Handbook in Monte Carlo Simulation: Applications in Financial Engineering, Risk Management, and Economics (Wiley Handbooks in Financial Engineering and Econometrics)

An accessible treatment of Monte Carlo methods, techniques, and applications in the field of finance and economics Providing readers with an in-depth and comprehensive guide, the __Handbook in Monte Carlo Simulation: Applications in Financial Engineering, Risk Management, and Economics__ presents a timely account of the applicationsof Monte Carlo methods in financial engineering and economics. Written by an international leading expert in thefield, the handbook illustrates the challenges confronting present-day financial practitioners and provides various applicationsof Monte Carlo techniques to answer these issues. The book is organized into five parts: introduction andmotivation; input analysis, modeling, and estimation; random variate and sample path generation; output analysisand variance reduction; and applications ranging from option pricing and risk management to optimization. The __Handbook in Monte Carlo Simulation__ features: * An introductory section for basic material on stochastic modeling and estimation aimed at readers who may need a summary or review of the essentials * Carefully crafted examples in order to spot potential pitfalls and drawbacks of each approach * An accessible treatment of advanced topics such as low-discrepancy sequences, stochastic optimization, dynamic programming, risk measures, and Markov chain Monte Carlo methods * Numerous pieces of R code used to illustrate fundamental ideas in concrete terms and encourage experimentation The __Handbook in Monte Carlo Simulation: Applications in Financial Engineering, Risk Management, and Economics__ is a complete reference for practitioners in the fields of finance, business, applied statistics, econometrics, and engineering, as well as a supplement for MBA and graduate-level courses on Monte Carlo methods and simulation. Cover 1 Title Page 5 Copyright Page 6 Contents 7 Preface 15 Part I Overview and Motivation 21 1 Introduction to Monte Carlo Methods 23 1.1 Historical origin of Monte Carlo simulation 24 1.2 Monte Carlo simulation vs. Monte Carlo sampling 27 1.3 System dynamics and the mechanics of Monte Carlo simulation 30 1.3.1 Discrete-time models 30 1.3.2 Continuous-time models 33 1.3.3 Discrete-event models 36 1.4 Simulation and optimization 42 1.4.1 Nonconvex optimization 43 1.4.2 Stochastic optimization 46 1.4.3 Stochastic dynamic programming 48 1.5 Pitfalls in Monte Carlo simulation 50 1.5.1 Technical issues 51 1.5.2 Philosophical issues 53 1.6 Software tools for Monte Carlo simulation 55 1.7 Prerequisites 57 1.7.1 Mathematical background 57 1.7.2 Financial background 58 1.7.3 Technical background 58 For further reading 59 References 59 2 Numerical Integration Methods 61 2.1 Classical quadrature formulas 63 2.1.1 The rectangle rule 64 2.1.2 Interpolatory quadrature formulas 65 2.1.3 An alternative derivation 66 2.2 Gaussian quadrature 68 2.2.1 Theory of Gaussian quadrature: The role of orthogonal polynomials 69 2.2.2 Gaussian quadrature in R 71 2.3 Extension to higher dimensions: Product rules 73 2.4 Alternative approaches for high-dimensional integration 75 2.4.1 Monte Carlo integration 76 2.4.2 Low-discrepancy sequences 79 2.4.3 Lattice methods 81 2.5 Relationship with moment matching 87 2.5.1 Binomial lattices 87 2.5.2 Scenario generation in stochastic programming 89 2.6 Numerical integration in R 89 For further reading 91 References 91 Part II Input Analysis: Modeling and Estimation 93 3 Stochastic Modeling in Finance and Economics 95 3.1 Introductory examples 97 3.1.1 Single-period portfolio optimization and modeling returns 98 3.1.2 Consumption-saving with uncertain labor income 101 3.1.3 Continuous-time models for asset prices and interest rates 103 3.2 Some common probability distributions 106 3.2.1 Bernoulli, binomial, and geometric variables 108 3.2.2 Exponential and Poisson distributions 112 3.2.3 Normal and related distributions 118 3.2.4 Beta distribution 125 3.2.5 Gamma distribution 127 3.2.6 Empirical distributions 128 3.3 Multivariate distributions: Covariance and correlation 132 3.3.1 Multivariate distributions 133 3.3.2 Covariance and Pearson's correlation 138 3.3.3 R functions for covariance and correlation 142 3.3.4 Some typical multivariate distributions 144 3.4 Modeling dependence with copulas 148 3.4.1 Kendall's tau and Spearman's rho 154 3.4.2 Tail dependence 156 3.5 Linear regression models: A probabilistic view 157 3.6 Time series models 158 3.6.1 Moving-average processes 162 3.6.2 Autoregressive processes 167 3.6.3 ARMA and ARIMA processes 171 3.6.4 Vector autoregressive models 175 3.6.5 Modeling stochastic volatility 177 3.7 Stochastic differential equations 179 3.7.1 From discrete to continuous time 180 3.7.2 Standard Wiener process 183 3.7.3 Stochastic integration and Itô's lemma 187 3.7.4 Geometric Brownian motion 193 3.7.5 Generalizations 196 3.8 Dimensionality reduction 198 3.8.1 Principal component analysis (PCA) 199 3.8.2 Factor models 209 3.9 Risk-neutral derivative pricing 212 3.9.1 Option pricing in the binomial model 213 3.9.2 A continuous-time model for option pricing: The Black-Scholes-Merton formula 216 3.9.3 Option pricing in incomplete markets 223 For further reading 225 References 226 4 Estimation and Fitting 229 4.1 Basic inferential statistics in R 231 4.1.1 Confidence intervals 231 4.1.2 Hypothesis testing 234 4.1.3 Correlation testing 238 4.2 Parameter estimation 239 4.2.1 Features of point estimators 241 4.2.2 The method of moments 242 4.2.3 The method of maximum likelihood 243 4.2.4 Distribution fitting in R 247 4.3 Checking the fit of hypothetical distributions 248 4.3.1 The chi-square test 249 4.3.2 The Kolmogorov-Smirnov test 251 4.3.3 Testing normality 252 4.4 Estimation of linear regression models by ordinary least squares 253 4.5 Fitting time series models 257 4.6 Subjective probability: The Bayesian view 259 4.6.1 Bayesian estimation 261 4.6.2 Bayesian learning and coin flipping 263 For further reading 268 References 269 Part III Sampling and Path Generation 271 5 Random Variate Generation 273 5.1 The structure of a Monte Carlo simulation 274 5.2 Generating pseudorandom numbers 276 5.2.1 Linear congruential generators 276 5.2.2 Desirable properties of random number generators 280 5.2.3 General structure of random number generators 284 5.2.4 Random number generators in R 286 5.3 The inverse transform method 287 5.4 The acceptance-rejection method 289 5.5 Generating normal variates 296 5.5.1 Sampling the standard normal distribution 296 5.5.2 Sampling a multivariate normal distribution 298 5.6 Other ad hoc methods 302 5.7 Sampling from copulas 303 For further reading 306 References 307 6 Sample Path Generation for Continuous-Time Models 309 6.1 Issues in path generation 310 6.1.1 Euler vs. Milstein schemes 313 6.1.2 Predictor-corrector methods 315 6.2 Simulating geometric Brownian motion 317 6.2.1 An application: Pricing a vanilla call option 319 6.2.2 Multidimensional GBM 321 6.2.3 The Brownian bridge 324 6.3 Sample paths of short-term interest rates 328 6.3.1 The Vasicek short-rate model 330 6.3.2 The Cox-Ingersoll-Ross short-rate model 332 6.4 Dealing with stochastic volatility 335 6.5 Dealing with jumps 336 For further reading 339 References 340 Part IV Output Analysis and Efficiency Improvement 343 7 Output Analysis 345 7.1 Pitfalls in output analysis 347 7.1.1 Bias and dependence issues: A financial example 350 7.2 Setting the number of replications 354 7.3 A world beyond averages 355 7.4 Good and bad news 357 For further reading 358 References 358 8 Variance Reduction Methods 361 8.1 Antithetic sampling 362 8.2 Common random numbers 368 8.3 Control variates 369 8.4 Conditional Monte Carlo 373 8.5 Stratified sampling 377 8.6 Importance sampling 384 8.6.1 Importance sampling and rare events 390 8.6.2 A digression: Moment and cumulant generating functions 393 8.6.3 Exponential tilting 394 For further reading 396 References 397 9 Low-Discrepancy Sequences 399 9.1 Low-discrepancy sequences 400 9.2 Halton sequences 402 9.3 Sobol low-discrepancy sequences 407 9.3.1 Sobol sequences and the algebra of polynomials 409 9.4 Randomized and scrambled low-discrepancy sequences 413 9.5 Sample path generation with low-discrepancy sequences 415 For further reading 419 References 420 Part V Miscellaneous Applications 423 10 Optimization 425 10.1 Classification of optimization problems 427 10.2 Optimization model building 441 10.2.1 Mean-variance portfolio optimization 441 10.2.2 Modeling with logical decision variables: Optimal portfolio tracking 442 10.2.3 A scenario-based model for the newsvendor problem 445 10.2.4 Fixed-mix asset allocation 446 10.2.5 Asset pricing 447 10.2.6 Parameter estimation and model calibration 450 10.3 Monte Carlo methods for global optimization 452 10.3.1 Local search and other metaheuristics 453 10.3.2 Simulated annealing 455 10.3.3 Genetic algorithms 459 10.3.4 Particle swarm optimization 461 10.4 Direct search and simulation-based optimization methods 464 10.4.1 Simplex search 465 10.4.2 Metamodeling 466 10.5 Stochastic programming models 468 10.5.1 Two-stage stochastic linear programming with recourse 468 10.5.2 A multistage model for portfolio management 472 10.5.3 Scenario generation and stability in stochastic programming 477 10.6 Stochastic dynamic programming 489 10.6.1 The shortest path problem 490 10.6.2 The functional equation of dynamic programming 493 10.6.3 Infinite-horizon stochastic optimization 497 10.6.4 Stochastic programming with recourse vs. dynamic programming 498 10.7 Numerical dynamic programming 500 10.7.1 Approximating the value function: A deterministic example 500 10.7.2 Value iteration for infinite-horizon problems 504 10.7.3 A numerical approach to consumption-saving 513 10.8 Approximate dynamic programming 526 10.8.1 A basic version of ADP 527 10.8.2 Post-decision state variables in ADP 530 10.8.3 Q-learning for a simple MDP 533 For further reading 539 References 540 11 Option Pricing 545 11.1 European-style multidimensional options in the BSM world 546 11.2 European-style path-dependent options in the BSM world 552 11.2.1 Pricing a barrier option 552 11.2.2 Pricing an arithmetic average Asian option 559 11.3 Pricing options with early exercise features 566 11.3.1 Sources of bias in pricing options with early exercise features 568 11.3.2 The scenario tree approach 569 11.3.3 The regression-based approach 572 11.4 A look outside the BSM world: Equity options under the Heston model 580 11.5 Pricing interest rate derivatives 583 11.5.1 Pricing bonds and bond options under the Vasicek model 585 11.5.2 Pricing a zero-coupon bond under the CIR model 587 For further reading 589 References 589 12 Sensitivity Estimation 593 12.1 Estimating option greeks by finite differences 595 12.2 Estimating option greeks by pathwise derivatives 601 12.3 Estimating option greeks by the likelihood ratio method 605 For further reading 609 References 610 13 Risk Measurement and Management 611 13.1 What is a risk measure? 613 13.2 Quantile-based risk measures: Value-at-risk 615 13.3 Issues in Monte Carlo estimation of V@R 621 13.4 Variance reduction methods for V@R 627 13.5 Mean-risk models in stochastic programming 633 13.6 Simulating delta hedging strategies 639 13.7 The interplay of financial and nonfinancial risks 645 For further reading 646 References 647 14 Markov Chain Monte Carlo and Bayesian Statistics 649 14.1 Acceptance-rejection sampling in Bayesian statistics 650 14.2 An introduction to Markov chains 651 14.3 The Metropolis–Hastings algorithm 656 14.3.1 The Gibbs sampler 660 14.4 A re-examination of simulated annealing 663 For further reading 666 References 667 Index 669 EULA 685 The Handbook in Monte Carlo Simulation Applications in Financial Engineering, Risk Management, and Economics features: An introductory section for basic material on stochastic modeling and estimation aimed at readers who may need a summary or review of the essentials Carefully crafted examples in order to spot potential pitfalls and drawbacks of each approach An accessible treatment of advanced topics such as low discrepancy sequences, stochastic optimization, dynamic programming, risk measures, and Markov chain Monte Carlo methods Numerous pieces of R code used to illustrate fundamental ideas in concrete terms and encourage experimentation