Blog For Fina With Finite Elements

Financial Engineering with Finite Elements Jurgen ¨ Topper

Contents Preface

xv

List of symbols PART I

PRELIMINARIES

1 Introduction

xvii 1 3

2 Some Prototype Models 2.1 Optimal price policy of a monopolist 2.2 The Black–Scholes option pricing model 2.3 Pricing American options 2.4 Multi-asset options with stochastic correlation 2.5 The steady-state distribution of the Vasicek interest rate process 2.6 Notes

7 7 8 10 12 14 16

3 The Conventional Approach: Finite Differences 3.1 General considerations for numerical computations 3.1.1 Evaluation criteria 3.1.2 Turning unbounded domains into bounded domains 3.2 Ordinary initial value problems 3.2.1 Basic concepts 3.2.2 Euler’s method 3.2.3 Taylor methods 3.2.4 Runge–Kutta methods 3.2.5 The backward Euler method 3.2.6 The Crank–Nicolson method 3.2.7 Predictor–corrector methods 3.2.8 Adaptive techniques 3.2.9 Methods for systems of equations

17 17 17 18 22 22 23 28 30 33 35 36 38 39

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Contents

3.3 Ordinary two-point boundary value problems 3.3.1 Introductory remarks 3.3.2 Finite difference methods 3.3.3 Shooting methods 3.4 Initial boundary value problems 3.4.1 The explicit scheme 3.4.2 The implicit scheme 3.4.3 The Crank–Nicolson method 3.4.4 Integrating early exercise 3.5 Notes PART II

FINITE ELEMENTS

46 46 46 49 49 49 51 52 52 53 55

4 Static 1D Problems 4.1 Basic features of finite element methods 4.2 The method of weighted residuals – one-element solutions 4.3 The Ritz variational method 4.4 The method of weighted residuals – a more general view 4.5 Multi-element solutions 4.5.1 The Galerkin method with linear elements 4.5.2 The Galerkin method with quadratic trial functions 4.5.3 The collocation method with cubic Hermite trial functions 4.6 Case studies 4.6.1 The Evans model of a monopolist 4.6.2 First exit time of a geometric Brownian motion 4.6.3 The steady-state distribution of the Ornstein–Uhlenbeck process 4.6.4 Convection-dominated problems 4.7 Convergence 4.8 Notes

57 57 57 72 74 75 76 89 93 99 99 99 101 102 106 107

5 Dynamic 1D Problems 5.1 Derivation of element equations 5.1.1 The Galerkin method 5.1.2 The collocation method 5.2 Case studies 5.2.1 Plain vanilla options 5.2.2 Hedging parameters 5.2.3 Various exotic options 5.2.4 The CEV model 5.2.5 Some practicalities: Dividends and settlement

109 109 109 114 115 115 123 132 142 150

6 Static 2D Problems 6.1 Introduction and overview 6.2 Construction of a mesh 6.3 The Galerkin method 6.3.1 The Galerkin method with linear elements (triangles) 6.3.2 The Galerkin method with linear elements (rectangular elements)

161 161 162 165 165 187

Contents

6.4 Case studies 6.4.1 Brownian motion leaving a disk 6.4.2 Ritz revisited 6.4.3 First exit time in a two-asset pricing problem 6.5 Notes

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187 187 188 191 194

7 Dynamic 2D Problems 7.1 Derivation of element equations 7.2 Case studies 7.2.1 Various rainbow options 7.2.2 Modeling volatility as a risk factor

195 195 197 197 203

8 Static 3D Problems 8.1 Derivation of element equations: The collocation method 8.2 Case studies 8.2.1 First exit time of purely Brownian motion 8.2.2 First exit time of geometric Brownian motion 8.3 Notes

207 207 209 209 211 213

9 Dynamic 3D Problems 9.1 Derivation of element equations: The collocation method 9.2 Case studies 9.2.1 Pricing and hedging a basket option 9.2.2 Basket options with barriers

215 215 216 216 218

10 Nonlinear Problems 10.1 Introduction 10.2 Case studies 10.2.1 Penalty methods 10.2.2 American options 10.2.3 Passport options 10.2.4 Uncertain volatility: Best and worst cases 10.2.5 Worst-case pricing of rainbow options 10.3 Notes

221 221 223 223 223 227 240 248 252

PART III

OUTLOOK

253

11 Future Directions of Research

255

PART IV

257

APPENDICES

A Some Useful Results from Analysis A.1 Important theorems from calculus A.1.1 Various concepts of continuity A.1.2 Taylor’s theorem A.1.3 Mean value theorems A.1.4 Various theorems

259 259 259 260 262 263

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Contents

A.2 Basic numerical tools A.2.1 Quadrature A.2.2 Solving nonlinear equations A.3 Differential equations A.3.1 Definition and classification A.3.2 Ordinary initial value problems A.3.3 Ordinary boundary value problems A.3.4 Partial differential equations of second order A.3.5 Parabolic problems A.3.6 Elliptic PDEs A.3.7 Hyperbolic PDEs A.3.8 Hyperbolic conservation laws A.4 Calculus of variations

264 264 268 270 270 272 279 285 287 295 296 297 299

B Some Useful Results from Stochastics B.1 Some important distributions B.1.1 The univariate normal distribution B.1.2 The bivariate normal distribution B.1.3 The multivariate normal distribution B.1.4 The lognormal distribution B.1.5 The
distribution B.1.6 The central χ 2 distribution B.1.7 The noncentral χ 2 distribution B.2 Some important processes B.2.1 Basic concepts B.2.2 Wiener process B.2.3 Brownian motion with drift B.2.4 Geometric Brownian motion B.2.5 Itˆo process B.2.6 Ornstein–Uhlenbeck process B.2.7 A process for commodities B.3 Results B.3.1 The transition probability density function B.3.2 The backward Kolmogorov equation B.3.3 The forward Kolmogorov equation B.3.4 Steady-state distributions B.3.5 First exit times B.3.6 Itˆo’s lemma B.4 Notes

305 305 305 305 307 307 307 309 310 310 310 312 313 313 314 314 314 314 314 315 316 321 322 325 326

C Some Useful Results from Linear Algebra C.1 Some basic facts C.2 Errors and norms C.3 Ill-conditioning

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Contents

C.4 C.5

Solving linear algebraic systems Notes

xiii

333 339

D A Quick Introduction to PDE2D

341

References

343

Index

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