Monte Carlo Methods and Models in Finance and Insurance

Transcription

1 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

2 Contents List of Algorithms xi 1 Introduction and User Guide ' Introduction and concept Contents How to use this book Further literature Acknowledgments 4 2 Generating Random Numbers Introduction How do we get random numbers? Quality criteria for RNGs Technical terms Examples of random number generators Linear congruential generators Multiple recursive generators Combined generators Lagged Fibonacci generators F2-linear generators Nonlinear RNGs More random number generators Improving RNGs Testing and analyzing RNGs Analyzing the lattice structure Equidistribution Diffusion capacity Statistical tests Generating random numbers with general distributions Inversion method Acceptance-rejection method Selected distributions Generating normally distributed random numbers Generating beta-distributed RNs Generating Weibull-distributed RNs Generating gamma-distributed RNs Generating chi-square-distributed RNs 42

3 VI 2.6 Multivariate random variables Multivariate normals Remark: Copulas Sampling from conditional distributions Quasirandom sequences as a substitute for random sequences Halton sequences Sobol sequences Randomized quasi-monte Carlo methods Hybrid Monte Carlo methods Quasirandom sequences and transformations into other random distributions Parallelization techniques Leap-frog method Sequence splitting Several RNGs Independent sequences 53 " Testing parallel RNGs 53 The Monte Carlo Method: Basic Principles Introduction The strong law of large numbers and the Monte Carlo method The strong law of large numbers The crude Monte Carlo method The Monte Carlo method: Some first applications Improving the speed of convergence of the Monte Carlo method: Variance reduction methods Antithetic variates Control variates Stratified sampling Variance reduction by conditional sampling Importance sampling Further aspects of variance reduction methods More methods Application of the variance reduction methods Continuous-Time Stochastic Processes: Continuous Paths Introduction Stochastic processes and their paths: Basic definitions The Monte Carlo method for stochastic processes Monte Carlo and stochastic processes Simulating paths of stochastic processes: Basics Variance reduction for stochastic processes Brownian motion and the Brownian bridge Ill Properties of Brownian motion Weak convergence and Donsker's theorem 116

4 Vll Brownian bridge Basics of Ito calculus The Ito integral The Ito formula Martingale representation and change of measure Stochastic differential equations Basic results on stochastic differential equations Linear stochastic differential equations The square-root stochastic differential equation The Feynman-Kac representation theorem Simulating solutions of stochastic differential'equations Introduction and basic aspects Numerical schemes for ordinary differential equations Numerical schemes for stochastic differential equations Convergence of numerical schemes for SDEs More numerical schemes for SDEs Efficiency of numerical schemes for SDEs Weak extrapolation methods The multilevel Monte Carlo method Which simulation methods for SDE should be chosen? Simulating Financial Models: Continuous Paths Introduction Basics of stock price modelling A Black-Scholes type stock price framework An important special case: The Black-Scholes model Completeness of the market model Basic facts of options An introduction to option pricing A short history of option pricing Option pricing via the replication principle Dividends in the Black-Scholes setting Option pricing and the Monte Carlo method in the Black- Scholes setting Path-independent European options Path-dependent European options More exotic options Data preprocessing by moment matching methods Weaknesses of the Black-Scholes model Local volatility models and the CEV model CEV option pricing with Monte Carlo methods An excursion: Calibrating a model Aspects of option pricing in incomplete markets Stochastic volatility and option pricing in the Heston model The Andersen algorithm for the Heston model

5 vm The Heath-Platen estimator in the Heston model Variance reduction principles in non-black-scholes models Stochastic local volatility models Monte Carlo option pricing: American and Bermudan options The Longstaff-Schwartz algorithm and regression-based variants for pricing Bermudan options Upper price bounds by dual methods Monte Carlo calculation of option price sensitivities The role of the price sensitivities Finite difference simulation The pathwise differentiation method The likelihood ratio method Combining the pathwise differentiation and the likelihood ratio methods by localization Numerical testing in the Black-Scholes setting Basics of interest rate modelling Different notions of interest rates Some popular interest rate products The short rate approach to interest rate modelling Change of numeraire and option pricing: The forward measure The Vasicek model The Cox-Ingersoll-Ross (CIR) model AfHne linear short rate models Perfect calibration: Deterministic shifts and the Hull- White approach Log-normal models and further short rate models The forward rate approach to interest rate modelling The continuous-time Ho-Lee model The Cheyette model LIBOR market models x Log-normal forward-libor modelling Relation between the swaptions and the cap market Aspects of Monte Carlo path simulations of forward- LIBOR rates and derivative pricing Monte Carlo pricing of Bermudan swaptions with a parametric exercise boundary and further comments Alternatives to log-normal forward-libor models Continuous-Time Stochastic Processes: Discontinuous Paths Introduction Poisson processes and Poisson random-measures: Definition and simulation Stochastic integrals with respect to Poisson processes Jump-diffusions: Basics, properties, and simulation 315

6 IX Simulating Gauss-Poisson jump-diffusions Euler-Maruyama scheme for jump-diffusions Levy processes: Properties and examples Definition and properties of Levy processes Examples of Levy processes Simulation of Levy processes Exact simulation and time discretization The Euler-Maruyama scheme for Levy processes Small jump approximation Simulation via series representation 333 Simulating Financial Models: Discontinuous Paths Introduction Merton's jump-diffusion model and stochastic volatility models with jumps Merton's jump-diffusion setting Jump-diffusion with double exponential jumps Stochastic volatility models with jumps Special Levy models and their simulation The Esscher transform The hyperbolic Levy model The variance gamma model Normal inverse Gaussian processes Further aspects of Levy type models 354 Simulating Actuarial Models Introduction Premium principles and risk measures Properties and examples of premium principles Monte Carlo simulation of premium principles Properties and examples of risk measures! Connection between premium principles and risk measures Monte Carlo simulation of risk measures Some applications of Monte Carlo methods in life insurance Mortality: Definitions and classical models Dynamic mortality models Life insurance contracts and premium calculation Pricing longevity products by Monte Carlo simulation Premium reserves and Thiele's differential equation Simulating dependent risks with copulas Definition and basic properties Examples and simulation of copulas Application in actuarial models Nonlife insurance 403

7 8.5.1 The individual model The collective model Rare event simulation and heavy-tailed distributions Dependent claims: An example with copulas Markov chain Monte Carlo and Bayesian estimation Basic properties of Markov chains Simulation of Markov chains Markov chain Monte Carlo methods MCMC methods and Bayesian estimation Examples of MCMC methods and Bayesian estimation in actuarial mathematics Asset-liability management and Solvency II Solvency II Asset-liability management (ALM) 435 References 441 Index 459