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Tools for computational finance

Author: Seydal, Rüdiger Series: Universitext Publisher: Springer, 2006.Edition: 3rd ed.Language: EnglishDescription: 300 p. : Graphs ; 24 cm.ISBN: 3540279237Type of document: BookBibliography/Index: Includes bibliographical references and index
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Item type Current location Collection Call number Status Date due Barcode Item holds
Book Europe Campus
Main Collection
Print HG179 .S49 2006
(Browse shelf)
001175532
Available 001175532
Total holds: 0

Includes bibliographical references and index

Digitized

Tools for Computational Finance Contents Prefaces ............................................................................................ V Contents ........................................................................................ XIII Notation ............................................................................................ XVII Chapter 1 Modeling Tools for Financial Options ..................................... 1 1.1 1.2 1.3 1.4 1.5 1.6 Options ......................................................................................... 1 Model of the Financial Market ........................................................8 Numerical Methods .................................................................... 10 The Binomial Method ................................................................. 12 Risk-Neutral Valuation ............................................................... 21 Stochastic Processes ................................................................... 25 1.6.1 Wiener Process ............................................................... 26 1.6.2 Stochastic Integral ........................................................... 28 1.7 Stochastic Differential Equations ................................................ 31 1.7.1 Itô Process ....................................................................... 31 1.7.2 Application to the Stock Market ...................................... 33 1.7.3 Risk-Neutral Valuation ................................................... 36 1.7.4 Mean Reversion .............................................................. 37 1.7.5 Vector-Valued SDEs ........................................................ 39 1.8 Itô Lemma and Implications ....................................................... 40 1.9 Jump Processes .......................................................................... 45 Notes and Comments ....................................................................... 48 Exercises .......................................................................................... 52 Chapter 2 Generating Random Numbers with Specified Distributions .. 61 2.1 Uniform Deviates ........................................................................ 62 2.1.1 Linear Congruential Generators ...................................... 62 2.1.2 Quality of Generators ...................................................... 63 2.1.3 Random Vectors and Lattice Structure ............................ 64 2.1.4 Fibonacci Generators ...................................................... 67 2.2 Transformed Random Variables .................................................. 69 2.2.1 Inversion ......................................................................... 69 2.2.2 Transformations in R1 ................................................................................................ 70 XIV Contents 2.2.3 Transformation in IRn .................................................................................................... 72 2.3 Normally Distributed Random Variables ........................................ 72 2.3.1 Method of Box and Muller ................................................. 72 2.3.2 Variant of Marsaglia .......................................................... 73 2.3.3 Correlated Random Variables ............................................ 75 2.4 Monte Carlo Integration ................................................................ 77 2.5 Sequences of Numbers with Low Discrepancy ................................ 79 2.5.1 Discrepancy ...................................................................... 79 2.5.2 Examples of Low-Discrepancy Sequences .......................... 82 Notes and Comments ......................................................................... 85 Exercises ........................................................................................... 87 Chapter 3 Simulation with Stochastic Differential Equations ............................................................................... 91 3.1 Approximation Error ..................................................................... 3.2 Stochastic Taylor Expansion ......................................................... 3.3 Examples of Numerical Methods ................................................... 3.4 Intermediate Values ...................................................................... 3.5 Monte Carlo Simulation ................................................................ 3.5.1 Integral Representation ..................................................... 3.5.2 The Basic Version for European Options ............................ 3.5.3 Bias ................................................................................. 3.5.4 Variance Reduction ........................................................... 3.5.5 American Options ............................................................. 3.5.6 Further Hints ................................................................... Notes and Comments ........................................................................ Exercises .......................................................................................... 92 95 98 102 102 103 104 107 108 111 116 117 119 Chapter 4 Standard Methods for Standard Options ....................... 123 4.1 Preparations ............................................................................... 4.2 Foundations of Finite-Difference Methods .................................... 4.2.1 Difference Approximation .................................................. 4.2.2 The Grid ........................................................................... 4.2.3 Explicit Method ................................................................ 4.2.4 Stability ............................................................................ 4.2.5 An Implicit Method ........................................................... 4.3 Crank-Nicolson Method ............................................................... 4.4 Boundary Conditions .................................................................. 4.5 American Options as Free Boundary Problems ............................. 4.5.1 Early-Exercise Curve ........................................................ 4.5.2 Free Boundary Problems ................................................... 4.5.3 Black-Scholes Inequality ................................................... 4.5.4 Obstacle Problems ............................................................ 4.5.5 Linear Complementarity for American Put Options ............. 124 126 126 127 128 130 133 135 138 140 141 143 146 148 151 Contents XV 4.6 Computation of American Options .............................................. 4.6.1 Discretization with Finite Differences ............................... 4.6.2 Iterative Solution ............................................................. 4.6.3 An Algorithm for Calculating American Options ................ 4.7 On the Accuracy ........................................................................ 4.7.1 Elementary Error Control ................................................ 4.7.2 Extrapolation .................................................................. 4.8 Analytic Methods ....................................................................... 4.8.1 Approximation Based on Interpolation ............................. 4.8.2 Quadratic Approximation ................................................ 4.8.3 Analytic Method of Lines .................................................. 4.8.4 Methods Evaluating Probabilities ..................................... Notes and Comments ...................................................................... Exercises ........................................................................................ 152 152 154 157 161 162 165 165 167 169 172 173 174 178 Chapter 5 Finite-Element Methods .................................................... 183 5.1 Weighted Residuals ..................................................................... 5.1.1 The Principle of Weighted Residuals ................................. 5.1.2 Examples of Weighting Functions .................................... 5.1.3 Examples of Basis Functions ........................................... 5.2 Galerkin Approach with Hat Functions ........................................ 5.2.1 Hat Functions ................................................................. 5.2.2 Assembling ..................................................................... 5.2.3 A Simple Application ....................................................... 5.3 Application to Standard Options .................................................. 5.4 Error Estimates .......................................................................... 5.4.1 Classical and Weak Solutions .......................................... 5.4.2 Approximation on Finite-Dimensional Subspaces ............. 5.4.3 Céa's Lemma ................................................................... Notes and Comments ........................................................................ Exercises .......................................................................................... 184 184 186 187 188 189 191 192 194 198 199 201 202 205 206 Chapter 6 Pricing of Exotic Options ................................................... 209 6.1 Exotic Options ............................................................................ 6.2 Options Depending on Several Assets .......................................... 6.3 Asian Options ............................................................................. 6.3.1 The Payoff ....................................................................... 6.3.2 Modeling in the Black-Scholes Framework ....................... 6.3.3 Reduction to a One-Dimensional Equation ....................... 6.3.4 Discrete Monitoring ......................................................... 6.4 Numerical Aspects ...................................................................... 6.4.1 Convection-Diffusion Problems ........................................ 6.4.2 Von Neumann Stability Analysis ...................................... 6.5 Upwind Schemes and Other Methods .......................................... 210 211 214 214 215 216 220 222 222 225 226 XVI Contents 6.5.1 Upwind Scheme .............................................................. 6.5.2 Dispersion ...................................................................... 6.6 High-Resolution Methods ........................................................... 6.6.1 The Lax-Wendroff Method ................................................ 6.6.2 Total Variation Diminishing ............................................. 6.6.3 Numerical Dissipation ..................................................... Notes and Comments ........................................................................ Exercises .......................................................................................... 226 230 231 231 232 233 235 237 Appendices ......................................................................................... 239 A Financial Derivatives ..................................................................... Al Investment and Risk ..................................................... A2 Financial Derivatives ..................................................... A3 Forwards and the No-Arbitrage Principle ....................... A4 The Black-Scholes Equation ................................................ A5 Early-Exercise Curve ..................................................... B Stochastic Tools ............................................................................ B1 Essentials of Stochastics ............................................... B2 Advanced Topics ........................................................... B3 State-Price Process ........................................................ C Numerical Methods ....................................................................... Cl Basic Numerical Tools ................................................... C2 Iterative Methods for Ax = b ................................................................ C3 Function Spaces ........................................................... D Complementary Material ............................................................... D1 Bounds for Options ....................................................... D2 Approximation Formula ...................................................... D3 Software ............................................................................. 239 239 240 243 244 249 253 253 257 260 265 265 270 272 277 277 279 281 References ..................................................................................... 283 Index .................................................................................................... 293

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