Monte Carlo Methods and Models in Finance and Insurance

485 Pages·2010·5.049 MB·English

by  Ralf Korn, Elke Korn, Gerald Kroisandt

Monte Carlo
Methods and
Models in Finance
and Insurance
CHAPMAN & HALL/CRC

Financial Mathematics Series
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The field of financial mathematics forms an ever-expanding slice of the financial sector. This series
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Centre for Financial Robert H. Smith School Center for Financial
Research of Business Engineering
Judge Business School University of Maryland Columbia University
University of Cambridge New York
Published Titles
American-Style Derivatives; Valuation and Computation, Jerome Detemple
Analysis, Geometry, and Modeling in Finance: Advanced Methods in Option Pricing,
 Pierre Henry-Labordère
Credit Risk: Models, Derivatives, and Management, Niklas Wagner
Engineering BGM, Alan Brace
Financial Modelling with Jump Processes, Rama Cont and Peter Tankov
Interest Rate Modeling: Theory and Practice, Lixin Wu
An Introduction to Credit Risk Modeling, Christian Bluhm, Ludger Overbeck, and Christoph Wagner
Introduction to Stochastic Calculus Applied to Finance, Second Edition,
 Damien Lamberton and Bernard Lapeyre
Monte Carlo Methods and Models in Finance and Insurance, Ralf Korn, Elke Korn,
 and Gerald Kroisandt
Numerical Methods for Finance, John A. D. Appleby, David C. Edelman, and John J. H. Miller
Portfolio Optimization and Performance Analysis, Jean-Luc Prigent
Quantitative Fund Management, M. A. H. Dempster, Georg Pflug, and Gautam Mitra
Robust Libor Modelling and Pricing of Derivative Products, John Schoenmakers
Stochastic Financial Models, Douglas Kennedy
Structured Credit Portfolio Analysis, Baskets & CDOs, Christian Bluhm and Ludger Overbeck
Understanding Risk: The Theory and Practice of Financial Risk Management, David Murphy
Unravelling the Credit Crunch, David Murphy
Proposals for the series should be submitted to one of the series editors above or directly to:
CRC Press, Taylor & Francis Group
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Monte Carlo
Methods and
Models in Finance
and Insurance
Ralf Korn
Elke Korn
Gerald Kroisandt
CRC Press
Taylor & Francis Group
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Boca Raton, FL 33487-2742
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Library of Congress Cataloging‑in‑Publication Data
Korn, Ralf.
Monte Carlo methods and models in finance and insurance / Ralf Korn, Elke Korn,
Gerald Kroisandt.
p. cm. -- (Financial mathematics series)
Includes bibliographical references and index.
ISBN 978-1-4200-7618-9 (hardcover : alk. paper)
1. Business mathematics. 2. Insurance--Mathematics. 3. Monte Carlo method. I.
Korn, Elke, 1962- II. Kroisandt, Gerald. III. Title. IV. Series.
HF5691.K713 2010
518’.282--dc22 2009045581
Visit the Taylor & Francis Web site at
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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
v
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 Quasirandomsequences 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 Parallelizationtechniques . . . . . . . . . . . . . . . . . . . . 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
3 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 ImprovingthespeedofconvergenceoftheMonteCarlomethod:
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
4 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 . . . . . . . . . . 111
4.4.1 Properties of Brownian motion . . . . . . . . . . . . . 113
4.4.2 Weak convergence and Donsker’s theorem . . . . . . . 116
vii
4.4.3 Brownian bridge . . . . . . . . . . . . . . . . . . . . . 120
4.5 Basics of Itˆo calculus . . . . . . . . . . . . . . . . . . . . . . 126
4.5.1 The Itoˆ integral. . . . . . . . . . . . . . . . . . . . . . 126
4.5.2 The Itoˆ 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-rootstochastic 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
5 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
viii
5.11.2 The Heath-Platen estimator in the Heston model . . . 232
5.12 Variance reduction principles in non-Black-Scholesmodels . 238
5.13 Stochastic local volatility models . . . . . . . . . . . . . . . . 239
5.14 Monte Carlo option pricing: American and Bermudan options 240
5.14.1 TheLongstaff-Schwartzalgorithmandregression-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 Affine 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 . . . . . . . . . . . . . . . . . . . . . 293
5.19.1 Log-normal forward-LIBORmodelling . . . . . . . . . 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-normalforward-LIBORmodels . . 308
6 Continuous-TimeStochasticProcesses: DiscontinuousPaths 309
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 309
6.2 PoissonprocessesandPoissonrandommeasures: Definition
and simulation . . . . . . . . . . . . . . . . . . . . . . . . . . 310
6.2.1 Stochastic integrals with respect to Poissonprocesses 312
6.3 Jump-diffusions: Basics, properties, and simulation . . . . . 315
ix
6.3.1 Simulating Gauss-Poissonjump-diffusions . . . . . . . 317
6.3.2 Euler-Maruyama scheme for jump-diffusions . . . . . . 319
6.4 L´evy processes: Properties and examples . . . . . . . . . . . 320
6.4.1 Definition and properties of L´evy processes . . . . . . 320
6.4.2 Examples of L´evy processes . . . . . . . . . . . . . . . 324
6.5 Simulation of L´evy processes . . . . . . . . . . . . . . . . . . 329
6.5.1 Exact simulation and time discretization . . . . . . . . 329
6.5.2 The Euler-Maruyamascheme for L´evy processes . . . 330
6.5.3 Small jump approximation . . . . . . . . . . . . . . . 331
6.5.4 Simulation via series representation. . . . . . . . . . . 333
7 Simulating Financial Models: Discontinuous Paths 335
7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 335
7.2 Merton’sjump-diffusionmodelandstochasticvolatilitymodels
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 L´evy models and their simulation . . . . . . . . . . . 340
7.3.1 The Esscher transform . . . . . . . . . . . . . . . . . . 341
7.3.2 The hyperbolic L´evy model . . . . . . . . . . . . . . . 342
7.3.3 The variance gamma model . . . . . . . . . . . . . . . 344
7.3.4 Normal inverse Gaussian processes . . . . . . . . . . . 352
7.3.5 Further aspects of L´evy type models . . . . . . . . . . 354
8 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