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Contents
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
1.2.1 Special kinds of boundary condition 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
2.3.1 Elliptic equations 15
2.3.2 Free boundary value problems 17
2.4 Parabolic partial differential equations 18
2.4.1 Special cases 20
2.5 Hyperbolic equations 20
2.5.1 Second-order equations 20
2.5.2 First-order equations 21
2.6 Systems of equations 22
2.6.1 Parabolic systems 22
2.6.2 First-order hyperbolic systems 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
4.4.1 Heat flow in a road with ends held at constant temperature 42
4.4.2 Heat flow in a rod whose ends are at a specified
variable temperature 42
4.4.3 Heat flow in an infinite rod 43
4.4.4 Eigenfunction expansions 43
4.5 Transformation techniques for the heat equation 44
4.5.1 Laplace transform 45
4.5.2 Fourier transform for the heat equation 45
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
5.2.1 An example 48
5.3 Second-order hyperbolic equations 50
5.3.1 Numerical integration along the characteristic lines 50
5.4 Applications to financial engineering 53
5.4.1 Generalisations 55
5.5 Systems of equations 55
5.5.1 An example 57
5.6 Propagation of discontinuities 57
5.6.1 Other problems 58
5.7 Summary and conclusions 59
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
6.5.1 Pad´e matrix approximations 68
6.5.2 Extrapolation 71
6.6 Nonlinear initial value problems 72
6.6.1 Predictor–corrector methods 73
6.6.2 Runge–Kutta methods 74
6.7 Scalar initial value problems 75
6.7.1 Exponentially fitted schemes 76
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
7.3.1 A test case 80
7.3.2 Toeplitz matrices 82
7.3.3 Semi-discretisation for convection-diffusion problems 82
7.3.4 Essentially positive matrices 84
7.4 Numerical approximation of first-order systems 85
7.4.1 Fully discrete schemes 86
7.4.2 Semi-linear problems 87
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
8.2.1 Consistency 93
8.2.2 Stability 93
8.2.3 Convergence 94
8.3 Stability and the Fourier transform 94
8.4 The discrete Fourier transform 96
8.4.1 Some other examples 98
8.5 Stability for initial boundary value problems 99
8.5.1 Gerschgorin’s circle theorem 100
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
9.3.1 Discontinuous initial conditions 106
9.4 A simple explicit scheme 106
9.5 Some common schemes for initial value problems 108
9.5.1 Some other schemes 110
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
9.8.1 General linear problems 112
9.8.2 Systems of equations 112
9.8.3 Nonlinear problems 114
9.8.4 Several independent variables 114
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
11.2.1 ‘Continuous’ exponential approximation 124
11.2.2 ‘Discrete’ exponential approximation 125
11.2.3 Where is exponential fitting being used? 128
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
12.4.1 Stability of the trinomial method 141
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
13.3.1 A general formulation 150
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
14.4.1 Double barrier call options 156
14.4.2 Single barrier call options 156
14.5 Time-dependent volatility 156
14.6 Some other kinds of exotic options 157
14.6.1 Plain vanilla power call options 158
14.6.2 Capped power call options 158
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
15.3.1 What is discrete monitoring? 168
15.3.2 Finite difference schemes and jumps in time 169
15.3.3 Lookback options and jumps 170
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
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