Handbook in Monte Carlo Simulation

PAOLO BRANDIMARTE is Full Professor of Quantitative Methods for Finance and Logistics in the Department of Mathematical Sciences at Politecnico di Torino in Italy. He has extensive teaching experience in engineering and economics faculties, including master’s- and PhD-level courses. Dr. Brandimarte is the author or coauthor of Introduction to Distribution Logistics, Quantitative Methods: An Introduction for Business Management, and Numerical Methods in Finance and Economics: A MATLAB-Based Introduction, Second Edition, all published by Wiley.

Preface xiii

Part I Overview and Motivation

1 Introduction to Monte Carlo Methods 3

1.1 Historical origin of Monte Carlo simulation 4

1.2 Monte Carlo Simulation vs. Monte Carlo Sampling 7

1.3 System dynamics and the mechanics of Monte Carlo simulation 10

1.4 Simulation and optimization 21

1.5 Pitfalls in Monte Carlo simulation 30

1.6 Software tools for Monte Carlo simulation 35

1.7 Prerequisites 37

For further reading 38

Chapter References 38

2 Numerical Integration Methods 41

2.1 Classical quadrature formulae 43

2.2 Gaussian quadrature 48

2.3 Extension to higher dimensions: Product rules 53

2.4 Alternative approaches for high-dimensional integration 55

2.5 Relationship with moment matching 67

2.6 Numerical integration in R 69

For further reading 71

Chapter References 71

Part II Input Analysis: Modeling and Estimation

3 Stochastic Modeling in Finance and Economics 75

3.1 Introductory examples 77

3.2 Some common probability distributions 86

3.3 Multivariate distributions: Covariance and correlation 111

3.4 Modeling dependence with copulae 127

3.5 Linear regression models: a probabilistic view 136

3.6 Time series models 137

3.7 Stochastic differential equations 158

3.8 Dimensionality reduction 177

S3.1 Risk-neutral derivative pricing 190

S3.1.1 Option pricing in the binomial model 192

S3.1.2 A continuous-time model for option pricing: The Black–Scholes–Merton formula 194

S3.1.3 Option pricing in incomplete markets 199

For further reading 202

Chapter References 203

4 Estimation and Fitting 205

4.1 Basic inferential statistics in R 207

4.2 Parameter estimation 215

4.3 Checking the fit of hypothetical distributions 224

4.4 Estimation of linear regression models by ordinary least squares 229

4.5 Fitting time series models 232

4.6 Subjective probability: the Bayesian view 235

For further reading 244

Chapter References 245

Part III Sampling and Path Generation

5 Random Variate Generation 249

5.1 The structure of a Monte Carlo simulation 250

5.2 Generating pseudo-random numbers 252

5.3 The inverse transform method 263

5.4 The acceptance–rejection method 265

5.5 Generating normal variates 269

5.6 Other ad hoc methods 274

5.7 Sampling from copulae 276

For further reading 277

Chapter References 279

6 Sample Path Generation for Continuous-Time Models 281

6.1 Issues in path generation 282

6.2 Simulating geometric Brownian motion 287

6.3 Sample paths of short-term interest rates 298

6.4 Dealing with stochastic volatility 306

6.5 Dealing with jumps 308

For further reading 310

Chapter References 311

Part IV Output Analysis and Efficiency Improvement

7 Output Analysis 315

7.1 Pitfalls in output analysis 317

7.2 Setting the number of replications 323

7.3 A world beyond averages 325

7.4 Good and bad news 327

For further reading 327

Chapter References 328

8 Variance Reduction Methods 329

8.1 Antithetic sampling 330

8.2 Common random numbers 336

8.3 Control variates 337

8.4 Conditional Monte Carlo 341

8.5 Stratified sampling 344

8.6 Importance sampling 350

For further reading 363

Chapter References 363

9 Low-Discrepancy Sequences 365

9.1 Low-discrepancy sequences 366

9.2 Halton sequences 367

9.3 Sobol low-discrepancy sequences 374

9.4 Randomized and scrambled low-discrepancy sequences 379

9.5 Sample path generation with low-discrepancy sequences 381

For further reading 385

Chapter References 385

Part V Miscellaneous Applications

10 Optimization 389

10.1 Classification of optimization problems 390

10.2 Optimization model building 405

10.3 Monte Carlo methods for global optimization 412

10.4 Direct search and simulation-based optimization methods 416

10.5 Stochastic programming models 420

10.6 Scenario generation and Monte Carlo methods for stochastic programming 428

10.7 Stochastic dynamic programming 433

10.8 Numerical dynamic programming 440

10.9 Approximate dynamic programming 451

For further reading 453

Chapter References 453

11 Option Pricing 455

11.1 European-style multidimensional options in the BSM world 456

11.2 European-style path-dependent options in the BSM world 462

11.3 Pricing options with early exercise features 475

11.4 A look outside the BSM world 487

11.5 Pricing interest-rate derivatives 490

For further reading 497

Chapter References 498

12 Sensitivity Estimation 501

12.1 Estimating option greeks by finite differences 503

12.2 Estimating option greeks by pathwise derivatives 509

12.3 Estimating option greeks by the likelihood ratio method 513

For further reading 517

Chapter References 518

13 Risk Measurement and Management 519

13.1 What is a risk measure? 520

13.2 Quantile-based risk measures: value at risk 522

13.3 Monte Carlo methods for V@R 533

13.4 Mean-risk models in stochastic programming 537

13.5 Simulating delta-hedging strategies 540

13.6 The interplay of financial and nonfinancial risks 546

For further reading 548

Chapter References 548

14 Markov Chain Monte Carlo and Bayesian Statistics 551

14.1 An introduction to Markov chains 552

14.2 The Metropolis–Hastings algorithm 555

14.3 A re-examination of simulated annealing 558

For further reading 560

Chapter References 561

Index 563