Chapman & Hall/CRC FINANCIAL MATHEMATICS SERIES Monte Carlo Methods and Models in Finance and Insurance Ralf Korn Elke Korn Gerald Kroisandt f r oc) CRC Press \ V^ J Taylor & Francis Croup ^^"^ Boca Raton London New York CRC Press is an imprint of the Taylor St Francis Croup, an informa business A CHAPMAN & HALL BOOK
Contents List of Algorithms xi 1 Introduction and User Guide ' 1 1.1 Introduction and concept 1 1.2 Contents 2 1.3 How to use this book 3 1.4 Further literature 3 1.5 Acknowledgments 4 2 Generating Random Numbers 5 2.1 Introduction 5 2.1.1 How do we get random numbers? 5 2.1.2 Quality criteria for RNGs 6 2.1.3 Technical terms 8 2.2 Examples of random number generators 8 2.2.1 Linear congruential generators 8 2.2.2 Multiple recursive generators 12 2.2.3 Combined generators 15 2.2.4 Lagged Fibonacci generators 16 2.2.5 F2-linear generators 17 2.2.6 Nonlinear RNGs 22 2.2.7 More random number generators 24 2.2.8 Improving RNGs 24 2.3 Testing and analyzing RNGs 25 2.3.1 Analyzing the lattice structure 25 2.3.2 Equidistribution 26 2.3.3 Diffusion capacity 27 2.3.4 Statistical tests 27 2.4 Generating random numbers with general distributions... 31 2.4.1 Inversion method 31 2.4.2 Acceptance-rejection method 33 2.5 Selected distributions 36 2.5.1 Generating normally distributed random numbers.. 36 2.5.2 Generating beta-distributed RNs 38 2.5.3 Generating Weibull-distributed RNs 38 2.5.4 Generating gamma-distributed RNs 39 2.5.5 Generating chi-square-distributed RNs 42
VI 2.6 Multivariate random variables 43 2.6.1 Multivariate normals 43 2.6.2 Remark: Copulas 44 2.6.3 Sampling from conditional distributions 44 2.7 Quasirandom sequences as a substitute for random sequences 45 2.7.1 Halton sequences 47 2.7.2 Sobol sequences 48 2.7.3 Randomized quasi-monte Carlo methods 49 2.7.4 Hybrid Monte Carlo methods 50 2.7.5 Quasirandom sequences and transformations into other random distributions 50 2.8 Parallelization techniques 51 2.8.1 Leap-frog method 51 2.8.2 Sequence splitting 52 2.8.3 Several RNGs 53 2.8.4 Independent sequences 53 " 2.8.5 Testing parallel RNGs 53 The Monte Carlo Method: Basic Principles 55 3.1 Introduction 55 3.2 The strong law of large numbers and the Monte Carlo method 56 3.2.1 The strong law of large numbers 56 3.2.2 The crude Monte Carlo method 57 3.2.3 The Monte Carlo method: Some first applications.. 60 3.3 Improving the speed of convergence of the Monte Carlo method: Variance reduction methods 65 3.3.1 Antithetic variates 66 3.3.2 Control variates 70 3.3.3 Stratified sampling 76 3.3.4 Variance reduction by conditional sampling 85 3.3.5 Importance sampling 87 3.4 Further aspects of variance reduction methods 97 3.4.1 More methods 97 3.4.2 Application of the variance reduction methods... 100 Continuous-Time Stochastic Processes: Continuous Paths 103 4.1 Introduction 103 4.2 Stochastic processes and their paths: Basic definitions... 103 4.3 The Monte Carlo method for stochastic processes 107 4.3.1 Monte Carlo and stochastic processes 107 4.3.2 Simulating paths of stochastic processes: Basics... 108 4.3.3 Variance reduction for stochastic processes 110 4.4 Brownian motion and the Brownian bridge Ill 4.4.1 Properties of Brownian motion 113 4.4.2 Weak convergence and Donsker's theorem 116
Vll 4.4.3 Brownian bridge 120 4.5 Basics of Ito calculus 126 4.5.1 The Ito integral 126 4.5.2 The Ito formula 133 4.5.3 Martingale representation and change of measure... 135 4.6 Stochastic differential equations 137 4.6.1 Basic results on stochastic differential equations... 137 4.6.2 Linear stochastic differential equations 139 4.6.3 The square-root stochastic differential equation... 141 4.6.4 The Feynman-Kac representation theorem 142 4.7 Simulating solutions of stochastic differential'equations... 145 4.7.1 Introduction and basic aspects 145 4.7.2 Numerical schemes for ordinary differential equations 146 4.7.3 Numerical schemes for stochastic differential equations 151 4.7.4 Convergence of numerical schemes for SDEs 156 4.7.5 More numerical schemes for SDEs 159 4.7.6 Efficiency of numerical schemes for SDEs 162 4.7.7 Weak extrapolation methods 163 4.7.8 The multilevel Monte Carlo method 167 4.8 Which simulation methods for SDE should be chosen?... 173 Simulating Financial Models: Continuous Paths 175 5.1 Introduction 175 5.2 Basics of stock price modelling 176 5.3 A Black-Scholes type stock price framework 177 5.3.1 An important special case: The Black-Scholes model. 180 5.3.2 Completeness of the market model 183 5.4 Basic facts of options 184 5.5 An introduction to option pricing 187 5.5.1 A short history of option pricing 187 5.5.2 Option pricing via the replication principle 187 5.5.3 Dividends in the Black-Scholes setting 195 5.6 Option pricing and the Monte Carlo method in the Black- Scholes setting 196 5.6.1 Path-independent European options 197 5.6.2 Path-dependent European options 199 5.6.3 More exotic options 210 5.6.4 Data preprocessing by moment matching methods.. 211 5.7 Weaknesses of the Black-Scholes model 213 5.8 Local volatility models and the CEV model 216 5.8.1 CEV option pricing with Monte Carlo methods... 219 5.9 An excursion: Calibrating a model 221 5.10 Aspects of option pricing in incomplete markets 222 5.11 Stochastic volatility and option pricing in the Heston model 224 5.11.1 The Andersen algorithm for the Heston model... 227
vm 5.11.2 The Heath-Platen estimator in the Heston model... 232 5.12 Variance reduction principles in non-black-scholes models. 238 5.13 Stochastic local volatility models 239 5.14 Monte Carlo option pricing: American and Bermudan options 240 5.14.1 The Longstaff-Schwartz algorithm and regression-based variants for pricing Bermudan options 243 5.14.2 Upper price bounds by dual methods 250 5.15 Monte Carlo calculation of option price sensitivities 257 5.15.1 The role of the price sensitivities 257 5.15.2 Finite difference simulation 258 5.15.3 The pathwise differentiation method 261 5.15.4 The likelihood ratio method 264 5.15.5 Combining the pathwise differentiation and the likelihood ratio methods by localization 265 5.15.6 Numerical testing in the Black-Scholes setting 267 5.16 Basics of interest rate modelling 269 5.16.1 Different notions of interest rates 270 5.16.2 Some popular interest rate products 271 5.17 The short rate approach to interest rate modelling 275 5.17.1 Change of numeraire and option pricing: The forward measure 276 5.17.2 The Vasicek model 278 5.17.3 The Cox-Ingersoll-Ross (CIR) model 281 5.17.4 AfHne linear short rate models 283 5.17.5 Perfect calibration: Deterministic shifts and the Hull- White approach.- 283 5.17.6 Log-normal models and further short rate models... 287 5.18 The forward rate approach to interest rate modelling... 288 5.18.1 The continuous-time Ho-Lee model 289 5.18.2 The Cheyette model 290 5.19 LIBOR market models x 293 5.19.1 Log-normal forward-libor modelling 294 5.19.2 Relation between the swaptions and the cap market. 297 5.19.3 Aspects of Monte Carlo path simulations of forward- LIBOR rates and derivative pricing 299 5.19.4 Monte Carlo pricing of Bermudan swaptions with a parametric exercise boundary and further comments. 305 5.19.5 Alternatives to log-normal forward-libor models.. 308 Continuous-Time Stochastic Processes: Discontinuous Paths 309 6.1 Introduction 309 6.2 Poisson processes and Poisson random-measures: Definition and simulation 310 6.2.1 Stochastic integrals with respect to Poisson processes 312 6.3 Jump-diffusions: Basics, properties, and simulation 315
IX 6.3.1 Simulating Gauss-Poisson jump-diffusions 317 6.3.2 Euler-Maruyama scheme for jump-diffusions 319 6.4 Levy processes: Properties and examples 320 6.4.1 Definition and properties of Levy processes 320 6.4.2 Examples of Levy processes 324 6.5 Simulation of Levy processes 329 6.5.1 Exact simulation and time discretization 329 6.5.2 The Euler-Maruyama scheme for Levy processes... 330 6.5.3 Small jump approximation 331 6.5.4 Simulation via series representation 333 Simulating Financial Models: Discontinuous Paths 335 7.1 Introduction 335 7.2 Merton's jump-diffusion model and stochastic volatility models with jumps 335 7.2.1 Merton's jump-diffusion setting 335 7.2.2 Jump-diffusion with double exponential jumps... 339 7.2.3 Stochastic volatility models with jumps 340 7.3 Special Levy models and their simulation 340 7.3.1 The Esscher transform 341 7.3.2 The hyperbolic Levy model 342 7.3.3 The variance gamma model 344 7.3.4 Normal inverse Gaussian processes 352 7.3.5 Further aspects of Levy type models 354 Simulating Actuarial Models 357 8.1 Introduction 357 8.2 Premium principles and risk measures 357 8.2.1 Properties and examples of premium principles... 358 8.2.2 Monte Carlo simulation of premium principles... 362 8.2.3 Properties and examples of risk measures! 362 8.2.4 Connection between premium principles and risk measures 365 8.2.5 Monte Carlo simulation of risk measures 366 8.3 Some applications of Monte Carlo methods in life insurance. 377 8.3.1 Mortality: Definitions and classical models 378 8.3.2 Dynamic mortality models 379 8.3.3 Life insurance contracts and premium calculation... 383 8.3.4 Pricing longevity products by Monte Carlo simulation 385 8.3.5 Premium reserves and Thiele's differential equation.. 387 8.4 Simulating dependent risks with copulas 390 8.4.1 Definition and basic properties 390 8.4.2 Examples and simulation of copulas 393 8.4.3 Application in actuarial models 402 8.5 Nonlife insurance 403
8.5.1 The individual model 404 8.5.2 The collective model 405 8.5.3 Rare event simulation and heavy-tailed distributions. 410 8.5.4 Dependent claims: An example with copulas 413 8.6 Markov chain Monte Carlo and Bayesian estimation 415 8.6.1 Basic properties of Markov chains 415 8.6.2 Simulation of Markov chains 419 8.6.3 Markov chain Monte Carlo methods 420 8.6.4 MCMC methods and Bayesian estimation 427 8.6.5 Examples of MCMC methods and Bayesian estimation in actuarial mathematics 429 8.7 Asset-liability management and Solvency II 433 8.7.1 Solvency II 433 8.7.2 Asset-liability management (ALM) 435 References 441 Index 459