness: ADI methods for asian option PDEs 253<br/>23.5 Cheyette interest models 253<br/>23.6 New developments 254<br/>23.7 Summary and conclusions 255<br/>24 Multi-Asset Options 257<br/>24.1 Introduction and objectives 257<br/>24.2 A taxonomy of multi-asset options 257<br/>24.2.1 Exchange options 260<br/>24.2.2 Rainbow options 261<br/>24.2.3 Basket options 262<br/>24.2.4 The best and worst 263<br/>24.2.5 Quotient options 263<br/>24.2.6 Foreign equity options 264<br/>24.2.7 Quanto options 264</p><p>24.2.8 Spread options 264<br/>24.2.9 Dual-strike options 265<br/>24.2.10 Out-perfomance options 265<br/>24.3 Common framework for multi-asset options 265<br/>24.4 An overview of finite difference schemes for multi-asset problems 266<br/>24.5 Numerical solution of elliptic equations 267<br/>24.6 Solving multi-asset Black–Scholes equations 269<br/>24.7 Special guidelines and caveats 270<br/>24.8 Summary and conclusions 271<br/>25 Finite Difference Methods for Fixed-Income Problems 273<br/>25.1 Introduction and objectives 273<br/>25.2 An introduction to interest rate modelling 273<br/>25.3 Single-factor models 274<br/>25.4 Some specific stochastic models 276<br/>25.4.1 The Merton model 277<br/>25.4.2 The Vasicek model 277<br/>25.4.3 Cox, Ingersoll and Ross (CIR) 277<br/>25.4.4 The Hull–White model 277<br/>25.4.5 Lognormal models 278<br/>25.5 An introduction to multidimensional models 278<br/>25.6 The thorny issue of boundary conditions 280<br/>25.6.1 One-factor models 280<br/>25.6.2 Multi-factor models 281<br/>25.7 Introduction to approximate methods for interest rate models 282<br/>25.7.1 One-factor models 282<br/>25.7.2 Many-factor models 283<br/>25.8 Summary and conclusions 283<br/>PART VI FREE AND MOVING BOUNDARY VALUE PROBLEMS 285<br/>26 Background to Free and Moving Boundary Value Problems 287<br/>26.1 Introduction and objectives 287<br/>26.2 Notation and definitions 287<br/>26.3 Some preliminary examples 288<br/>26.3.1 Single-phase melting ice 288<br/>26.3.2 One-factor option modelling: American exercise style 289<br/>26.3.3 Two-phase melting ice 290<br/>26.3.4 The inverse Stefan problem 290<br/>26.3.5 Two and three space dimensions 291<br/>26.3.6 Oxygen diffusion 293<br/>26.4 Solutions in financial engineering: A preview 293<br/>26.4.1 What kinds of early exercise features? 293<br/>26.4.2 What kinds of numerical techniques? 294<br/>26.5 Summary and conclusions 294</p><p>27 Numerical Methods for Free Boundary Value Problems:<br/>Front-Fixing Methods 295<br/>27.1 Introduction and objectives 295<br/>27.2 An introduction to front-fixing methods 295<br/>27.3 A crash course on partial derivatives 295<br/>27.4 Functions and implicit forms 297<br/>27.5 Front fixing for the heat equation 299<br/>27.6 Front fixing for general problems 300<br/>27.7 Multidimensional problems 300<br/>27.8 Front fixing and American options 303<br/>27.9 Other finite difference schemes 305<br/>27.9.1 The method of lines and predictor–corrector 305<br/>27.10 Summary and conclusions 306<br/>28 Viscosity Solutions and Penalty Methods for American Option Problems 307<br/>28.1 Introduction and objectives 307<br/>28.2 Definitions and main results for parabolic problems 307<br/>28.2.1 Semi-continuity 307<br/>28.2.2 Viscosity solutions of nonlinear parabolic problems 308<br/>28.3 An introduction to semi-linear equations and penalty method 310<br/>28.4 Implicit, explicit and semi-implicit schemes 311<br/>28.5 Multi-asset American options 312<br/>28.6 Summary and conclusions 314<br/>29 Variational Formulation of American Option Problems 315<br/>29.1 Introduction and objectives 315<br/>29.2 A short history of variational inequalities 316<br/>29.3 A first parabolic variational inequality 316<br/>29.4 Functional analysis background 318<br/>29.5 Kinds of variational inequalities 319<br/>29.5.1 Diffusion with semi-permeable membrane 319<br/>29.5.2 A one-dimensional finite element approximation 320<br/>29.6 Variational inequalities using Rothe’s methods 323<br/>29.7 American options and variational inequalities 324<br/>29.8 Summary and conclusions 324<br/>PART VII DESIGN AND IMPLEMENTATION IN C++ 325<br/>30 Finding the Appropriate Finite Difference Schemes for your Financial<br/>Engineering Problem 327<br/>30.1 Introduction and objectives 327<br/>30.2 The financial model 328<br/>30.3 The viewpoints in the continuous model 328<br/>30.3.1 Payoff functions 329<br/>30.3.2 Boundary conditions 330<br/>30.3.3 Transformations 331</p><p>30.4 The viewpoints in the discrete model 332<br/>30.4.1 Functional and non-functional requirements 332<br/>30.4.2 Approximating the spatial derivatives in the PDE 333<br/>30.4.3 Time discretisation in the PDE 334<br/>30.4.4 Payoff functions 334<br/>30.4.5 Boundary conditions 335<br/>30.5 Auxiliary numerical methods 335<br/>30.6 New Developments 336<br/>30.7 Summary and conclusions 336<br/>31 Design and Implementation of First-Order Problems 337<br/>31.1 Introduction and objectives 337<br/>31.2 Software requirements 337<br/>31.3 Modular decomposition 338<br/>31.4 Useful C++ data structures 339<br/>31.5 One-factor models 339<br/>31.5.1 Main program and output 342<br/>31.6 Multi-factor models 343<br/>31.7 Generalisations and applications to quantitative finance 346<br/>31.8 Summary and conclusions 347<br/>31.9 Appendix: Useful data structures in C++ 348<br/>32 Moving to Black–Scholes 353<br/>32.1 Introduction and objectives 353<br/>32.2 The PDE model 354<br/>32.3 The FDM model 355<br/>32.4 Algorithms and data structures 355<br/>32.5 The C++ model 356<br/>32.6 Test case: The two-dimensional heat equation 357<br/>32.7 Finite difference solution 357<br/>32.8 Moving to software and method implementation 358<br/>32.8.1 Defining the continuous problem 358<br/>32.8.2 Creating a mesh 358<br/>32.8.3 Choosing a scheme 360<br/>32.8.4 Termination criterion 361<br/>32.9 Generalisations 361<br/>32.9.1 More general PDEs 361<br/>32.9.2 Other finite difference schemes 361<br/>32.9.3 Flexible software solutions 361<br/>32.10 Summary and conclusions 362<br/>33 C++ Class Hierarchies for One-Factor and Two-Factor Payoffs 363<br/>33.1 Introduction and objectives 363<br/>33.2 Abstract and concrete payoff classes 364<br/>33.3 Using payoff classes 367<br/>33.4 Lightweight payoff classes 368<br/>33.5 Super-lightweight payoff functions 369</p><p>33.6 Payoff functions for multi-asset option problems 371<br/>33.7 Caveat: non-smooth payoff and convergence degradation 373<br/>33.8 Summary and conclusions 374<br/>Appendices 375<br/>A1 An introduction to integral and partial integro-differential equations 375<br/>A2 An introduction to the finite element method 393<br/>Bibliography 409<br/>Index 417</p><div class="zoom"></div></div>

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