Numerical Methods and Optimization in Finance [2nd ed.]

Description

Contents......Page 3 Figures......Page 9 Tables......Page 15 Algorithms......Page 16 Foreword......Page 18 --- Fundamentals......Page 20 Growth of computing power......Page 21 Computational finance......Page 22 Principles......Page 24 Software......Page 25 Approximations & accuracy......Page 28 Theme of the book......Page 33 Representation of real numbers......Page 34 Example of limitations of floating point arithmetic......Page 36 Measuring errors......Page 37 Approximating 1st-order derivatives......Page 38 How to choose......Page 39 Truncation error for forward difference......Page 40 Example of numerically unstable algorithm......Page 41 Example of ill-conditioned problem......Page 42 Condition number of a matrix......Page 43 Comments and examples......Page 44 Criteria for comparison......Page 45 Order of complexity and classification......Page 46 Operation count for basic linear algebra operations......Page 47 Linear Equations & Least Squares......Page 48 Triangular systems......Page 49 LU factorization......Page 50 Cholesky factorization......Page 52 Singular value decomposition......Page 54 Iterative methods......Page 55 Jacobi, Gauss–Seidel, and SOR......Page 56 Successive overrelaxation......Page 57 Convergence of iterative methods......Page 58 General structure of algorithms for iterative methods......Page 59 Block iterative methods......Page 61 Tridiagonal systems......Page 62 Irregular sparse matrices......Page 64 Structural properties of sparse matrices......Page 65 The Least Squares problem......Page 67 Method of normal equations......Page 68 Least Squares via QR factorization......Page 71 Least Squares via SVD decomposition......Page 72 Solving linear systems in R......Page 73 Least Squares......Page 75 Example of a numerical solution......Page 77 First numerical approximation......Page 78 Second numerical approximation......Page 79 Classification of differential equations......Page 80 The Black–Scholes equation......Page 81 Initial & boundary conditions and definition of the grid......Page 83 Implementation of θ-method with MatLab......Page 87 Stability......Page 89 Coordinate transformation of space variables......Page 92 American options......Page 95 Note on MatLab’s spdiags function......Page 102 Matching moments......Page 104 Growing the tree......Page 105 Implementing a tree......Page 106 Vectorization......Page 107 Binomial expansion......Page 108 Dividends......Page 110 Greeks from the tree......Page 112 --- Simulation......Page 115 Monte Carlo methods & sampling......Page 116 Uniform random number generators Congruential generators......Page 117 Nonuniform distributions The inversion method......Page 120 Acceptance–rejection method......Page 122 Specialized methods for selected distributions Normal distribution......Page 124 Higher order moments and the Cornish–Fisher expansion......Page 126 Further distributions......Page 127 Sampling from a discrete set Discrete uniform selection......Page 129 Roulette wheel selection......Page 130 Sampling errors—and how to reduce them The basic problem......Page 131 Quasi-Monte Carlo......Page 132 Stratified sampling......Page 133 Variance reduction......Page 134 Bootstrap......Page 135 analysis......Page 140 function......Page 141 function......Page 142 Remedies......Page 144 Transformation methods Linear correlation......Page 146 Rank correlation......Page 151 Markov chains Concepts......Page 157 The Metropolis algorithm......Page 159 Copula models Concepts......Page 161 Simulation using copulas......Page 163 Setting the stage......Page 166 Single-period simulations Terminal asset prices......Page 167 1-over-N portfolios......Page 168 European options......Page 170 VaR of a covered put portfolio......Page 172 Simple price processes......Page 174 Moving averages......Page 176 Autoregressive models......Page 177 Autoregressive moving average (ARMA) models......Page 178 Simulating ARMA models......Page 179 Models with long-term memory......Page 180 Time-varying volatility The concepts......Page 182 Autocorrelated time-varying volatility......Page 183 Simulating GARCH processes......Page 186 Selected further autoregressive volatility models......Page 188 Adaptive expectations and patterns in price processes Price–earnings models......Page 191 Models with learning......Page 192 Historical simulation Backtesting......Page 193 Bootstrap......Page 194 Agent-based models and complexity......Page 198 Constant proportion portfolio insurance (CPPI) Basic concepts......Page 202 Bootstrap......Page 204 VaR estimation with Extreme Value Theory Basic concepts......Page 205 Using Extreme Value Theory......Page 206 Option pricing......Page 208 Modeling prices......Page 209 Pricing models......Page 212 Greeks......Page 221 Quasi-Monte Carlo......Page 223 --- Optimization......Page 230 What to optimize?......Page 231 Solving the model Problems......Page 232 Evaluating solutions......Page 234 Examples Portfolio optimization with alternative risk measures......Page 236 Robust/resistant regression......Page 237 Calibration of option-pricing models......Page 238 Calibration of yield structure models......Page 239 Summary......Page 240 A naïve approach......Page 241 Graphical solution......Page 242 Bracketing......Page 243 Bisection......Page 244 Fixed point method......Page 245 Convergence......Page 247 Newton’s method......Page 250 Comments......Page 252 Classical unconstrained optimization......Page 253 Convergence......Page 254 Newton’s method......Page 255 Golden section search......Page 256 Unconstrained optimization in multiple dimensions Steepest descent method......Page 257 Newton’s method......Page 259 Quasi-Newton method......Page 260 Direct search methods......Page 262 Practical issues with MATLAB......Page 266 Nonlinear Least Squares Problem statement and notation......Page 268 Gauss–Newton method......Page 269 Levenberg–Marquardt method......Page 270 General considerations......Page 272 Fixed point methods......Page 274 Newton’s method......Page 275 Quasi-Newton methods......Page 280 Further approaches......Page 281 Synoptic view of solution methods......Page 282 Heuristics......Page 284 What is a heuristic?......Page 285 Iterative search......Page 286 Stochastic Local Search......Page 287 Simulated Annealing......Page 288 Threshold Accepting......Page 289 Genetic Algorithms......Page 290 Differential Evolution......Page 291 PSO......Page 292 Hybrids......Page 293 Constraints......Page 295 The stochastics of heuristic search Stochastic solutions and computational resources......Page 296 Illustrative experiment......Page 298 Efficient implementations......Page 300 Parameter settings......Page 304 Implementing heuristic methods with MATLAB......Page 305 The problems......Page 307 Threshold Accepting......Page 309 Genetic Algorithm......Page 314 Differential Evolution......Page 317 Particle Swarm Optimization......Page 319 Appendix 12.B Parallel computations in MATLAB......Page 320 Parallel execution of restart loops......Page 322 Local Search......Page 325 Genetic Algorithm......Page 326 Particle Swarm Optimization......Page 327 Restarts......Page 328 Optimization Models......Page 330 The problem: choosing few from many The subset-sum problem......Page 331 Evaluating a solution......Page 332 Knowing the solution......Page 333 Solution strategies Being thorough......Page 334 Being constructive......Page 335 Being random......Page 336 Getting better......Page 338 Heuristics On heuristics......Page 341 Local Search......Page 342 Threshold Accepting......Page 347 Stochastics of LS and TA......Page 351 Application: selecting variables in a regression Linear models......Page 353 Fast least squares......Page 354 Selection criterion......Page 355 Putting it all together......Page 356 Models......Page 358 Local-Search algorithms......Page 359 The investment problem......Page 365 Mean–variance optimization The model......Page 367 Examples of mean–variance models......Page 368 True, estimated, and realized frontiers......Page 376 Repairing matrices......Page 378 Asset selection with Local Search......Page 387 Scenario Optimization with Threshold Accepting......Page 393 Portfolio optimization with TA: examples......Page 402 Diagnostics for techniques based on Local Search......Page 421 Portfolios under Value-at-Risk Why Value-at-Risk matters......Page 423 Setting up experiments......Page 424 Numerical results......Page 425 Computing returns......Page 429 Scoping rules in R & objective functions......Page 430 Vectorized objective functions......Page 432 Neighborhood for switching elements......Page 434 What is (the problem with) backtesting?......Page 437 The ugly: intentional overfitting......Page 438 The bad: unintentional overfitting and other difficulties......Page 444 The good: getting insights (and confidence) in strategies......Page 446 What data to use?......Page 447 function......Page 449 Simple backtests......Page 450 Robert Shiller’s Irrational-Exuberance data......Page 460 Kenneth French’s data library......Page 469 Momentum......Page 474 Portfolio optimization......Page 480 Notes on zoo......Page 489 Parallel computations in R......Page 490 Term structure models Yield curves......Page 497 The Nelson–Siegel model......Page 502 Calibration strategies......Page 506 Experiments......Page 528 Robust and resistant regression......Page 532 The regression model......Page 535 Estimation......Page 537 An example......Page 542 Numerical experiments......Page 545 Final remarks......Page 550 Estimating Time Series Models Adventures with time series estimation......Page 552 The case of GARCH models......Page 553 Numerical experiments with Differential Evolution......Page 555 Appendix 16.A Maximizing the Sharpe ratio......Page 559 Calibrating Option Pricing Models......Page 560 Implied volatility with Black–Scholes......Page 561 The smile......Page 563 A pricing equation......Page 564 Numerical integration......Page 569 Techniques......Page 589 Organizing the problem and implementation......Page 591 Two experiments......Page 598 Final remarks......Page 602 Quadrature rules for infinity......Page 603 NMOF Package......Page 606 Biblio......Page 608 Index......Page 618