{"product_id":"finite-difference-methods-in-financial-engineering-isbn-9780470858820","title":"Finite Difference Methods in Financial Engineering","description":"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.  \u003cp\u003eIn 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:\u003c\/p\u003e \u003cul\u003e \u003cli\u003eCrank-Nicolson, exponentially fitted and higher-order schemes for one-factor and multi-factor options\u003c\/li\u003e \u003cli\u003eEarly exercise features and approximation using front-fixing, penalty and variational methods\u003c\/li\u003e \u003cli\u003eModelling stochastic volatility models using Splitting methods\u003c\/li\u003e \u003cli\u003eCritique of ADI and Crank-Nicolson schemes; when they work and when they don't work\u003c\/li\u003e \u003cli\u003eModelling jumps using Partial Integro Differential Equations (PIDE)\u003c\/li\u003e \u003cli\u003eFree and moving boundary value problems in QF\u003c\/li\u003e \u003c\/ul\u003e \u003cp\u003eIncluded 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.\u003c\/p\u003e \u003cp\u003e\u003cb\u003e0 Goals of this Book and Global Overview 1\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e0.1 What is this book? 1\u003c\/p\u003e \u003cp\u003e0.2 Why has this book been written? 2\u003c\/p\u003e \u003cp\u003e0.3 For whom is this book intended? 2\u003c\/p\u003e \u003cp\u003e0.4 Why should I read this book? 2\u003c\/p\u003e \u003cp\u003e0.5 The structure of this book 3\u003c\/p\u003e \u003cp\u003e0.6 What this book does not cover 4\u003c\/p\u003e \u003cp\u003e0.7 Contact, feedback and more information 4\u003c\/p\u003e \u003cp\u003e\u003cb\u003ePart I The Continuous Theory of Partial Differential Equations 5\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e\u003cb\u003e1 An Introduction to Ordinary Differential Equations 7\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e1.1 Introduction and objectives 7\u003c\/p\u003e \u003cp\u003e1.2 Two-point boundary value problem 8\u003c\/p\u003e \u003cp\u003e1.3 Linear boundary value problems 9\u003c\/p\u003e \u003cp\u003e1.4 Initial value problems 10\u003c\/p\u003e \u003cp\u003e1.5 Some special cases 10\u003c\/p\u003e \u003cp\u003e1.6 Summary and conclusions 11\u003c\/p\u003e \u003cp\u003e\u003cb\u003e2 An Introduction to Partial Differential Equations 13\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e2.1 Introduction and objectives 13\u003c\/p\u003e \u003cp\u003e2.2 Partial differential equations 13\u003c\/p\u003e \u003cp\u003e2.3 Specialisations 15\u003c\/p\u003e \u003cp\u003e2.4 Parabolic partial differential equations 18\u003c\/p\u003e \u003cp\u003e2.5 Hyperbolic equations 20\u003c\/p\u003e \u003cp\u003e2.6 Systems of equations 22\u003c\/p\u003e \u003cp\u003e2.7 Equations containing integrals 23\u003c\/p\u003e \u003cp\u003e2.8 Summary and conclusions 24\u003c\/p\u003e \u003cp\u003e\u003cb\u003e3 Second-Order Parabolic Differential Equations 25\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e3.1 Introduction and objectives 25\u003c\/p\u003e \u003cp\u003e3.2 Linear parabolic equations 25\u003c\/p\u003e \u003cp\u003e3.3 The continuous problem 26\u003c\/p\u003e \u003cp\u003e3.4 The maximum principle for parabolic equations 28\u003c\/p\u003e \u003cp\u003e3.5 A special case: one-factor generalised Black–Scholes models 29\u003c\/p\u003e \u003cp\u003e3.6 Fundamental solution and the Green’s function 30\u003c\/p\u003e \u003cp\u003e3.7 Integral representation of the solution of parabolic PDEs 31\u003c\/p\u003e \u003cp\u003e3.8 Parabolic equations in one space dimension 33\u003c\/p\u003e \u003cp\u003e3.9 Summary and conclusions 35\u003c\/p\u003e \u003cp\u003e\u003cb\u003e4 An Introduction to the Heat Equation in One Dimension 37\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e4.1 Introduction and objectives 37\u003c\/p\u003e \u003cp\u003e4.2 Motivation and background 38\u003c\/p\u003e \u003cp\u003e4.3 The heat equation and financial engineering 39\u003c\/p\u003e \u003cp\u003e4.4 The separation of variables technique 40\u003c\/p\u003e \u003cp\u003e4.5 Transformation techniques for the heat equation 44\u003c\/p\u003e \u003cp\u003e4.6 Summary and conclusions 46\u003c\/p\u003e \u003cp\u003e\u003cb\u003e5 An Introduction to the Method of Characteristics 47\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e5.1 Introduction and objectives 47\u003c\/p\u003e \u003cp\u003e5.2 First-order hyperbolic equations 47\u003c\/p\u003e \u003cp\u003e5.3 Second-order hyperbolic equations 50\u003c\/p\u003e \u003cp\u003e5.4 Applications to financial engineering 53\u003c\/p\u003e \u003cp\u003e5.5 Systems of equations 55\u003c\/p\u003e \u003cp\u003e5.6 Propagation of discontinuities 57\u003c\/p\u003e \u003cp\u003e5.7 Summary and conclusions 59\u003c\/p\u003e \u003cp\u003e\u003cb\u003ePart II Finite Difference Methods: the Fundamentals 61\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e\u003cb\u003e6 An Introduction to the Finite Difference Method 63\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e6.1 Introduction and objectives 63\u003c\/p\u003e \u003cp\u003e6.2 Fundamentals of numerical differentiation 63\u003c\/p\u003e \u003cp\u003e6.3 Caveat: accuracy and round-off errors 65\u003c\/p\u003e \u003cp\u003e6.4 Where are divided differences used in instrument pricing? 67\u003c\/p\u003e \u003cp\u003e6.5 Initial value problems 67\u003c\/p\u003e \u003cp\u003e6.6 Nonlinear initial value problems 72\u003c\/p\u003e \u003cp\u003e6.7 Scalar initial value problems 75\u003c\/p\u003e \u003cp\u003e6.8 Summary and conclusions 76\u003c\/p\u003e \u003cp\u003e\u003cb\u003e7 An Introduction to the Method of Lines 79\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e7.1 Introduction and objectives 79\u003c\/p\u003e \u003cp\u003e7.2 Classifying semi-discretisation methods 79\u003c\/p\u003e \u003cp\u003e7.3 Semi-discretisation in space using FDM 80\u003c\/p\u003e \u003cp\u003e7.4 Numerical approximation of first-order systems 85\u003c\/p\u003e \u003cp\u003e7.5 Summary and conclusions 89\u003c\/p\u003e \u003cp\u003e\u003cb\u003e8 General Theory of the Finite Difference Method 91\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e8.1 Introduction and objectives 91\u003c\/p\u003e \u003cp\u003e8.2 Some fundamental concepts 91\u003c\/p\u003e \u003cp\u003e8.3 Stability and the Fourier transform 94\u003c\/p\u003e \u003cp\u003e8.4 The discrete Fourier transform 96\u003c\/p\u003e \u003cp\u003e8.5 Stability for initial boundary value problems 99\u003c\/p\u003e \u003cp\u003e8.6 Summary and conclusions 101\u003c\/p\u003e \u003cp\u003e\u003cb\u003e9 Finite Difference Schemes for First-Order Partial Differential Equations 103\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e9.1 Introduction and objectives 103\u003c\/p\u003e \u003cp\u003e9.2 Scoping the problem 103\u003c\/p\u003e \u003cp\u003e9.3 Why first-order equations are different: Essential difficulties 105\u003c\/p\u003e \u003cp\u003e9.4 A simple explicit scheme 106\u003c\/p\u003e \u003cp\u003e9.5 Some common schemes for initial value problems 108\u003c\/p\u003e \u003cp\u003e9.6 Some common schemes for initial boundary value problems 110\u003c\/p\u003e \u003cp\u003e9.7 Monotone and positive-type schemes 110\u003c\/p\u003e \u003cp\u003e9.8 Extensions, generalisations and other applications 111\u003c\/p\u003e \u003cp\u003e9.9 Summary and conclusions 115\u003c\/p\u003e \u003cp\u003e\u003cb\u003e10 FDM for the One-Dimensional Convection–Diffusion Equation 117\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e10.1 Introduction and objectives 117\u003c\/p\u003e \u003cp\u003e10.2 Approximation of derivatives on the boundaries 118\u003c\/p\u003e \u003cp\u003e10.3 Time-dependent convection–diffusion equations 120\u003c\/p\u003e \u003cp\u003e10.4 Fully discrete schemes 120\u003c\/p\u003e \u003cp\u003e10.5 Specifying initial and boundary conditions 121\u003c\/p\u003e \u003cp\u003e10.6 Semi-discretisation in space 121\u003c\/p\u003e \u003cp\u003e10.7 Semi-discretisation in time 122\u003c\/p\u003e \u003cp\u003e10.8 Summary and conclusions 122\u003c\/p\u003e \u003cp\u003e\u003cb\u003e11 Exponentially Fitted Finite Difference Schemes 123\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e11.1 Introduction and objectives 123\u003c\/p\u003e \u003cp\u003e11.2 Motivating exponential fitting 123\u003c\/p\u003e \u003cp\u003e11.3 Exponential fitting and time-dependent convection-diffusion 128\u003c\/p\u003e \u003cp\u003e11.4 Stability and convergence analysis 129\u003c\/p\u003e \u003cp\u003e11.5 Approximating the derivative of the solution 131\u003c\/p\u003e \u003cp\u003e11.6 Special limiting cases 132\u003c\/p\u003e \u003cp\u003e11.7 Summary and conclusions 132\u003c\/p\u003e \u003cp\u003e\u003cb\u003ePart III Applying FDM to One-factor Instrument Pricing 135\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e\u003cb\u003e12 Exact Solutions and Explicit Finite Difference Method for One-Factor Models 137\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e12.1 Introduction and objectives 137\u003c\/p\u003e \u003cp\u003e12.2 Exact solutions and benchmark cases 137\u003c\/p\u003e \u003cp\u003e12.3 Perturbation analysis and risk engines 139\u003c\/p\u003e \u003cp\u003e12.4 The trinomial method: Preview 139\u003c\/p\u003e \u003cp\u003e12.5 Using exponential fitting with explicit time marching 142\u003c\/p\u003e \u003cp\u003e12.6 Approximating the Greeks 142\u003c\/p\u003e \u003cp\u003e12.7 Summary and conclusions 144\u003c\/p\u003e \u003cp\u003e12.8 Appendix: the formula for Vega 144\u003c\/p\u003e \u003cp\u003e\u003cb\u003e13 An Introduction to the Trinomial Method 147\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e13.1 Introduction and objectives 147\u003c\/p\u003e \u003cp\u003e13.2 Motivating the trinomial method 147\u003c\/p\u003e \u003cp\u003e13.3 Trinomial method: Comparisons with other methods 149\u003c\/p\u003e \u003cp\u003e13.4 The trinomial method for barrier options 151\u003c\/p\u003e \u003cp\u003e13.5 Summary and conclusions 152\u003c\/p\u003e \u003cp\u003e\u003cb\u003e14 Exponentially Fitted Difference Schemes for Barrier Options 153\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e14.1 Introduction and objectives 153\u003c\/p\u003e \u003cp\u003e14.2 What are barrier options? 153\u003c\/p\u003e \u003cp\u003e14.3 Initial boundary value problems for barrier options 154\u003c\/p\u003e \u003cp\u003e14.4 Using exponential fitting for barrier options 154\u003c\/p\u003e \u003cp\u003e14.5 Time-dependent volatility 156\u003c\/p\u003e \u003cp\u003e14.6 Some other kinds of exotic options 157\u003c\/p\u003e \u003cp\u003e14.7 Comparisons with exact solutions 159\u003c\/p\u003e \u003cp\u003e14.8 Other schemes and approximations 162\u003c\/p\u003e \u003cp\u003e14.9 Extensions to the model 162\u003c\/p\u003e \u003cp\u003e14.10 Summary and conclusions 163\u003c\/p\u003e \u003cp\u003e\u003cb\u003e15 Advanced Issues in Barrier and Lookback Option Modelling 165\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e15.1 Introduction and objectives 165\u003c\/p\u003e \u003cp\u003e15.2 Kinds of boundaries and boundary conditions 165\u003c\/p\u003e \u003cp\u003e15.3 Discrete and continuous monitoring 168\u003c\/p\u003e \u003cp\u003e15.4 Continuity corrections for discrete barrier options 171\u003c\/p\u003e \u003cp\u003e15.5 Complex barrier options 171\u003c\/p\u003e \u003cp\u003e15.6 Summary and conclusions 173\u003c\/p\u003e \u003cp\u003e\u003cb\u003e16 The Meshless (Meshfree) Method in Financial Engineering 175\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e16.1 Introduction and objectives 175\u003c\/p\u003e \u003cp\u003e16.2 Motivating the meshless method 175\u003c\/p\u003e \u003cp\u003e16.3 An introduction to radial basis functions 177\u003c\/p\u003e \u003cp\u003e16.4 Semi-discretisations and convection–diffusion equations 177\u003c\/p\u003e \u003cp\u003e16.5 Applications of the one-factor Black–Scholes equation 179\u003c\/p\u003e \u003cp\u003e16.6 Advantages and disadvantages of meshless 180\u003c\/p\u003e \u003cp\u003e16.7 Summary and conclusions 181\u003c\/p\u003e \u003cp\u003e\u003cb\u003e17 Extending the Black–Scholes Model: Jump Processes 183\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e17.1 Introduction and objectives 183\u003c\/p\u003e \u003cp\u003e17.2 Jump–diffusion processes 183\u003c\/p\u003e \u003cp\u003e17.3 Partial integro-differential equations and financial applications 186\u003c\/p\u003e \u003cp\u003e17.4 Numerical solution of PIDE: Preliminaries 187\u003c\/p\u003e \u003cp\u003e17.5 Techniques for the numerical solution of PIDEs 188\u003c\/p\u003e \u003cp\u003e17.6 Implicit and explicit methods 188\u003c\/p\u003e \u003cp\u003e17.7 Implicit–explicit Runge–Kutta methods 189\u003c\/p\u003e \u003cp\u003e17.8 Using operator splitting 189\u003c\/p\u003e \u003cp\u003e17.9 Splitting and predictor–corrector methods 190\u003c\/p\u003e \u003cp\u003e17.10 Summary and conclusions 191\u003c\/p\u003e \u003cp\u003e\u003cb\u003ePart IV FDM for Multidimensional Problems 193\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e\u003cb\u003e18 Finite Difference Schemes for Multidimensional Problems 195\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e18.1 Introduction and objectives 195\u003c\/p\u003e \u003cp\u003e18.2 Elliptic equations 195\u003c\/p\u003e \u003cp\u003e18.3 Diffusion and heat equations 202\u003c\/p\u003e \u003cp\u003e18.4 Advection equation in two dimensions 205\u003c\/p\u003e \u003cp\u003e18.5 Convection–diffusion equation 207\u003c\/p\u003e \u003cp\u003e18.6 Summary and conclusions 208\u003c\/p\u003e \u003cp\u003e\u003cb\u003e19 An Introduction to Alternating Direction Implicit and Splitting Methods 209\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e19.1 Introduction and objectives 209\u003c\/p\u003e \u003cp\u003e19.2 What is ADI, really? 210\u003c\/p\u003e \u003cp\u003e19.3 Improvements on the basic ADI scheme 212\u003c\/p\u003e \u003cp\u003e19.4 ADI for first-order hyperbolic equations 215\u003c\/p\u003e \u003cp\u003e19.5 ADI classico and three-dimensional problems 217\u003c\/p\u003e \u003cp\u003e19.6 The Hopscotch method 218\u003c\/p\u003e \u003cp\u003e19.7 Boundary conditions 219\u003c\/p\u003e \u003cp\u003e19.8 Summary and conclusions 221\u003c\/p\u003e \u003cp\u003e\u003cb\u003e20 Advanced Operator Splitting Methods: Fractional Steps 223\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e20.1 Introduction and objectives 223\u003c\/p\u003e \u003cp\u003e20.2 Initial examples 223\u003c\/p\u003e \u003cp\u003e20.3 Problems with mixed derivatives 224\u003c\/p\u003e \u003cp\u003e20.4 Predictor–corrector methods (approximation correctors) 226\u003c\/p\u003e \u003cp\u003e20.5 Partial integro-differential equations 227\u003c\/p\u003e \u003cp\u003e20.6 More general results 228\u003c\/p\u003e \u003cp\u003e20.7 Summary and conclusions 228\u003c\/p\u003e \u003cp\u003e\u003cb\u003e21 Modern Splitting Methods 229\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e21.1 Introduction and objectives 229\u003c\/p\u003e \u003cp\u003e21.2 Systems of equations 229\u003c\/p\u003e \u003cp\u003e21.3 A different kind of splitting: The IMEX schemes 232\u003c\/p\u003e \u003cp\u003e21.4 Applicability of IMEX schemes to Asian option pricing 234\u003c\/p\u003e \u003cp\u003e21.5 Summary and conclusions 235\u003c\/p\u003e \u003cp\u003e\u003cb\u003ePart V Applying FDM to Multi-factor Instrument Pricing 237\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e\u003cb\u003e22 Options with Stochastic Volatility: The Heston Model 239\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e22.1 Introduction and objectives 239\u003c\/p\u003e \u003cp\u003e22.2 An introduction to Ornstein–Uhlenbeck processes 239\u003c\/p\u003e \u003cp\u003e22.3 Stochastic differential equations and the Heston model 240\u003c\/p\u003e \u003cp\u003e22.4 Boundary conditions 241\u003c\/p\u003e \u003cp\u003e22.5 Using finite difference schemes: Prologue 243\u003c\/p\u003e \u003cp\u003e22.6 A detailed example 243\u003c\/p\u003e \u003cp\u003e22.7 Summary and conclusions 246\u003c\/p\u003e \u003cp\u003e\u003cb\u003e23 Finite Difference Methods for Asian Options and Other ‘Mixed’ Problems 249\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e23.1 Introduction and objectives 249\u003c\/p\u003e \u003cp\u003e23.2 An introduction to Asian options 249\u003c\/p\u003e \u003cp\u003e23.3 My first PDE formulation 250\u003c\/p\u003e \u003cp\u003e23.4 Using operator splitting methods 251\u003c\/p\u003e \u003cp\u003e23.5 Cheyette interest models 253\u003c\/p\u003e \u003cp\u003e23.6 New developments 254\u003c\/p\u003e \u003cp\u003e23.7 Summary and conclusions 255\u003c\/p\u003e \u003cp\u003e\u003cb\u003e24 Multi-Asset Options 257\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e24.1 Introduction and objectives 257\u003c\/p\u003e \u003cp\u003e24.2 A taxonomy of multi-asset options 257\u003c\/p\u003e \u003cp\u003e24.3 Common framework for multi-asset options 265\u003c\/p\u003e \u003cp\u003e24.4 An overview of finite difference schemes for multi-asset problems 266\u003c\/p\u003e \u003cp\u003e24.5 Numerical solution of elliptic equations 267\u003c\/p\u003e \u003cp\u003e24.6 Solving multi-asset Black–Scholes equations 269\u003c\/p\u003e \u003cp\u003e24.7 Special guidelines and caveats 270\u003c\/p\u003e \u003cp\u003e24.8 Summary and conclusions 271\u003c\/p\u003e \u003cp\u003e\u003cb\u003e25 Finite Difference Methods for Fixed-Income Problems 273\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e25.1 Introduction and objectives 273\u003c\/p\u003e \u003cp\u003e25.2 An introduction to interest rate modelling 273\u003c\/p\u003e \u003cp\u003e25.3 Single-factor models 274\u003c\/p\u003e \u003cp\u003e25.4 Some specific stochastic models 276\u003c\/p\u003e \u003cp\u003e25.5 An introduction to multidimensional models 278\u003c\/p\u003e \u003cp\u003e25.6 The thorny issue of boundary conditions 280\u003c\/p\u003e \u003cp\u003e25.7 Introduction to approximate methods for interest rate models 282\u003c\/p\u003e \u003cp\u003e25.8 Summary and conclusions 283\u003c\/p\u003e \u003cp\u003e\u003cb\u003ePart VI Free and Moving Boundary Value Problems 285\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e\u003cb\u003e26 Background to Free and Moving Boundary Value Problems 287\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e26.1 Introduction and objectives 287\u003c\/p\u003e \u003cp\u003e26.2 Notation and definitions 287\u003c\/p\u003e \u003cp\u003e26.3 Some preliminary examples 288\u003c\/p\u003e \u003cp\u003e26.4 Solutions in financial engineering: A preview 293\u003c\/p\u003e \u003cp\u003e26.5 Summary and conclusions 294\u003c\/p\u003e \u003cp\u003e\u003cb\u003e27 Numerical Methods for Free Boundary Value Problems: Front-Fixing Methods 295\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e27.1 Introduction and objectives 295\u003c\/p\u003e \u003cp\u003e27.2 An introduction to front-fixing methods 295\u003c\/p\u003e \u003cp\u003e27.3 A crash course on partial derivatives 295\u003c\/p\u003e \u003cp\u003e27.4 Functions and implicit forms 297\u003c\/p\u003e \u003cp\u003e27.5 Front fixing for the heat equation 299\u003c\/p\u003e \u003cp\u003e27.6 Front fixing for general problems 300\u003c\/p\u003e \u003cp\u003e27.7 Multidimensional problems 300\u003c\/p\u003e \u003cp\u003e27.8 Front fixing and American options 303\u003c\/p\u003e \u003cp\u003e27.9 Other finite difference schemes 305\u003c\/p\u003e \u003cp\u003e27.10 Summary and conclusions 306\u003c\/p\u003e \u003cp\u003e\u003cb\u003e28 Viscosity Solutions and Penalty Methods for American Option Problems 307\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e28.1 Introduction and objectives 307\u003c\/p\u003e \u003cp\u003e28.2 Definitions and main results for parabolic problems 307\u003c\/p\u003e \u003cp\u003e28.3 An introduction to semi-linear equations and penalty method 310\u003c\/p\u003e \u003cp\u003e28.4 Implicit, explicit and semi-implicit schemes 311\u003c\/p\u003e \u003cp\u003e28.5 Multi-asset American options 312\u003c\/p\u003e \u003cp\u003e28.6 Summary and conclusions 314\u003c\/p\u003e \u003cp\u003e\u003cb\u003e29 Variational Formulation of American Option Problems 315\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e29.1 Introduction and objectives 315\u003c\/p\u003e \u003cp\u003e29.2 A short history of variational inequalities 316\u003c\/p\u003e \u003cp\u003e29.3 A first parabolic variational inequality 316\u003c\/p\u003e \u003cp\u003e29.4 Functional analysis background 318\u003c\/p\u003e \u003cp\u003e29.5 Kinds of variational inequalities 319\u003c\/p\u003e \u003cp\u003e29.6 Variational inequalities using Rothe’s methods 323\u003c\/p\u003e \u003cp\u003e29.7 American options and variational inequalities 324\u003c\/p\u003e \u003cp\u003e29.8 Summary and conclusions 324\u003c\/p\u003e \u003cp\u003e\u003cb\u003ePart VII Design and Implementation in C++ 325\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e\u003cb\u003e30 Finding the Appropriate Finite Difference Schemes for your Financial Engineering Problem 327\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e30.1 Introduction and objectives 327\u003c\/p\u003e \u003cp\u003e30.2 The financial model 328\u003c\/p\u003e \u003cp\u003e30.3 The viewpoints in the continuous model 328\u003c\/p\u003e \u003cp\u003e30.4 The viewpoints in the discrete model 332\u003c\/p\u003e \u003cp\u003e30.5 Auxiliary numerical methods 335\u003c\/p\u003e \u003cp\u003e30.6 New Developments 336\u003c\/p\u003e \u003cp\u003e30.7 Summary and conclusions 336\u003c\/p\u003e \u003cp\u003e\u003cb\u003e31 Design and Implementation of First-Order Problems 337\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e31.1 Introduction and objectives 337\u003c\/p\u003e \u003cp\u003e31.2 Software requirements 337\u003c\/p\u003e \u003cp\u003e31.3 Modular decomposition 338\u003c\/p\u003e \u003cp\u003e31.4 Useful C++ data structures 339\u003c\/p\u003e \u003cp\u003e31.5 One-factor models 339\u003c\/p\u003e \u003cp\u003e31.6 Multi-factor models 343\u003c\/p\u003e \u003cp\u003e31.7 Generalisations and applications to quantitative finance 346\u003c\/p\u003e \u003cp\u003e31.8 Summary and conclusions 347\u003c\/p\u003e \u003cp\u003e31.9 Appendix: Useful data structures in C++ 348\u003c\/p\u003e \u003cp\u003e\u003cb\u003e32 Moving to Black–Scholes 353\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e32.1 Introduction and objectives 353\u003c\/p\u003e \u003cp\u003e32.2 The PDE model 354\u003c\/p\u003e \u003cp\u003e32.3 The FDM model 355\u003c\/p\u003e \u003cp\u003e32.4 Algorithms and data structures 355\u003c\/p\u003e \u003cp\u003e32.5 The C++ model 356\u003c\/p\u003e \u003cp\u003e32.6 Test case: The two-dimensional heat equation 357\u003c\/p\u003e \u003cp\u003e32.7 Finite difference solution 357\u003c\/p\u003e \u003cp\u003e32.8 Moving to software and method implementation 358\u003c\/p\u003e \u003cp\u003e32.9 Generalisations 361\u003c\/p\u003e \u003cp\u003e32.10 Summary and conclusions 362\u003c\/p\u003e \u003cp\u003e\u003cb\u003e33 C++ Class Hierarchies for One-Factor and Two-Factor Payoffs 363\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e33.1 Introduction and objectives 363\u003c\/p\u003e \u003cp\u003e33.2 Abstract and concrete payoff classes 364\u003c\/p\u003e \u003cp\u003e33.3 Using payoff classes 367\u003c\/p\u003e \u003cp\u003e33.4 Lightweight payoff classes 368\u003c\/p\u003e \u003cp\u003e33.5 Super-lightweight payoff functions 369\u003c\/p\u003e \u003cp\u003e33.6 Payoff functions for multi-asset option problems 371\u003c\/p\u003e \u003cp\u003e33.7 Caveat: non-smooth payoff and convergence degradation 373\u003c\/p\u003e \u003cp\u003e33.8 Summary and conclusions 374\u003c\/p\u003e \u003cp\u003eAppendices 375\u003c\/p\u003e \u003cp\u003eA1 An introduction to integral and partial integro-differential equations 375\u003c\/p\u003e \u003cp\u003eA2 An introduction to the finite element method 393\u003c\/p\u003e \u003cp\u003eBibliography 409\u003c\/p\u003e \u003cp\u003eIndex 417\u003c\/p\u003e  \u003cb\u003eDaniel Duffy\u003c\/b\u003e 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.\u003cbr\u003e Daniel Duffy is founder of Datasim Education and Datasim Component Technology, two companies involved in training, consultancy and software development.  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.  \u003cp\u003eIn 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:\u003c\/p\u003e \u003cul\u003e \u003cli\u003eCrank-Nicolson, exponentially fitted and higher-order schemes for one-factor and multi-factor options\u003c\/li\u003e \u003cli\u003eEarly exercise features and approximation using front-fixing, penalty and variational methods\u003c\/li\u003e \u003cli\u003eModelling stochastic volatility models using Splitting methods\u003c\/li\u003e \u003cli\u003eCritique of ADI and Crank-Nicolson schemes; when they work and when they don’t work\u003c\/li\u003e \u003cli\u003eModelling jumps using Partial Integro Differential Equations (PIDE)\u003c\/li\u003e \u003cli\u003eFree and moving boundary value problems in QF\u003c\/li\u003e \u003c\/ul\u003e \u003cp\u003eIncluded 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.\u003c\/p\u003e","brand":"Wiley","offers":[{"title":"Default Title","offer_id":47989214281957,"sku":"NP9780470858820","price":138.0,"currency_code":"USD","in_stock":false}],"thumbnail_url":"\/\/cdn.shopify.com\/s\/files\/1\/1842\/7735\/files\/9780470858820.jpg?v=1761783234","url":"https:\/\/k12savings.com\/products\/finite-difference-methods-in-financial-engineering-isbn-9780470858820","provider":"K12savings","version":"1.0","type":"link"}