Monte Carlo

Methods and

Models in Finance

and Insurance

CHAPMAN & HALL/CRC

Financial Mathematics Series

Aims and scope:

The eld of nancial mathematics forms an ever-expanding slice of the nancial sector. This series

aims to capture new developments and summarize what is known over the whole spectrum of this

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Centre for Financial

Research

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Engineering

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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 Pug, 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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1-4 Singer Street

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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

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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 Howtousethisbook 3

1.4 Furtherliterature 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 QualitycriteriaforRNGs 6

2.1.3 Technicalterms 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 F

2

-lineargenerators 17

2.2.6 NonlinearRNGs 22

2.2.7 More random number generators . . . . . . . . . . . . 24

2.2.8 ImprovingRNGs 24

2.3 TestingandanalyzingRNGs 25

2.3.1 Analyzingthelatticestructure 25

2.3.2 Equidistribution 26

2.3.3 Diﬀusioncapacity 27

2.3.4 Statisticaltests 27

2.4 Generating random numbers with general distributions . . . 31

2.4.1 Inversionmethod 31

2.4.2 Acceptance-rejection method . . . . . . . . . . . . . . 33

2.5 Selecteddistributions 36

2.5.1 Generating normally distributed random numbers . . 36

2.5.2 Generating beta-distributed RNs . . . . . . . . . . . . 38

2.5.3 Generating Weibull-distributedRNs 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 Quasirandom sequences as a substitute for random sequences 45

2.7.1 Haltonsequences 47

2.7.2 Sobolsequences 48

2.7.3 Randomized quasi-Monte Carlo methods . . . . . . . . 49

2.7.4 HybridMonteCarlomethods 50

2.7.5 Quasirandom sequences and transformations into

other random distributions . . . . . . . . . . . . . . . 50

2.8 Parallelizationtechniques 51

2.8.1 Leap-frogmethod 51

2.8.2 Sequencesplitting 52

2.8.3 SeveralRNGs 53

2.8.4 Independentsequences 53

2.8.5 TestingparallelRNGs 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 Improving the speed of convergence of the Monte Carlo method:

Variancereductionmethods 65

3.3.1 Antitheticvariates 66

3.3.2 Controlvariates 70

3.3.3 Stratiﬁed sampling . . . . . . . . . . . . . . . . . . . . 76

3.3.4 Variance reduction by conditional sampling . . . . . . 85

3.3.5 Importancesampling 87

3.4 Further aspects of variance reduction methods . . . . . . . . 97

3.4.1 Moremethods 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 PropertiesofBrownianmotion 113

4.4.2 Weak convergence and Donsker’s theorem . . . . . . . 116

vii

4.4.3 Brownianbridge 120

4.5 Basics of Itˆocalculus 126

4.5.1 The Itˆointegral 126

4.5.2 The Itˆoformula 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

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 Basicfactsofoptions 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-

Scholessetting 196

5.6.1 Path-independent European options . . . . . . . . . . 197

5.6.2 Path-dependent European options . . . . . . . . . . . 199

5.6.3 Moreexoticoptions 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-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

5.17.2TheVasicekmodel 278

5.17.3 The Cox-Ingersoll-Ross (CIR) model . . . . . . . . . . 281

5.17.4Aﬃnelinearshortratemodels 283

5.17.5 Perfect calibration: Deterministic shifts and the Hull-

Whiteapproach 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

5.18.2TheCheyettemodel 290

5.19LIBORmarketmodels 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

6 Continuous-Time Stochastic Processes: Discontinuous Paths 309

6.1 Introduction 309

6.2 Poisson processes and Poisson random measures: Deﬁnition

andsimulation 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 L´evy processes: Properties and examples . . . . . . . . . . . 320

6.4.1 Deﬁnition and properties of L´evy processes . . . . . . 320

6.4.2 Examples of L´evyprocesses 324

6.5 Simulation of L´evyprocesses 329

6.5.1 Exact simulation and time discretization . . . . . . . . 329

6.5.2 The Euler-Maruyama scheme 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’s jump-diﬀusion model and stochastic volatility models

withjumps 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´evymodelsandtheirsimulation 340

7.3.1 TheEsschertransform 341

7.3.2 The hyperbolic L´evymodel 342

7.3.3 Thevariancegammamodel 344

7.3.4 Normal inverse Gaussian processes . . . . . . . . . . . 352

7.3.5 Further aspects of L´evytypemodels 354

8 Simulating Actuarial Models 357

8.1 Introduction 357

8.2 Premiumprinciplesandriskmeasures 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 Nonlifeinsurance 403

x

8.5.1 Theindividualmodel 404

8.5.2 Thecollectivemodel 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

inactuarialmathematics 429

8.7 Asset-liability management and Solvency II . . . . . . . . . . 433

8.7.1 SolvencyII 433

8.7.2 Asset-liability management (ALM) . . . . . . . . . . . 435

References 441

Index 459

List of Algorithms

2.1 LCG 9

2.2 MRG 13

2.3 F

2

-lineargenerators 18

2.4 Linear feedback shift registergenerators 18

2.5 Example of a combined LFSR generator . . . . . . . . . . . . 20

2.6 GFSR 21

2.7 Inversivecongruentialgenerators 23

2.8 Inversionmethod 32

2.9 Acceptance-rejection method . . . . . . . . . . . . . . . . . . 34

2.10 Approximation of the standard normal c.d.f. . . . . . . . . . . 36

2.11 Beasley-Springer-Moro algorithm for the inverse standard

normal 37

2.12TheBox-Mullermethod 38

2.13 J¨ohnk’sbetagenerator 39

2.14 Cheng’s algorithm for gamma-distributed RNs with a>1 40

2.15 Algorithm for gamma-distributed RNs with 0 <a<1 41

2.16 J¨ohnk’s generator for gamma-distributed RNs with

0 <a<1 42

2.17Chi-square-distributedRNs 42

2.18 Chi-square-distributed RNs with the help of normally

distributedRNs 43

2.19 Cholesky factorization . . . . . . . . . . . . . . . . . . . . . . 44

2.20 One-dimensional Halton sequences (Van-der-Corput

sequences) 47

2.21 t-dimensional Halton sequences 48

3.1 The(crude)MonteCarlomethod 57

4.1 Simulation of a discrete-time stochastic process with

independentincrements 109

4.2 Simulation of a continuous-time stochastic process with

continuouspaths 110

4.3 Simulation of a Brownian motion . . . . . . . . . . . . . . . . 115

4.4 Forward simulation of a Brownian bridge . . . . . . . . . . . 123

4.5 Backward simulation of a Brownian bridge from a to b with

n =2

K

timepoints 124

4.6 Variance reduction by Brownian bridge . . . . . . . . . . . . . 125

4.7 TheEuler-Maruyamascheme 152

4.8 TheMilsteinscheme 155

xi

xii

4.9 TheMilsteinorder2scheme 160

4.10 Talay-Tubaro extrapolation . . . . . . . . . . . . . . . . . . . 165

4.11ThestatisticalRombergmethod 167

4.12 Multilevel Monte Carlo simulation . . . . . . . . . . . . . . . 169

4.13 Adaptive multilevel Monte Carlo simulation . . . . . . . . . . 170

5.1 Simulating a stock price path in the Black-Scholes model . . 181

5.2 Option pricing via Monte Carlo simulation . . . . . . . . . . . 196

5.3 MC pricing of path-independent options . . . . . . . . . . . . 197

5.4 Corrected geometric mean control for ﬁxed-strike average

options 201

5.5 Monte Carlo for discrete barrier options . . . . . . . . . . . . 204

5.6 Conditional MC for double-barrier knock-out options . . . . . 206

5.7 MC pricing of out-barrier options with the bridge technique . 207

5.8 Double-barrier knock-out call pricing using Moon’s method for

piecewise constant barriers . . . . . . . . . . . . . . . . . . . . 210

5.9 Preprocessing a d-dimensional normal sample . . . . . . . . . 213

5.10MonteCarlopricingintheCEVmodel 219

5.11 Calibrating σ intheBlack-Scholesmodel 221

5.12 Simulating price paths in the Heston model (naive way) . . . 226

5.13 Quadratic exponential (QE) method for simulating the

volatility in the Heston model . . . . . . . . . . . . . . . . . . 230

5.14 Stock price paths in the Heston model . . . . . . . . . . . . . 232

5.15 Call pricing with the Heath-Platen estimator . . . . . . . . . 236

5.16 Algorithmic framework for pricing American (Bermudan)

options by Monte Carlo methods . . . . . . . . . . . . . . . . 243

5.17 Dynamic programming for calculating the price of

Bermudanoptions 244

5.18 Longstaﬀ-Schwartz algorithm for pricing Bermudan options . 247

5.19 Andersen-Broadie algorithm to obtain upper bounds for the

priceofaBermudanoption 253

5.20OptionpricingintheCheyettemodel 293

5.21 Simulation of paths of forward-LIBOR rates under the spot-

LIBORmeasure 300

6.1 Simulation of a compound Poisson process . . . . . . . . . . . 312

6.2 Simulation of a stochastic integral with respect to a

compoundPoissonprocess 316

6.3 Simulation of a Gauss-Poisson jump-diﬀusion process with a

ﬁxedtimediscretization 318

6.4 Euler-Maruyama scheme for jump-diﬀusions . . . . . . . . . . 320

6.5 Exact simulation of discretized L´evy processes . . . . . . . . . 329

6.6 Euler-Maruyama scheme for L´evyprocesses 331

7.1 A path in the Merton jump-diﬀusion model . . . . . . . . . . 337

7.2 Simulating a variance gamma process by subordination . . . . 346

7.3 Variance gamma path by diﬀerences . . . . . . . . . . . . . . 347

7.4 A VG path by the DGBS algorithm . . . . . . . . . . . . . . 349

xiii

7.5 Pricing up-and-in calls by the DGBS method . . . . . . . . . 352

7.6 SimulationofanNIGpath 354

8.1 Crude Monte Carlo simulation of the α-quantile 366

8.2 Importance sampling for quantiles . . . . . . . . . . . . . . . 369

8.3 Value-at-Risk via importance sampling of the delta-gamma

approximation 373

8.4 Modelling dynamic mortality by extrapolation . . . . . . . . 380

8.5 Stochastic dynamic mortality modelling . . . . . . . . . . . . 380

8.6 Simulating dynamic mortality rates in the stochastic Gompertz

model 382

8.7 Measure calibration in the stochastic Gompertz model with a

tradedlongevitybond 387

8.8 Simulation of the (mean) prospective reserve of an insurance

contract 389

8.9 SimulatingwithaGaussiancopula 396

8.10 Simulating with a t-copula 399

8.11 Simulating with an Archimedean copula . . . . . . . . . . . . 401

8.12Copulaframeworkfordependentrisks 403

8.13 Simulating a path of a mixed Poisson process . . . . . . . . . 407

8.14 Simulating a path of an inhomogeneous Poisson process . . . 408

8.15 Simulating a path of a Cox process . . . . . . . . . . . . . . . 408

8.16 Simulating the Asmussen-Kroese estimator . . . . . . . . . . 412

8.17 Simulating a path of a homogeneous Markov chain with a

precalculatedtransitionmatrix 420

8.18 Simulation of a Markov chain . . . . . . . . . . . . . . . . . . 420

8.19 Metropolis-Hastings algorithm . . . . . . . . . . . . . . . . . 421

8.20Gibbssampler 425

Chapter 1

Introduction and User Guide

1.1 Introduction and concept

Monte Carlo methods are ubiquitous in applications in the ﬁnance and

insurance industry. They are often the only accessible tool for ﬁnancial engi-

neers and actuaries when it comes to complicated price or risk computations,

in particular for those that are based on many underlyings. However, as they

tend to be slow, it is very important to have a big tool box for speeding them

up or – equivalently – for increasing their accuracy. Further, recent years have

seen a lot of developments in Monte Carlo methods with a high potential for

success in applications. Some of them are highly speciﬁed (such as the Ander-

sen algorithm in the Heston setting), others are general algorithmic principles

(such as the multilevel Monte Carlo approach). However, they are often only

available in working papers or as technical mathematical publications.

On the other hand, there is still a lack of understanding of the theory of

ﬁnance and insurance mathematics when new numerical methods are applied

to those areas. Even in very recent papers one sees presentations of big break-

throughs when indeed the methods are applied to already solved problems or

to problems that do not make sense when viewed from the ﬁnancial or insur-

ance mathematics side.

We therefore have chosen an approach that combines the presentation of

the application background in ﬁnance and insurance together with the theory

and application of Monte Carlo methods in one book. To do this and still keep

a reasonable page limit, compromises in the presentation are unavoidable. In

particular, we will not to give strict formal proofs of results. However, one

can often use the arguments given in the book to construct a rigorous proof

in a straightforward way. If short and nontechnical arguments are not easy

to provide, then the related references are given. This will keep the book at

a reasonable length and allow for ﬂuent reading and for getting fast to the

point while avoiding burdening the reader with too many technicalities. Also,

our intention is to give the reader a feeling for the methods and the topics via

simple pedagogical examples and numerical and graphical illustrations. On

this basis we try to be as rigorous and detailed as possible. This in particular

means that we introduce the ﬁnancial and actuarial models in great detail

and also comment on the necessity of technicalities.

1

2 Monte Carlo Methods and Models in Finance and Insurance

In our approach, we have chosen to separately present the Monte Carlo

techniques, the stochastic process basics, and the theoretical background and

intuition behind ﬁnancial and actuarial mathematics. This has the advantage

that the standard Monte Carlo tools can easily be identiﬁed and do not need

to be separated from the application by the reader. Also, it allows the reader

to concentrate on the main principles of ﬁnancial and insurance mathematics

in a compact way. Mostly, the chapters are as self-contained as possible,

although the later ones are often building up on the earlier ones. Of course,

all ingredients come together when the applications of Monte Carlo methods

are presented within the areas of ﬁnance and insurance.

1.2 Contents

We have chosen to start the book with a survey on random number gen-

eration and the use of random number generators as the basis for all Monte

Carlo methods. It is indeed important to know whether one is using a good

random number generator. Of course, one should not implement a new one

as there are many excellent ones freely available. But the user should be able

to identify the type of generator that his preferred programming package is

oﬀering or the internal system of the company is using. Modern aspects such

as parallelization of the generation of random numbers are also touched and

are important for speeding up Monte Carlo methods.

This chapter is followed by an introduction to the Monte Carlo method,

its theoretical background, and the presentation of various methods to speed

them up by what is called variance reduction. The application of this method

to stochastic processes of diﬀusion type is the next step. For this, basics of

diﬀusion processes and the Itˆo calculus are provided together with numerical

methods for solving stochastic diﬀerential equations, a tool that is essential for

simulating paths of e.g. stock prices or interest rates. Here, we already present

some very recent methods such as the statistical Romberg or the multilevel

Monte Carlo method.

The ﬁfth chapter contains an introduction to both classical stock option

pricing, more recent stock price models in the diﬀusion context, and interest

rate models. Here, many nonstandard models (such as stochastic volatility

models) are presented. Further, we give a lot of applications of Monte Carlo

methods in option pricing and interest rate product pricing. Some of the

methods are standard applications of what have been presented in the pre-

ceding chapters, some are tailored to the ﬁnancial applications, and some have

been developed more recently.

In the sixth and seventh chapters we leave the diﬀusion framework. Stochas-

tic processes that contain jumps such as jump-diﬀusions or L´evy processes en-

Introduction and User Guide 3

ter the scene as the building blocks for modelling the uncertainty inherent in

ﬁnancial markets. Developing eﬃcient Monte Carlo methods for L´evy models

is a very active area that still is at its beginning. We therefore only present

the basics, but also include some spectacular examples of specially tailored

algorithms such as the variance gamma bridge sampling method.

Finally, in Chapter 8, some applications of Monte Carlo methods in ac-

tuarial mathematics are presented. As actuarial (or insurance) mathematics

has many aspects that we do not touch, we have only chosen to present some

main areas such as premium priniciples, life insurance, nonlife insurance, and

asset-liability management.

1.3 How to use this book

This book is intended as an introduction to both Monte Carlo methods and

ﬁnancial and actuarial models. Although we often avoid technicalities, it is

our aim to go for more than just standard models and methods. We believe

that the book can be used as an introductory text to ﬁnance and insurance

for the numerical analyst, as an introduction to Monte Carlo methods for the

practitioners working in banks and insurance companies, as an introduction

to both areas for students, but also as a source for new ideas of presentation

that can be used in lectures, as a source for new models and Monte Carlo

methods even for specialists in the areas. And ﬁnally the book can be used as

a cooking book via the collection of the diﬀerent algorithms that are explicitly

stated where they are developed.

There are diﬀerent ways to read the contents of this book. Although we

recommend to have a look at Chapter 2 and the aspects of generation of

random numbers, one can directly start with the presentation of the Monte

Carlo method and its variants in Chapters 3 and 4. If one is interested in the

applications in ﬁnance then Chapter 5 is a must. Also, it contains methods

and ideas that will again be used in Chapters 6 through 8.

If one is interested in special models then they are usually dealt with in

a self-contained way. If one is only interested in a special algorithm for a

particular problem then this one can be found via the table of algorithms.

1.4 Further literature

All topics covered in this book are popular subjects of applied mathematics.

Therefore, there is a huge amount of monographs, survey papers, and lecture

4 Monte Carlo Methods and Models in Finance and Insurance

notes. We have tried to incorporate the main contributions to these areas. Of

course, there exist related monographs on Monte Carlo methods. The book

that comes closest to ours in terms of applications in ﬁnance is the excellent

monograph by Glasserman (2004) which is a standard reference for ﬁnancial

engineers and researchers. We have also beneﬁtted a lot from reading it, but

have of course tried to avoid copying it. In particular, we have chosen to

present ﬁnancial and actuarial models in greater detail. A further recent and

excellent reference is the book by Asmussen and Glynn (2007) that has a

broader scope than the applications in ﬁnance and insurance and that needs

a higher level of preknowledge as our book. In comparison to both these

references, we concentrate more on the presentation of the models.

More classic and recent texts dealing with Monte Carlo simulation are Ru-

binstein (1981), Hammersley and Handscomb (1964), or Ugur (2009) who also

considers numerical methods in ﬁnance diﬀerent from Monte Carlo.

1.5 Acknowledgments

As with all books written there are many major and minor contributors

besides the authors. We have beneﬁtted from many discussions, lectures,

and results from friends and colleagues. Also, our experiences from recent

years gained from lecture series, student comments, and in particular industry

projects at the Fraunhofer Institute for Industrial Mathematics ITWM at

Kaiserslautern entered the presentations in the book.

We are happy to thank Christina Andersson, Roman Horsky and Henning

Marxen for careful proof reading and suggestions. Assistance in providing

numbers and code from Georgi Dimitroﬀ, Nora Imkeller, and Susanne Wendel

is gratefully acknowledged. Further, the authors thank Hansj¨org Albrecher

and participants of the 22nd Summer School of the SWISS Association of

Actuaries at the University of Lausanne for many useful comments, good

questions, and discussions.

Finally, the staﬀ at Taylor & Francis/CRC Press has been very friendly

and supportive. In particular, we thank (in alphabetical order) Rob Calver,

Kevin Craig, Shashi Kumar, Linda Leggio, Sarah Morris, Katy Smith, and

Jessica Vakili for their great help and encouragement.

Chapter 2

Generating Random Numbers

2.1 Introduction

2.1.1 How do we get random numbers?

Stochastic simulations and especially the Monte Carlo method use random

variables (RVs). So the ability to provide random numbers (RNs) with a

speciﬁed distribution becomes necessary. The main problem is to ﬁnd num-

bers that are really random and unpredictable. Of course, throwing dice is

much too slow for most applications as usually a lot of RNs are needed. An

alternative is to use physical phenomena like radioactive decay, which is often

considered as a synonym for randomness, and then to transform measurements

into RNs. With the right equipment this works much faster. But how can one

ensure that the required distribution is mimicked? Indeed, modern research

makes it possible to use physical devices for random number generation by

transforming the measurements so that they deliver a good approximation to a

ﬁxed distribution, but those devices still are too slow for extensive simulations.

Another disadvantage is that the sequence of RNs cannot be reproduced un-

less they have been stored. Reproducibility is an important feature for Monte

Carlo methods, e.g. for variance reduction techniques, or simply for debugging

reasons. However, physical random number generators (RNGs) are useful for

applications in cryptography or gambling machines, when you have to make

sure that the numbers are absolutely unpredictable.

The RNs for Monte Carlo simulations are usually generated by a numerical

algorithm. This makes the sequence of numbers deterministic which is the

reason why those RNs are often called pseudorandom numbers.Butifwe

look at them without knowing the algorithm they appear to be random. And

in many statistical tests they behave like true random numbers. Of course,

as they are deterministic, they will fail some tests. When choosing a RNG we

should make sure that the RNs are thoroughly tested and we should be aware

which tests they fail.

First of all, we concentrate on generating uniformly distributed RNs, es-

pecially uniformly distributed real RNs on the interval from zero to one,

u ∼ U [0, 1], U (0, 1), U (0, 1] or U [0, 1). Theoretically, whether 1 or 0 are

included or not does not matter, as the probability for a single number is

5

6 Monte Carlo Methods and Models in Finance and Insurance

zero. But the user should be aware whether the chosen RNG delivers zeroes

andonesasthismightleadtoprogramme crashes,ase.g.ln(0).

In a second step these uniformly distributed RNs will be transformed into

RNs of a desired distribution (such as e.g. the normal or the gamma one).

Nowadays there exist a lot of diﬀerent methods to produce RNs and the

research in this area is still very active. The reader should be aware that the

top algorithms of today might be old-fashioned tomorrow. Also the subject

of writing a good RNG is so involved that it cannot be covered here in all its

details. So, we will not describe the perfect RNG in this chapter (in fact it

does not exist!). Here we will give the basics for understanding a RNG and

being able to judge its quality. Also, we will try to give you some orientation to

choose the one that suits your problem. After reading this chapter you should

not go and write your own RNG, except for research reasons and to collect

experience. It is better to search for a good, well-programmed and suitable

up-to-date RNG, or to check the built-in one in your computer package, accept

it as usable, and be happy with it (for some time) or otherwise, if it is no good,

mistrust the simulation results.

Although implementing RNGs is such an enormous and complicated ﬁeld,

we will present some simple algorithms here so that you can experiment a

little bit for yourself. But do not see those algorithms as recommendations,

the only aim is to get familiar with the technical terms.

2.1.2 Quality criteria for RNGs

For a reliable simulation good random numbers are imperative. A bad RNG

could cause totally stupid simulation results which can lead to false conclu-

sions. Before using an built-in RNG of a software programme, it should be

checked ﬁrst. Here are some quality criteria the user should be aware of.

• Of course, uniformly distributed RNs ∼ U [0, 1] should be evenly dis-

tributed in the interval [0, 1], but the structure should not be too regular.

Otherwise they will not be considered as random. However, in some simula-

tion situations we are able to take advantage of regular structures, especially

when working with smooth functions. In these cases we can use quasirandom

sequences, which do not look random at all, but are very evenly distributed.

• As in Monte Carlo simulations, where we often need lots of RNs, it should

be possible to produce them very fast and eﬃciently without using too much

memory. So, speed and memory requirements matter.

• RN algorithms work with a ﬁnite set of data due to the construction prin-

ciple of the algorithm and due to the ﬁnite representation of real numbers in

the computer. So the RNG will eventually repeat its sequence of numbers.

The maximal length of this sequence before it is repeated is called the period

of the RNG. For the extensive simulations of today huge amounts of RNs are

needed. Therefore the period must be long enough to avoid using the same

Generating Random Numbers 7

RNs again. A rule of thumb says that the period length of the RNG should

be at least as long as the square of the quantity of RNs needed. Otherwise

the deterministic aspect of the RNG comes into play and makes the RNs cor-

related. In former times RNGs with a period of about 10

19

were considered

as good, but nowadays those algorithms are considered as insuﬃcient. Even

worse, there are still RNGs with a period of only 10

5

in use, especially built-in

RNGs, which are often the cause for severely bad Monte Carlo simulations.

• In cryptography the unpredictability of the RNs is essential. This is not

really necessary for Monte Carlo methods. It is more important that RNs ap-

pear to be random and independent, and pass statistical tests about being

independent, identically distributed (i.i.d.) RNs ∼ U [0, 1].

• Within Monte Carlo methods the sequence of RNs should be reproducible.

First because of debugging reasons. If we get strange simulation results, we

will be able to examine the sequence of RNs. Also it is quite common in

some applications, e.g. in sensitivity analysis, to use the same sequence again.

Another advantage is the possibility to compare diﬀerent calculation methods

or similar ﬁnancial products in a more eﬃcient way by using the same RNs.

• The algorithm should be programmed so that it delivers the same RNs in

every computer, also called portability. It should be possible to repeat cal-

culations in other machines and always obtain the same result. The same seed

for initialization should always return the same RN sequence.

• To make calculations very fast it is often desirable to use a computer with

parallel processors. So the possibility of parallelization is another impor-

tant point of consideration. One option is e.g. to jump ahead quickly many

steps in the sequence and produce disjoint substreams of the stream of RNs.

Another option is to ﬁnd a family of RNGs which work equally well for a large

set of diﬀerent parameters.

• The structure of the random points is very important. A typical feature of

some RNGs is that if d-dimensional vectors are constructed out of consecutive

RNs, then those points lie on hyperplanes. Bad RNGs have only a few hy-

perplanes in some dimensions which are far apart, leaving huge gaps without

any random vectors in space. Further, one should be aware if there is a severe

correlation between some RNs. For example, if the algorithm constructs a

new RN out of two previous RNs, groups of three RNs could be linked too

closely. This correlation can lead to false simulation results if the model has

a structure that is similar.

• Sometimes it is necessary to choose random number generators which are

easy to implement, i.e. the code is simple and short. This is a more general

point and applies to all parts of the simulation. Very often it is useful to have

a second simulation routine to check important simulation results. In this

case, code which is easy to read increases the possibility that it is bug-free.

8 Monte Carlo Methods and Models in Finance and Insurance

2.1.3 Technical terms

All RNGs that are based on deterministic recurrences can be described as

a quintuple (S,μ,f,U,g) (see L’Ecuyer [1994]), where

•Sis the ﬁnite set of states, the so-called state space.

• s

n

∈Sis a particular state at iteration step n.

• μ is the probability measure for selecting s

0

, the initial state, from S. s

0

is

called the seed of the RNG.

• The function f, also called the transition function, describes the algo-

rithm, s

n+1

= f (s

n

).

•Uis the output space. We will mainly consider [0, 1], [0, 1), (0, 1] or (0, 1).

• The output function g : S→Umaps the state s

n

∈Sinto a number

u

n

∈U, the ﬁnal random number that we are interested in.

REMARK 2.1 1. The importance of the seed that invokes the RNG

should not be underestimated. One cause of strange simulation results is

often that the RNG has not been seeded or has been started with a zero

vector. Another reason is that some seeds have to be avoided as they lead

to bad RN sequences, e.g., a seed that contains mainly zeroes might lead to

RNs that do not diﬀer much. Further, it is advisable to save the seed for

debugging.

2. As real numbers can only be presented with ﬁnite accuracy in computers,

the set of states S must be ﬁnite and so the range g (S) ⊂Uis also ﬁnite. As

weareinterestedinanevendistribution,wewantthegapsin[0, 1) not to be

too large. Therefore, S has to be a large set.

3. Another consequence of S being ﬁnite is that s

n+ρ

= s

n

for some integer

ρ, meaning the sequence will repeat itself eventually. The smallest number ρ

for which this happens is called the period of the RNG. It cannot be larger

than |S|, another reason for the set of states being huge. In some generators

the period might be diﬀerent for diﬀerent seeds and the RN cycles are then

disjoint. So we have to be careful with which seed we start.

2.2 Examples of random number generators

2.2.1 Linear congruential generators

The linear congruential method was one of the ﬁrst RNGs. Linear congru-

ential generators (LCGs) were ﬁrst introduced by Lehmer (1949) and were

very popular for many years.

Generating Random Numbers 9

Algorithm 2.1 LCG

s

n+1

=(as

n

+ c)modm, n ∈ N,

where

• m ∈ N \{0} is called the modulus,

• a ∈ N is called the multiplier, a<m,

• c ∈ N is called the increment, c<m,

• s

0

∈ N is called the seed, s

0

<m.

Numbers in [0, 1) are obtained via

u

n

=

s

n

m

.

These generators are often denoted by LCG(m, a, c). The state space can

be described as

S⊆N, S = {0, 1, ,m− 1} or {1, 2, ,m−1}.

For the increment c = 0 the generator cannot be started with s

0

=0. Then0

is excluded from the set of states.

Choosing coeﬃcients

Often the modulus m is chosen as a prime. Then all calculations are done

in the ﬁnite ﬁeld Z

m

. Preferred moduli are Mersenne Primes, primes of the

form 2

k

− 1, e.g. 2

31

− 1 = 2147483647, the largest 32-bit signed integer.

Sometimes a power of 2 is chosen as the modulus, because calculations can

be made faster by exploiting the binary structure of computers. But then the

lower order bits of the RNs are highly correlated, especially the last bit of

the states s

n

is either constant or strictly alternates between 0 and 1. This

means that all our integer RNs s

n

are either even or odd or we have a regular

even/odd pattern.

If the period of such a generator is m or m −1(inthecaseofc =0)wecall

this property full period as this is the maximum possible period length. In

order to achieve a long period it is advisable to choose a very large number as

the modulus. Further, the increment and the multiplier have to be selected

to create a generator with full period. If c =0,m is a prime, a is a primitive

root modulo m, then the generator has full period. For c = 0 the criteria for

full period are more complicated (see e.g. Knuth [1998]): the only common

divisor of m and c should be 1, every prime number that divides m should also

divide a −1; if m is divisible by 4, then a − 1 should also have the divisor 4.

10 Monte Carlo Methods and Models in Finance and Insurance

Implementation issues

Our next task is the implementation of the algorithm. If the modulus m is

near the maximum integer and also the multiplier a is quite large, then the

calculation step done with integer arithmetics can lead to an overﬂow. We

then get errors or, in the better case, our code will be machine dependent

and is no longer portable. To circumvent this diﬃculty there are several

possibilities:

• Instead of integer arithmetic we can work with ﬂoating-point arithmetic. For

example, in 64-bit ﬂoating-point arithmetic integers up to 2

53

are represented

exactly. So we just take care that a (m − 1) ≤ 2

53

.

• As we work in the ring Z

m

, we could consider multiplying with −¯a instead

of multiplying with a,where¯a = m − a is the additive inverse of a.

• Another technique is called approximate factoring.Ifm is large and

a not too large, e.g. a<

√

m, then we decompose m = aq + r, i.e. r = m

mod a , q = m/a. The complicated part of the iteration step is done by

(as

n

)modm =

a (s

n

mod q) −rz/q if it is ≥ 0

a (s

n

mod q) −rz/q+ m otherwise

(2.1)

As the parameter r is small this method avoids overﬂows. This trick also

works when a = m/k with an integer k<

√

m.

• The powers-of-two decomposition can be applied when a can be written

as a sum or a diﬀerence of a small number of powers of 2, e.g. a =2

3

+2

8

.

Then the modulus is decomposed as m =2

k

−h with integers k, h ∈ N.Now

(as

n

)modm can be eﬃciently calculated using shifts, masks, additions,

subtractions, and a single multiplication by h.

Disadvantages of LCGs

LCGs should no longer be used for important simulations. First, much

better RNGs exist, and then they also have a lot of disadvantages:

• There exists a high sequential correlation between the RNs, so they do

not mimic randomness very well. For example, if the multiplier a is small

compared to the modulus m,thenverysmallRNsarealways followed by

another small RN. In a simulation this means that rare events could be too

close together or happen too often.

• The modulus m cannot be larger than the largest integer available in the

programming language and so the period, which is maximally m, is usually

not very long. In 32-bit computers the period is often only 2

31

−1 and this is

simply too short nowadays.

• We take a closer look at the set of all possible t-dimensional vectors ﬁlled

with consecutive RNs, the ﬁnite set Ψ

t

:= {(u

1

,u

2

, ,u

t

) |s

0

∈S}⊆[0, 1)

t

Methods and

Models in Finance

and Insurance

CHAPMAN & HALL/CRC

Financial Mathematics Series

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Monte Carlo

Methods and

Models in Finance

and Insurance

Ralf Korn

Elke Korn

Gerald Kroisandt

CRC Press

Taylor & Francis Group

6000 Broken Sound Parkway NW, Suite 300

Boca Raton, FL 33487-2742

© 2010 by Taylor and Francis Group, LLC

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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.

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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 Howtousethisbook 3

1.4 Furtherliterature 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 QualitycriteriaforRNGs 6

2.1.3 Technicalterms 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 F

2

-lineargenerators 17

2.2.6 NonlinearRNGs 22

2.2.7 More random number generators . . . . . . . . . . . . 24

2.2.8 ImprovingRNGs 24

2.3 TestingandanalyzingRNGs 25

2.3.1 Analyzingthelatticestructure 25

2.3.2 Equidistribution 26

2.3.3 Diﬀusioncapacity 27

2.3.4 Statisticaltests 27

2.4 Generating random numbers with general distributions . . . 31

2.4.1 Inversionmethod 31

2.4.2 Acceptance-rejection method . . . . . . . . . . . . . . 33

2.5 Selecteddistributions 36

2.5.1 Generating normally distributed random numbers . . 36

2.5.2 Generating beta-distributed RNs . . . . . . . . . . . . 38

2.5.3 Generating Weibull-distributedRNs 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 Quasirandom sequences as a substitute for random sequences 45

2.7.1 Haltonsequences 47

2.7.2 Sobolsequences 48

2.7.3 Randomized quasi-Monte Carlo methods . . . . . . . . 49

2.7.4 HybridMonteCarlomethods 50

2.7.5 Quasirandom sequences and transformations into

other random distributions . . . . . . . . . . . . . . . 50

2.8 Parallelizationtechniques 51

2.8.1 Leap-frogmethod 51

2.8.2 Sequencesplitting 52

2.8.3 SeveralRNGs 53

2.8.4 Independentsequences 53

2.8.5 TestingparallelRNGs 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 Improving the speed of convergence of the Monte Carlo method:

Variancereductionmethods 65

3.3.1 Antitheticvariates 66

3.3.2 Controlvariates 70

3.3.3 Stratiﬁed sampling . . . . . . . . . . . . . . . . . . . . 76

3.3.4 Variance reduction by conditional sampling . . . . . . 85

3.3.5 Importancesampling 87

3.4 Further aspects of variance reduction methods . . . . . . . . 97

3.4.1 Moremethods 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 PropertiesofBrownianmotion 113

4.4.2 Weak convergence and Donsker’s theorem . . . . . . . 116

vii

4.4.3 Brownianbridge 120

4.5 Basics of Itˆocalculus 126

4.5.1 The Itˆointegral 126

4.5.2 The Itˆoformula 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

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 Basicfactsofoptions 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-

Scholessetting 196

5.6.1 Path-independent European options . . . . . . . . . . 197

5.6.2 Path-dependent European options . . . . . . . . . . . 199

5.6.3 Moreexoticoptions 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-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

5.17.2TheVasicekmodel 278

5.17.3 The Cox-Ingersoll-Ross (CIR) model . . . . . . . . . . 281

5.17.4Aﬃnelinearshortratemodels 283

5.17.5 Perfect calibration: Deterministic shifts and the Hull-

Whiteapproach 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

5.18.2TheCheyettemodel 290

5.19LIBORmarketmodels 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

6 Continuous-Time Stochastic Processes: Discontinuous Paths 309

6.1 Introduction 309

6.2 Poisson processes and Poisson random measures: Deﬁnition

andsimulation 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 L´evy processes: Properties and examples . . . . . . . . . . . 320

6.4.1 Deﬁnition and properties of L´evy processes . . . . . . 320

6.4.2 Examples of L´evyprocesses 324

6.5 Simulation of L´evyprocesses 329

6.5.1 Exact simulation and time discretization . . . . . . . . 329

6.5.2 The Euler-Maruyama scheme 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’s jump-diﬀusion model and stochastic volatility models

withjumps 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´evymodelsandtheirsimulation 340

7.3.1 TheEsschertransform 341

7.3.2 The hyperbolic L´evymodel 342

7.3.3 Thevariancegammamodel 344

7.3.4 Normal inverse Gaussian processes . . . . . . . . . . . 352

7.3.5 Further aspects of L´evytypemodels 354

8 Simulating Actuarial Models 357

8.1 Introduction 357

8.2 Premiumprinciplesandriskmeasures 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 Nonlifeinsurance 403

x

8.5.1 Theindividualmodel 404

8.5.2 Thecollectivemodel 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

inactuarialmathematics 429

8.7 Asset-liability management and Solvency II . . . . . . . . . . 433

8.7.1 SolvencyII 433

8.7.2 Asset-liability management (ALM) . . . . . . . . . . . 435

References 441

Index 459

List of Algorithms

2.1 LCG 9

2.2 MRG 13

2.3 F

2

-lineargenerators 18

2.4 Linear feedback shift registergenerators 18

2.5 Example of a combined LFSR generator . . . . . . . . . . . . 20

2.6 GFSR 21

2.7 Inversivecongruentialgenerators 23

2.8 Inversionmethod 32

2.9 Acceptance-rejection method . . . . . . . . . . . . . . . . . . 34

2.10 Approximation of the standard normal c.d.f. . . . . . . . . . . 36

2.11 Beasley-Springer-Moro algorithm for the inverse standard

normal 37

2.12TheBox-Mullermethod 38

2.13 J¨ohnk’sbetagenerator 39

2.14 Cheng’s algorithm for gamma-distributed RNs with a>1 40

2.15 Algorithm for gamma-distributed RNs with 0 <a<1 41

2.16 J¨ohnk’s generator for gamma-distributed RNs with

0 <a<1 42

2.17Chi-square-distributedRNs 42

2.18 Chi-square-distributed RNs with the help of normally

distributedRNs 43

2.19 Cholesky factorization . . . . . . . . . . . . . . . . . . . . . . 44

2.20 One-dimensional Halton sequences (Van-der-Corput

sequences) 47

2.21 t-dimensional Halton sequences 48

3.1 The(crude)MonteCarlomethod 57

4.1 Simulation of a discrete-time stochastic process with

independentincrements 109

4.2 Simulation of a continuous-time stochastic process with

continuouspaths 110

4.3 Simulation of a Brownian motion . . . . . . . . . . . . . . . . 115

4.4 Forward simulation of a Brownian bridge . . . . . . . . . . . 123

4.5 Backward simulation of a Brownian bridge from a to b with

n =2

K

timepoints 124

4.6 Variance reduction by Brownian bridge . . . . . . . . . . . . . 125

4.7 TheEuler-Maruyamascheme 152

4.8 TheMilsteinscheme 155

xi

xii

4.9 TheMilsteinorder2scheme 160

4.10 Talay-Tubaro extrapolation . . . . . . . . . . . . . . . . . . . 165

4.11ThestatisticalRombergmethod 167

4.12 Multilevel Monte Carlo simulation . . . . . . . . . . . . . . . 169

4.13 Adaptive multilevel Monte Carlo simulation . . . . . . . . . . 170

5.1 Simulating a stock price path in the Black-Scholes model . . 181

5.2 Option pricing via Monte Carlo simulation . . . . . . . . . . . 196

5.3 MC pricing of path-independent options . . . . . . . . . . . . 197

5.4 Corrected geometric mean control for ﬁxed-strike average

options 201

5.5 Monte Carlo for discrete barrier options . . . . . . . . . . . . 204

5.6 Conditional MC for double-barrier knock-out options . . . . . 206

5.7 MC pricing of out-barrier options with the bridge technique . 207

5.8 Double-barrier knock-out call pricing using Moon’s method for

piecewise constant barriers . . . . . . . . . . . . . . . . . . . . 210

5.9 Preprocessing a d-dimensional normal sample . . . . . . . . . 213

5.10MonteCarlopricingintheCEVmodel 219

5.11 Calibrating σ intheBlack-Scholesmodel 221

5.12 Simulating price paths in the Heston model (naive way) . . . 226

5.13 Quadratic exponential (QE) method for simulating the

volatility in the Heston model . . . . . . . . . . . . . . . . . . 230

5.14 Stock price paths in the Heston model . . . . . . . . . . . . . 232

5.15 Call pricing with the Heath-Platen estimator . . . . . . . . . 236

5.16 Algorithmic framework for pricing American (Bermudan)

options by Monte Carlo methods . . . . . . . . . . . . . . . . 243

5.17 Dynamic programming for calculating the price of

Bermudanoptions 244

5.18 Longstaﬀ-Schwartz algorithm for pricing Bermudan options . 247

5.19 Andersen-Broadie algorithm to obtain upper bounds for the

priceofaBermudanoption 253

5.20OptionpricingintheCheyettemodel 293

5.21 Simulation of paths of forward-LIBOR rates under the spot-

LIBORmeasure 300

6.1 Simulation of a compound Poisson process . . . . . . . . . . . 312

6.2 Simulation of a stochastic integral with respect to a

compoundPoissonprocess 316

6.3 Simulation of a Gauss-Poisson jump-diﬀusion process with a

ﬁxedtimediscretization 318

6.4 Euler-Maruyama scheme for jump-diﬀusions . . . . . . . . . . 320

6.5 Exact simulation of discretized L´evy processes . . . . . . . . . 329

6.6 Euler-Maruyama scheme for L´evyprocesses 331

7.1 A path in the Merton jump-diﬀusion model . . . . . . . . . . 337

7.2 Simulating a variance gamma process by subordination . . . . 346

7.3 Variance gamma path by diﬀerences . . . . . . . . . . . . . . 347

7.4 A VG path by the DGBS algorithm . . . . . . . . . . . . . . 349

xiii

7.5 Pricing up-and-in calls by the DGBS method . . . . . . . . . 352

7.6 SimulationofanNIGpath 354

8.1 Crude Monte Carlo simulation of the α-quantile 366

8.2 Importance sampling for quantiles . . . . . . . . . . . . . . . 369

8.3 Value-at-Risk via importance sampling of the delta-gamma

approximation 373

8.4 Modelling dynamic mortality by extrapolation . . . . . . . . 380

8.5 Stochastic dynamic mortality modelling . . . . . . . . . . . . 380

8.6 Simulating dynamic mortality rates in the stochastic Gompertz

model 382

8.7 Measure calibration in the stochastic Gompertz model with a

tradedlongevitybond 387

8.8 Simulation of the (mean) prospective reserve of an insurance

contract 389

8.9 SimulatingwithaGaussiancopula 396

8.10 Simulating with a t-copula 399

8.11 Simulating with an Archimedean copula . . . . . . . . . . . . 401

8.12Copulaframeworkfordependentrisks 403

8.13 Simulating a path of a mixed Poisson process . . . . . . . . . 407

8.14 Simulating a path of an inhomogeneous Poisson process . . . 408

8.15 Simulating a path of a Cox process . . . . . . . . . . . . . . . 408

8.16 Simulating the Asmussen-Kroese estimator . . . . . . . . . . 412

8.17 Simulating a path of a homogeneous Markov chain with a

precalculatedtransitionmatrix 420

8.18 Simulation of a Markov chain . . . . . . . . . . . . . . . . . . 420

8.19 Metropolis-Hastings algorithm . . . . . . . . . . . . . . . . . 421

8.20Gibbssampler 425

Chapter 1

Introduction and User Guide

1.1 Introduction and concept

Monte Carlo methods are ubiquitous in applications in the ﬁnance and

insurance industry. They are often the only accessible tool for ﬁnancial engi-

neers and actuaries when it comes to complicated price or risk computations,

in particular for those that are based on many underlyings. However, as they

tend to be slow, it is very important to have a big tool box for speeding them

up or – equivalently – for increasing their accuracy. Further, recent years have

seen a lot of developments in Monte Carlo methods with a high potential for

success in applications. Some of them are highly speciﬁed (such as the Ander-

sen algorithm in the Heston setting), others are general algorithmic principles

(such as the multilevel Monte Carlo approach). However, they are often only

available in working papers or as technical mathematical publications.

On the other hand, there is still a lack of understanding of the theory of

ﬁnance and insurance mathematics when new numerical methods are applied

to those areas. Even in very recent papers one sees presentations of big break-

throughs when indeed the methods are applied to already solved problems or

to problems that do not make sense when viewed from the ﬁnancial or insur-

ance mathematics side.

We therefore have chosen an approach that combines the presentation of

the application background in ﬁnance and insurance together with the theory

and application of Monte Carlo methods in one book. To do this and still keep

a reasonable page limit, compromises in the presentation are unavoidable. In

particular, we will not to give strict formal proofs of results. However, one

can often use the arguments given in the book to construct a rigorous proof

in a straightforward way. If short and nontechnical arguments are not easy

to provide, then the related references are given. This will keep the book at

a reasonable length and allow for ﬂuent reading and for getting fast to the

point while avoiding burdening the reader with too many technicalities. Also,

our intention is to give the reader a feeling for the methods and the topics via

simple pedagogical examples and numerical and graphical illustrations. On

this basis we try to be as rigorous and detailed as possible. This in particular

means that we introduce the ﬁnancial and actuarial models in great detail

and also comment on the necessity of technicalities.

1

2 Monte Carlo Methods and Models in Finance and Insurance

In our approach, we have chosen to separately present the Monte Carlo

techniques, the stochastic process basics, and the theoretical background and

intuition behind ﬁnancial and actuarial mathematics. This has the advantage

that the standard Monte Carlo tools can easily be identiﬁed and do not need

to be separated from the application by the reader. Also, it allows the reader

to concentrate on the main principles of ﬁnancial and insurance mathematics

in a compact way. Mostly, the chapters are as self-contained as possible,

although the later ones are often building up on the earlier ones. Of course,

all ingredients come together when the applications of Monte Carlo methods

are presented within the areas of ﬁnance and insurance.

1.2 Contents

We have chosen to start the book with a survey on random number gen-

eration and the use of random number generators as the basis for all Monte

Carlo methods. It is indeed important to know whether one is using a good

random number generator. Of course, one should not implement a new one

as there are many excellent ones freely available. But the user should be able

to identify the type of generator that his preferred programming package is

oﬀering or the internal system of the company is using. Modern aspects such

as parallelization of the generation of random numbers are also touched and

are important for speeding up Monte Carlo methods.

This chapter is followed by an introduction to the Monte Carlo method,

its theoretical background, and the presentation of various methods to speed

them up by what is called variance reduction. The application of this method

to stochastic processes of diﬀusion type is the next step. For this, basics of

diﬀusion processes and the Itˆo calculus are provided together with numerical

methods for solving stochastic diﬀerential equations, a tool that is essential for

simulating paths of e.g. stock prices or interest rates. Here, we already present

some very recent methods such as the statistical Romberg or the multilevel

Monte Carlo method.

The ﬁfth chapter contains an introduction to both classical stock option

pricing, more recent stock price models in the diﬀusion context, and interest

rate models. Here, many nonstandard models (such as stochastic volatility

models) are presented. Further, we give a lot of applications of Monte Carlo

methods in option pricing and interest rate product pricing. Some of the

methods are standard applications of what have been presented in the pre-

ceding chapters, some are tailored to the ﬁnancial applications, and some have

been developed more recently.

In the sixth and seventh chapters we leave the diﬀusion framework. Stochas-

tic processes that contain jumps such as jump-diﬀusions or L´evy processes en-

Introduction and User Guide 3

ter the scene as the building blocks for modelling the uncertainty inherent in

ﬁnancial markets. Developing eﬃcient Monte Carlo methods for L´evy models

is a very active area that still is at its beginning. We therefore only present

the basics, but also include some spectacular examples of specially tailored

algorithms such as the variance gamma bridge sampling method.

Finally, in Chapter 8, some applications of Monte Carlo methods in ac-

tuarial mathematics are presented. As actuarial (or insurance) mathematics

has many aspects that we do not touch, we have only chosen to present some

main areas such as premium priniciples, life insurance, nonlife insurance, and

asset-liability management.

1.3 How to use this book

This book is intended as an introduction to both Monte Carlo methods and

ﬁnancial and actuarial models. Although we often avoid technicalities, it is

our aim to go for more than just standard models and methods. We believe

that the book can be used as an introductory text to ﬁnance and insurance

for the numerical analyst, as an introduction to Monte Carlo methods for the

practitioners working in banks and insurance companies, as an introduction

to both areas for students, but also as a source for new ideas of presentation

that can be used in lectures, as a source for new models and Monte Carlo

methods even for specialists in the areas. And ﬁnally the book can be used as

a cooking book via the collection of the diﬀerent algorithms that are explicitly

stated where they are developed.

There are diﬀerent ways to read the contents of this book. Although we

recommend to have a look at Chapter 2 and the aspects of generation of

random numbers, one can directly start with the presentation of the Monte

Carlo method and its variants in Chapters 3 and 4. If one is interested in the

applications in ﬁnance then Chapter 5 is a must. Also, it contains methods

and ideas that will again be used in Chapters 6 through 8.

If one is interested in special models then they are usually dealt with in

a self-contained way. If one is only interested in a special algorithm for a

particular problem then this one can be found via the table of algorithms.

1.4 Further literature

All topics covered in this book are popular subjects of applied mathematics.

Therefore, there is a huge amount of monographs, survey papers, and lecture

4 Monte Carlo Methods and Models in Finance and Insurance

notes. We have tried to incorporate the main contributions to these areas. Of

course, there exist related monographs on Monte Carlo methods. The book

that comes closest to ours in terms of applications in ﬁnance is the excellent

monograph by Glasserman (2004) which is a standard reference for ﬁnancial

engineers and researchers. We have also beneﬁtted a lot from reading it, but

have of course tried to avoid copying it. In particular, we have chosen to

present ﬁnancial and actuarial models in greater detail. A further recent and

excellent reference is the book by Asmussen and Glynn (2007) that has a

broader scope than the applications in ﬁnance and insurance and that needs

a higher level of preknowledge as our book. In comparison to both these

references, we concentrate more on the presentation of the models.

More classic and recent texts dealing with Monte Carlo simulation are Ru-

binstein (1981), Hammersley and Handscomb (1964), or Ugur (2009) who also

considers numerical methods in ﬁnance diﬀerent from Monte Carlo.

1.5 Acknowledgments

As with all books written there are many major and minor contributors

besides the authors. We have beneﬁtted from many discussions, lectures,

and results from friends and colleagues. Also, our experiences from recent

years gained from lecture series, student comments, and in particular industry

projects at the Fraunhofer Institute for Industrial Mathematics ITWM at

Kaiserslautern entered the presentations in the book.

We are happy to thank Christina Andersson, Roman Horsky and Henning

Marxen for careful proof reading and suggestions. Assistance in providing

numbers and code from Georgi Dimitroﬀ, Nora Imkeller, and Susanne Wendel

is gratefully acknowledged. Further, the authors thank Hansj¨org Albrecher

and participants of the 22nd Summer School of the SWISS Association of

Actuaries at the University of Lausanne for many useful comments, good

questions, and discussions.

Finally, the staﬀ at Taylor & Francis/CRC Press has been very friendly

and supportive. In particular, we thank (in alphabetical order) Rob Calver,

Kevin Craig, Shashi Kumar, Linda Leggio, Sarah Morris, Katy Smith, and

Jessica Vakili for their great help and encouragement.

Chapter 2

Generating Random Numbers

2.1 Introduction

2.1.1 How do we get random numbers?

Stochastic simulations and especially the Monte Carlo method use random

variables (RVs). So the ability to provide random numbers (RNs) with a

speciﬁed distribution becomes necessary. The main problem is to ﬁnd num-

bers that are really random and unpredictable. Of course, throwing dice is

much too slow for most applications as usually a lot of RNs are needed. An

alternative is to use physical phenomena like radioactive decay, which is often

considered as a synonym for randomness, and then to transform measurements

into RNs. With the right equipment this works much faster. But how can one

ensure that the required distribution is mimicked? Indeed, modern research

makes it possible to use physical devices for random number generation by

transforming the measurements so that they deliver a good approximation to a

ﬁxed distribution, but those devices still are too slow for extensive simulations.

Another disadvantage is that the sequence of RNs cannot be reproduced un-

less they have been stored. Reproducibility is an important feature for Monte

Carlo methods, e.g. for variance reduction techniques, or simply for debugging

reasons. However, physical random number generators (RNGs) are useful for

applications in cryptography or gambling machines, when you have to make

sure that the numbers are absolutely unpredictable.

The RNs for Monte Carlo simulations are usually generated by a numerical

algorithm. This makes the sequence of numbers deterministic which is the

reason why those RNs are often called pseudorandom numbers.Butifwe

look at them without knowing the algorithm they appear to be random. And

in many statistical tests they behave like true random numbers. Of course,

as they are deterministic, they will fail some tests. When choosing a RNG we

should make sure that the RNs are thoroughly tested and we should be aware

which tests they fail.

First of all, we concentrate on generating uniformly distributed RNs, es-

pecially uniformly distributed real RNs on the interval from zero to one,

u ∼ U [0, 1], U (0, 1), U (0, 1] or U [0, 1). Theoretically, whether 1 or 0 are

included or not does not matter, as the probability for a single number is

5

6 Monte Carlo Methods and Models in Finance and Insurance

zero. But the user should be aware whether the chosen RNG delivers zeroes

andonesasthismightleadtoprogramme crashes,ase.g.ln(0).

In a second step these uniformly distributed RNs will be transformed into

RNs of a desired distribution (such as e.g. the normal or the gamma one).

Nowadays there exist a lot of diﬀerent methods to produce RNs and the

research in this area is still very active. The reader should be aware that the

top algorithms of today might be old-fashioned tomorrow. Also the subject

of writing a good RNG is so involved that it cannot be covered here in all its

details. So, we will not describe the perfect RNG in this chapter (in fact it

does not exist!). Here we will give the basics for understanding a RNG and

being able to judge its quality. Also, we will try to give you some orientation to

choose the one that suits your problem. After reading this chapter you should

not go and write your own RNG, except for research reasons and to collect

experience. It is better to search for a good, well-programmed and suitable

up-to-date RNG, or to check the built-in one in your computer package, accept

it as usable, and be happy with it (for some time) or otherwise, if it is no good,

mistrust the simulation results.

Although implementing RNGs is such an enormous and complicated ﬁeld,

we will present some simple algorithms here so that you can experiment a

little bit for yourself. But do not see those algorithms as recommendations,

the only aim is to get familiar with the technical terms.

2.1.2 Quality criteria for RNGs

For a reliable simulation good random numbers are imperative. A bad RNG

could cause totally stupid simulation results which can lead to false conclu-

sions. Before using an built-in RNG of a software programme, it should be

checked ﬁrst. Here are some quality criteria the user should be aware of.

• Of course, uniformly distributed RNs ∼ U [0, 1] should be evenly dis-

tributed in the interval [0, 1], but the structure should not be too regular.

Otherwise they will not be considered as random. However, in some simula-

tion situations we are able to take advantage of regular structures, especially

when working with smooth functions. In these cases we can use quasirandom

sequences, which do not look random at all, but are very evenly distributed.

• As in Monte Carlo simulations, where we often need lots of RNs, it should

be possible to produce them very fast and eﬃciently without using too much

memory. So, speed and memory requirements matter.

• RN algorithms work with a ﬁnite set of data due to the construction prin-

ciple of the algorithm and due to the ﬁnite representation of real numbers in

the computer. So the RNG will eventually repeat its sequence of numbers.

The maximal length of this sequence before it is repeated is called the period

of the RNG. For the extensive simulations of today huge amounts of RNs are

needed. Therefore the period must be long enough to avoid using the same

Generating Random Numbers 7

RNs again. A rule of thumb says that the period length of the RNG should

be at least as long as the square of the quantity of RNs needed. Otherwise

the deterministic aspect of the RNG comes into play and makes the RNs cor-

related. In former times RNGs with a period of about 10

19

were considered

as good, but nowadays those algorithms are considered as insuﬃcient. Even

worse, there are still RNGs with a period of only 10

5

in use, especially built-in

RNGs, which are often the cause for severely bad Monte Carlo simulations.

• In cryptography the unpredictability of the RNs is essential. This is not

really necessary for Monte Carlo methods. It is more important that RNs ap-

pear to be random and independent, and pass statistical tests about being

independent, identically distributed (i.i.d.) RNs ∼ U [0, 1].

• Within Monte Carlo methods the sequence of RNs should be reproducible.

First because of debugging reasons. If we get strange simulation results, we

will be able to examine the sequence of RNs. Also it is quite common in

some applications, e.g. in sensitivity analysis, to use the same sequence again.

Another advantage is the possibility to compare diﬀerent calculation methods

or similar ﬁnancial products in a more eﬃcient way by using the same RNs.

• The algorithm should be programmed so that it delivers the same RNs in

every computer, also called portability. It should be possible to repeat cal-

culations in other machines and always obtain the same result. The same seed

for initialization should always return the same RN sequence.

• To make calculations very fast it is often desirable to use a computer with

parallel processors. So the possibility of parallelization is another impor-

tant point of consideration. One option is e.g. to jump ahead quickly many

steps in the sequence and produce disjoint substreams of the stream of RNs.

Another option is to ﬁnd a family of RNGs which work equally well for a large

set of diﬀerent parameters.

• The structure of the random points is very important. A typical feature of

some RNGs is that if d-dimensional vectors are constructed out of consecutive

RNs, then those points lie on hyperplanes. Bad RNGs have only a few hy-

perplanes in some dimensions which are far apart, leaving huge gaps without

any random vectors in space. Further, one should be aware if there is a severe

correlation between some RNs. For example, if the algorithm constructs a

new RN out of two previous RNs, groups of three RNs could be linked too

closely. This correlation can lead to false simulation results if the model has

a structure that is similar.

• Sometimes it is necessary to choose random number generators which are

easy to implement, i.e. the code is simple and short. This is a more general

point and applies to all parts of the simulation. Very often it is useful to have

a second simulation routine to check important simulation results. In this

case, code which is easy to read increases the possibility that it is bug-free.

8 Monte Carlo Methods and Models in Finance and Insurance

2.1.3 Technical terms

All RNGs that are based on deterministic recurrences can be described as

a quintuple (S,μ,f,U,g) (see L’Ecuyer [1994]), where

•Sis the ﬁnite set of states, the so-called state space.

• s

n

∈Sis a particular state at iteration step n.

• μ is the probability measure for selecting s

0

, the initial state, from S. s

0

is

called the seed of the RNG.

• The function f, also called the transition function, describes the algo-

rithm, s

n+1

= f (s

n

).

•Uis the output space. We will mainly consider [0, 1], [0, 1), (0, 1] or (0, 1).

• The output function g : S→Umaps the state s

n

∈Sinto a number

u

n

∈U, the ﬁnal random number that we are interested in.

REMARK 2.1 1. The importance of the seed that invokes the RNG

should not be underestimated. One cause of strange simulation results is

often that the RNG has not been seeded or has been started with a zero

vector. Another reason is that some seeds have to be avoided as they lead

to bad RN sequences, e.g., a seed that contains mainly zeroes might lead to

RNs that do not diﬀer much. Further, it is advisable to save the seed for

debugging.

2. As real numbers can only be presented with ﬁnite accuracy in computers,

the set of states S must be ﬁnite and so the range g (S) ⊂Uis also ﬁnite. As

weareinterestedinanevendistribution,wewantthegapsin[0, 1) not to be

too large. Therefore, S has to be a large set.

3. Another consequence of S being ﬁnite is that s

n+ρ

= s

n

for some integer

ρ, meaning the sequence will repeat itself eventually. The smallest number ρ

for which this happens is called the period of the RNG. It cannot be larger

than |S|, another reason for the set of states being huge. In some generators

the period might be diﬀerent for diﬀerent seeds and the RN cycles are then

disjoint. So we have to be careful with which seed we start.

2.2 Examples of random number generators

2.2.1 Linear congruential generators

The linear congruential method was one of the ﬁrst RNGs. Linear congru-

ential generators (LCGs) were ﬁrst introduced by Lehmer (1949) and were

very popular for many years.

Generating Random Numbers 9

Algorithm 2.1 LCG

s

n+1

=(as

n

+ c)modm, n ∈ N,

where

• m ∈ N \{0} is called the modulus,

• a ∈ N is called the multiplier, a<m,

• c ∈ N is called the increment, c<m,

• s

0

∈ N is called the seed, s

0

<m.

Numbers in [0, 1) are obtained via

u

n

=

s

n

m

.

These generators are often denoted by LCG(m, a, c). The state space can

be described as

S⊆N, S = {0, 1, ,m− 1} or {1, 2, ,m−1}.

For the increment c = 0 the generator cannot be started with s

0

=0. Then0

is excluded from the set of states.

Choosing coeﬃcients

Often the modulus m is chosen as a prime. Then all calculations are done

in the ﬁnite ﬁeld Z

m

. Preferred moduli are Mersenne Primes, primes of the

form 2

k

− 1, e.g. 2

31

− 1 = 2147483647, the largest 32-bit signed integer.

Sometimes a power of 2 is chosen as the modulus, because calculations can

be made faster by exploiting the binary structure of computers. But then the

lower order bits of the RNs are highly correlated, especially the last bit of

the states s

n

is either constant or strictly alternates between 0 and 1. This

means that all our integer RNs s

n

are either even or odd or we have a regular

even/odd pattern.

If the period of such a generator is m or m −1(inthecaseofc =0)wecall

this property full period as this is the maximum possible period length. In

order to achieve a long period it is advisable to choose a very large number as

the modulus. Further, the increment and the multiplier have to be selected

to create a generator with full period. If c =0,m is a prime, a is a primitive

root modulo m, then the generator has full period. For c = 0 the criteria for

full period are more complicated (see e.g. Knuth [1998]): the only common

divisor of m and c should be 1, every prime number that divides m should also

divide a −1; if m is divisible by 4, then a − 1 should also have the divisor 4.

10 Monte Carlo Methods and Models in Finance and Insurance

Implementation issues

Our next task is the implementation of the algorithm. If the modulus m is

near the maximum integer and also the multiplier a is quite large, then the

calculation step done with integer arithmetics can lead to an overﬂow. We

then get errors or, in the better case, our code will be machine dependent

and is no longer portable. To circumvent this diﬃculty there are several

possibilities:

• Instead of integer arithmetic we can work with ﬂoating-point arithmetic. For

example, in 64-bit ﬂoating-point arithmetic integers up to 2

53

are represented

exactly. So we just take care that a (m − 1) ≤ 2

53

.

• As we work in the ring Z

m

, we could consider multiplying with −¯a instead

of multiplying with a,where¯a = m − a is the additive inverse of a.

• Another technique is called approximate factoring.Ifm is large and

a not too large, e.g. a<

√

m, then we decompose m = aq + r, i.e. r = m

mod a , q = m/a. The complicated part of the iteration step is done by

(as

n

)modm =

a (s

n

mod q) −rz/q if it is ≥ 0

a (s

n

mod q) −rz/q+ m otherwise

(2.1)

As the parameter r is small this method avoids overﬂows. This trick also

works when a = m/k with an integer k<

√

m.

• The powers-of-two decomposition can be applied when a can be written

as a sum or a diﬀerence of a small number of powers of 2, e.g. a =2

3

+2

8

.

Then the modulus is decomposed as m =2

k

−h with integers k, h ∈ N.Now

(as

n

)modm can be eﬃciently calculated using shifts, masks, additions,

subtractions, and a single multiplication by h.

Disadvantages of LCGs

LCGs should no longer be used for important simulations. First, much

better RNGs exist, and then they also have a lot of disadvantages:

• There exists a high sequential correlation between the RNs, so they do

not mimic randomness very well. For example, if the multiplier a is small

compared to the modulus m,thenverysmallRNsarealways followed by

another small RN. In a simulation this means that rare events could be too

close together or happen too often.

• The modulus m cannot be larger than the largest integer available in the

programming language and so the period, which is maximally m, is usually

not very long. In 32-bit computers the period is often only 2

31

−1 and this is

simply too short nowadays.

• We take a closer look at the set of all possible t-dimensional vectors ﬁlled

with consecutive RNs, the ﬁnite set Ψ

t

:= {(u

1

,u

2

, ,u

t

) |s

0

∈S}⊆[0, 1)

t

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