Monte Carlo Methods and Models in Finance and Insurance doc

Monte Carlo
Methods and
Models in Finance
and Insurance


CHAPMAN & HALL/CRC
Financial Mathematics Series
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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

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Monte Carlo
Methods and
Models in Finance
and Insurance

Ralf Korn
Elke Korn
Gerald Kroisandt


CRC Press
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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
Visit the Taylor & Francis Web site at

and the CRC Press Web site at


2009045581


Contents

List of Algorithms

1 Introduction and User Guide
1.1 Introduction and concept .
1.2 Contents . . . . . . . . . .
1.3 How to use this book . . .
1.4 Further literature . . . . .
1.5 Acknowledgments . . . . .

xi

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2

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2 Generating Random Numbers
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . .
2.1.1 How do we get random numbers? . . . . . . . . . .
2.1.2 Quality criteria for RNGs . . . . . . . . . . . . . .
2.1.3 Technical terms . . . . . . . . . . . . . . . . . . . .
2.2 Examples of random number generators . . . . . . . . . .
2.2.1 Linear congruential generators . . . . . . . . . . .
2.2.2 Multiple recursive generators . . . . . . . . . . . .
2.2.3 Combined generators . . . . . . . . . . . . . . . . .
2.2.4 Lagged Fibonacci generators . . . . . . . . . . . .
2.2.5 F2 -linear generators . . . . . . . . . . . . . . . . .
2.2.6 Nonlinear RNGs . . . . . . . . . . . . . . . . . . .
2.2.7 More random number generators . . . . . . . . . .
2.2.8 Improving RNGs . . . . . . . . . . . . . . . . . . .
2.3 Testing and analyzing RNGs . . . . . . . . . . . . . . . .
2.3.1 Analyzing the lattice structure . . . . . . . . . . .
2.3.2 Equidistribution . . . . . . . . . . . . . . . . . . .
2.3.3 Diffusion capacity . . . . . . . . . . . . . . . . . .
2.3.4 Statistical tests . . . . . . . . . . . . . . . . . . . .
2.4 Generating random numbers with general distributions .
2.4.1 Inversion method . . . . . . . . . . . . . . . . . . .
2.4.2 Acceptance-rejection method . . . . . . . . . . . .
2.5 Selected distributions . . . . . . . . . . . . . . . . . . . .
2.5.1 Generating normally distributed random numbers
2.5.2 Generating beta-distributed RNs . . . . . . . . . .
2.5.3 Generating Weibull-distributed RNs . . . . . . . .

2.5.4 Generating gamma-distributed RNs . . . . . . . .
2.5.5 Generating chi-square-distributed RNs . . . . . . .

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v


vi
2.6

2.7

2.8

Multivariate random variables . . . . . . . . . . . . . . . . .
2.6.1 Multivariate normals . . . . . . . . . . . . . . . . . . .
2.6.2 Remark: Copulas . . . . . . . . . . . . . . . . . . . . .
2.6.3 Sampling from conditional distributions . . . . . . . .
Quasirandom sequences as a substitute for random sequences
2.7.1 Halton sequences . . . . . . . . . . . . . . . . . . . . .
2.7.2 Sobol sequences . . . . . . . . . . . . . . . . . . . . . .
2.7.3 Randomized quasi-Monte Carlo methods . . . . . . . .
2.7.4 Hybrid Monte Carlo methods . . . . . . . . . . . . . .
2.7.5 Quasirandom sequences and transformations into
other random distributions . . . . . . . . . . . . . . .
Parallelization techniques . . . . . . . . . . . . . . . . . . . .
2.8.1 Leap-frog method . . . . . . . . . . . . . . . . . . . .
2.8.2 Sequence splitting . . . . . . . . . . . . . . . . . . . .
2.8.3 Several RNGs . . . . . . . . . . . . . . . . . . . . . . .
2.8.4 Independent sequences . . . . . . . . . . . . . . . . . .

2.8.5 Testing parallel RNGs . . . . . . . . . . . . . . . . . .

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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:
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
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2 Stochastic processes and their paths: Basic definitions . . . .
4.3 The Monte Carlo method for stochastic processes . . . . . .
4.3.1 Monte Carlo and stochastic processes . . . . . . . . .
4.3.2 Simulating paths of stochastic processes: Basics . . . .
4.3.3 Variance reduction for stochastic processes . . . . . .
4.4 Brownian motion and the Brownian bridge . . . . . . . . . .
4.4.1 Properties of Brownian motion . . . . . . . . . . . . .
4.4.2 Weak convergence and Donsker’s theorem . . . . . . .

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vii

4.5

4.6

4.7

4.8

4.4.3 Brownian bridge . . . . . . . . . . . . . . . . . . . . .
Basics of Itˆ calculus . . . . . . . . . . . . . . . . . . . . . .
o
4.5.1 The Itˆ integral . . . . . . . . . . . . . . . . . . . . . .
o
4.5.2 The Itˆ formula . . . . . . . . . . . . . . . . . . . . . .
o
4.5.3 Martingale representation and change of measure . . .
Stochastic differential equations . . . . . . . . . . . . . . . .

4.6.1 Basic results on stochastic differential equations . . . .
4.6.2 Linear stochastic differential equations . . . . . . . . .
4.6.3 The square-root stochastic differential equation . . . .
4.6.4 The Feynman-Kac representation theorem . . . . . . .
Simulating solutions of stochastic differential equations . . .
4.7.1 Introduction and basic aspects . . . . . . . . . . . . .
4.7.2 Numerical schemes for ordinary differential equations
4.7.3 Numerical schemes for stochastic differential equations
4.7.4 Convergence of numerical schemes for SDEs . . . . . .
4.7.5 More numerical schemes for SDEs . . . . . . . . . . .
4.7.6 Efficiency of numerical schemes for SDEs . . . . . . .
4.7.7 Weak extrapolation methods . . . . . . . . . . . . . .
4.7.8 The multilevel Monte Carlo method . . . . . . . . . .
Which simulation methods for SDE should be chosen? . . .

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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 BlackScholes 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-Scholes models . 238
5.13 Stochastic local volatility models . . . . . . . . . . . . . . . . 239
5.14 Monte Carlo option pricing: American and Bermudan options 240
5.14.1 The Longstaff-Schwartz algorithm and regression-based
variants for pricing Bermudan options . . . . . . . . . 243
5.14.2 Upper price bounds by dual methods . . . . . . . . . . 250
5.15 Monte Carlo calculation of option price sensitivities . . . . . 257
5.15.1 The role of the price sensitivities . . . . . . . . . . . . 257
5.15.2 Finite difference simulation . . . . . . . . . . . . . . . 258
5.15.3 The pathwise differentiation method . . . . . . . . . . 261
5.15.4 The likelihood ratio method . . . . . . . . . . . . . . . 264
5.15.5 Combining the pathwise differentiation and the
likelihood ratio methods by localization . . . . . . . . 265
5.15.6 Numerical testing in the Black-Scholes setting . . . . . 267
5.16 Basics of interest rate modelling . . . . . . . . . . . . . . . . 269
5.16.1 Different notions of interest rates . . . . . . . . . . . . 270
5.16.2 Some popular interest rate products . . . . . . . . . . 271
5.17 The short rate approach to interest rate modelling . . . . . . 275
5.17.1 Change of numeraire and option pricing: The forward
measure . . . . . . . . . . . . . . . . . . . . . . . . . . 276
5.17.2 The Vasicek model . . . . . . . . . . . . . . . . . . . . 278
5.17.3 The Cox-Ingersoll-Ross (CIR) model . . . . . . . . . . 281
5.17.4 Affine linear short rate models . . . . . . . . . . . . . 283
5.17.5 Perfect calibration: Deterministic shifts and the HullWhite 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-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 forwardLIBOR 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
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.2 Poisson processes and Poisson random measures: Definition
and simulation . . . . . . . . . . . . . . . . . . . . . . . . . .
6.2.1 Stochastic integrals with respect to Poisson processes
6.3 Jump-diffusions: Basics, properties, and simulation . . . . .

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6.4

6.5

6.3.1 Simulating Gauss-Poisson jump-diffusions . . . .
6.3.2 Euler-Maruyama scheme for jump-diffusions . . .
L´vy processes: Properties and examples . . . . . . . .
e
6.4.1 Definition and properties of L´vy processes . . .
e
6.4.2 Examples of L´vy processes . . . . . . . . . . . .
e
Simulation of L´vy processes . . . . . . . . . . . . . . .
e
6.5.1 Exact simulation and time discretization . . . . .
6.5.2 The Euler-Maruyama scheme for L´vy processes
e
6.5.3 Small jump approximation . . . . . . . . . . . .
6.5.4 Simulation via series representation . . . . . . . .

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7 Simulating Financial Models: Discontinuous Paths
335
7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 335
7.2 Merton’s jump-diffusion model and stochastic volatility models
with jumps . . . . . . . . . . . . . . . . . . . . . . . . . . . . 335
7.2.1 Merton’s jump-diffusion setting . . . . . . . . . . . . . 335
7.2.2 Jump-diffusion with double exponential jumps . . . . 339
7.2.3 Stochastic volatility models with jumps . . . . . . . . 340
7.3 Special L´vy models and their simulation . . . . . . . . . . . 340
e
7.3.1 The Esscher transform . . . . . . . . . . . . . . . . . . 341
7.3.2 The hyperbolic L´vy model . . . . . . . . . . . . . . . 342
e
7.3.3 The variance gamma model . . . . . . . . . . . . . . . 344
7.3.4 Normal inverse Gaussian processes . . . . . . . . . . . 352
7.3.5 Further aspects of L´vy type models . . . . . . . . . . 354
e
8 Simulating Actuarial Models
8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . .
8.2 Premium principles and risk measures . . . . . . . . . . . . .
8.2.1 Properties and examples of premium principles . . . .
8.2.2 Monte Carlo simulation of premium principles . . . .
8.2.3 Properties and examples of risk measures . . . . . . .
8.2.4 Connection between premium principles and risk
measures . . . . . . . . . . . . . . . . . . . . . . . . .
8.2.5 Monte Carlo simulation of risk measures . . . . . . . .
8.3 Some applications of Monte Carlo methods in life insurance .
8.3.1 Mortality: Definitions and classical models . . . . . .
8.3.2 Dynamic mortality models . . . . . . . . . . . . . . . .
8.3.3 Life insurance contracts and premium calculation . . .

8.3.4 Pricing longevity products by Monte Carlo simulation
8.3.5 Premium reserves and Thiele’s differential equation . .
8.4 Simulating dependent risks with copulas . . . . . . . . . . .
8.4.1 Definition and basic properties . . . . . . . . . . . . .
8.4.2 Examples and simulation of copulas . . . . . . . . . .
8.4.3 Application in actuarial models . . . . . . . . . . . . .
8.5 Nonlife insurance . . . . . . . . . . . . . . . . . . . . . . . .

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8.6

8.7

8.5.1 The individual model . . . . . . . . . . . . . . . . . .
8.5.2 The collective model . . . . . . . . . . . . . . . . . . .
8.5.3 Rare event simulation and heavy-tailed distributions .
8.5.4 Dependent claims: An example with copulas . . . . .
Markov chain Monte Carlo and Bayesian estimation . . . . .
8.6.1 Basic properties of Markov chains . . . . . . . . . . .
8.6.2 Simulation of Markov chains . . . . . . . . . . . . . .
8.6.3 Markov chain Monte Carlo methods . . . . . . . . . .
8.6.4 MCMC methods and Bayesian estimation . . . . . . .
8.6.5 Examples of MCMC methods and Bayesian estimation
in actuarial mathematics . . . . . . . . . . . . . . . .
Asset-liability management and Solvency II . . . . . . . . . .
8.7.1 Solvency II . . . . . . . . . . . . . . . . . . . . . . . .
8.7.2 Asset-liability management (ALM) . . . . . . . . . . .

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References

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Index

459


List of Algorithms

2.1
2.2
2.3
2.4
2.5
2.6
2.7
2.8
2.9
2.10
2.11
2.12
2.13
2.14
2.15

2.16
2.17
2.18
2.19
2.20
2.21
3.1
4.1
4.2
4.3
4.4
4.5
4.6
4.7
4.8

LCG . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
MRG . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
F2 -linear generators . . . . . . . . . . . . . . . . . . . . . . .
Linear feedback shift register generators . . . . . . . . . . . .
Example of a combined LFSR generator . . . . . . . . . . . .
GFSR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Inversive congruential generators . . . . . . . . . . . . . . . .
Inversion method . . . . . . . . . . . . . . . . . . . . . . . . .
Acceptance-rejection method . . . . . . . . . . . . . . . . . .
Approximation of the standard normal c.d.f. . . . . . . . . . .
Beasley-Springer-Moro algorithm for the inverse standard
normal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
The Box-Muller method . . . . . . . . . . . . . . . . . . . . .
Jăhnks beta generator . . . . . . . . . . . . . . . . . . . . . .

o
Cheng’s algorithm for gamma-distributed RNs with a > 1 . .
Algorithm for gamma-distributed RNs with 0 < a < 1 . . . .
Jăhnks generator for gamma-distributed RNs with
o
0Chi-square-distributed RNs . . . . . . . . . . . . . . . . . . .
Chi-square-distributed RNs with the help of normally
distributed RNs . . . . . . . . . . . . . . . . . . . . . . . . . .
Cholesky factorization . . . . . . . . . . . . . . . . . . . . . .
One-dimensional Halton sequences (Van-der-Corput
sequences) . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
t-dimensional Halton sequences . . . . . . . . . . . . . . . . .
The (crude) Monte Carlo method . . . . . . . . . . . . . . . .
Simulation of a discrete-time stochastic process with
independent increments . . . . . . . . . . . . . . . . . . . . .
Simulation of a continuous-time stochastic process with
continuous paths . . . . . . . . . . . . . . . . . . . . . . . . .
Simulation of a Brownian motion . . . . . . . . . . . . . . . .
Forward simulation of a Brownian bridge . . . . . . . . . . .
Backward simulation of a Brownian bridge from a to b with
n = 2K time points . . . . . . . . . . . . . . . . . . . . . . . .
Variance reduction by Brownian bridge . . . . . . . . . . . . .
The Euler-Maruyama scheme . . . . . . . . . . . . . . . . . .
The Milstein scheme . . . . . . . . . . . . . . . . . . . . . . .

9
13
18
18

20
21
23
32
34
36
37
38
39
40
41
42
42
43
44
47
48
57
109
110
115
123
124
125
152
155

xi

xii
4.9
4.10
4.11
4.12
4.13
5.1
5.2
5.3
5.4
5.5
5.6
5.7
5.8
5.9
5.10
5.11
5.12
5.13
5.14
5.15
5.16
5.17
5.18
5.19
5.20
5.21
6.1
6.2
6.3

6.4
6.5
6.6
7.1
7.2
7.3
7.4

The Milstein order 2 scheme . . . . . . . . . . . . . . . . . . .
Talay-Tubaro extrapolation . . . . . . . . . . . . . . . . . . .
The statistical Romberg method . . . . . . . . . . . . . . . .
Multilevel Monte Carlo simulation . . . . . . . . . . . . . . .
Adaptive multilevel Monte Carlo simulation . . . . . . . . . .
Simulating a stock price path in the Black-Scholes model . .
Option pricing via Monte Carlo simulation . . . . . . . . . . .
MC pricing of path-independent options . . . . . . . . . . . .
Corrected geometric mean control for fixed-strike average
options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Monte Carlo for discrete barrier options . . . . . . . . . . . .
Conditional MC for double-barrier knock-out options . . . . .
MC pricing of out-barrier options with the bridge technique .
Double-barrier knock-out call pricing using Moon’s method for
piecewise constant barriers . . . . . . . . . . . . . . . . . . . .
Preprocessing a d-dimensional normal sample . . . . . . . . .
Monte Carlo pricing in the CEV model . . . . . . . . . . . . .
Calibrating σ in the Black-Scholes model . . . . . . . . . . .
Simulating price paths in the Heston model (naive way) . . .
Quadratic exponential (QE) method for simulating the
volatility in the Heston model . . . . . . . . . . . . . . . . . .
Stock price paths in the Heston model . . . . . . . . . . . . .

Call pricing with the Heath-Platen estimator . . . . . . . . .
Algorithmic framework for pricing American (Bermudan)
options by Monte Carlo methods . . . . . . . . . . . . . . . .
Dynamic programming for calculating the price of
Bermudan options . . . . . . . . . . . . . . . . . . . . . . . .
Longstaff-Schwartz algorithm for pricing Bermudan options .
Andersen-Broadie algorithm to obtain upper bounds for the
price of a Bermudan option . . . . . . . . . . . . . . . . . . .
Option pricing in the Cheyette model . . . . . . . . . . . . .
Simulation of paths of forward-LIBOR rates under the spotLIBOR measure . . . . . . . . . . . . . . . . . . . . . . . . .
Simulation of a compound Poisson process . . . . . . . . . . .
Simulation of a stochastic integral with respect to a
compound Poisson process . . . . . . . . . . . . . . . . . . . .
Simulation of a Gauss-Poisson jump-diffusion process with a
fixed time discretization . . . . . . . . . . . . . . . . . . . . .
Euler-Maruyama scheme for jump-diffusions . . . . . . . . . .
Exact simulation of discretized L´vy processes . . . . . . . . .
e
Euler-Maruyama scheme for L´vy processes . . . . . . . . . .
e
A path in the Merton jump-diffusion model . . . . . . . . . .
Simulating a variance gamma process by subordination . . . .
Variance gamma path by differences . . . . . . . . . . . . . .
A VG path by the DGBS algorithm . . . . . . . . . . . . . .

160
165
167
169
170

181
196
197
201
204
206
207
210
213
219
221
226
230
232
236
243
244
247
253
293
300
312
316
318
320
329
331
337
346
347

349


xiii
7.5
7.6
8.1
8.2
8.3
8.4
8.5
8.6
8.7
8.8
8.9
8.10
8.11
8.12
8.13
8.14
8.15
8.16
8.17
8.18
8.19
8.20

Pricing up-and-in calls by the DGBS method . . . . . . . . .
Simulation of an NIG path . . . . . . . . . . . . . . . . . . .
Crude Monte Carlo simulation of the α-quantile . . . . . . . .

Importance sampling for quantiles . . . . . . . . . . . . . . .
Value-at-Risk via importance sampling of the delta-gamma
approximation . . . . . . . . . . . . . . . . . . . . . . . . . .
Modelling dynamic mortality by extrapolation . . . . . . . .
Stochastic dynamic mortality modelling . . . . . . . . . . . .
Simulating dynamic mortality rates in the stochastic Gompertz
model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Measure calibration in the stochastic Gompertz model with a
traded longevity bond . . . . . . . . . . . . . . . . . . . . . .
Simulation of the (mean) prospective reserve of an insurance
contract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Simulating with a Gaussian copula . . . . . . . . . . . . . . .
Simulating with a t-copula . . . . . . . . . . . . . . . . . . . .
Simulating with an Archimedean copula . . . . . . . . . . . .
Copula framework for dependent risks . . . . . . . . . . . . .
Simulating a path of a mixed Poisson process . . . . . . . . .
Simulating a path of an inhomogeneous Poisson process . . .
Simulating a path of a Cox process . . . . . . . . . . . . . . .
Simulating the Asmussen-Kroese estimator . . . . . . . . . .
Simulating a path of a homogeneous Markov chain with a
precalculated transition matrix . . . . . . . . . . . . . . . . .
Simulation of a Markov chain . . . . . . . . . . . . . . . . . .
Metropolis-Hastings algorithm . . . . . . . . . . . . . . . . .
Gibbs sampler . . . . . . . . . . . . . . . . . . . . . . . . . . .

352
354
366
369
373

380
380
382
387
389
396
399
401
403
407
408
408
412
420
420
421
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 engineers 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 Andersen 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 breakthroughs when indeed the methods are applied to already solved problems or
to problems that do not make sense when viewed from the financial or insurance 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 generation 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 Itˆ calculus are provided together with numerical
o
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 preceding 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. Stochastic processes that contain jumps such as jump-diffusions or L´vy processes ene


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´vy models
e
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 actuarial 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 Rubinstein (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 Hansjărg Albrecher
o
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
2.1.1

Introduction
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 numbers 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 unless 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. But if we
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, especially 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
and ones as this might lead to programme crashes, as e.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 conclusions. 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 distributed in the interval [0, 1], but the structure should not be too regular.

Otherwise they will not be considered as random. However, in some simulation 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 principle 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 correlated. In former times RNGs with a period of about 1019 were considered
as good, but nowadays those algorithms are considered as insufficient. Even
worse, there are still RNGs with a period of only 105 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 appear 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 calculations 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 important 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 hyperplanes 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
• S is the finite set of states, the so-called state space.
• sn ∈ S is a particular state at iteration step n.
• μ is the probability measure for selecting s0 , the initial state, from S. s0 is
called the seed of the RNG.
• The function f , also called the transition function, describes the algorithm, sn+1 = f (sn ).
• U is the output space. We will mainly consider [0, 1], [0, 1), (0, 1] or (0, 1).
• The output function g : S → U maps the state sn ∈ S into a number
un ∈ 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) ⊂ U is also finite. As
we are interested in an even distribution, we want the gaps in [0, 1) not to be
too large. Therefore, S has to be a large set.
3. Another consequence of S being finite is that sn+ρ = sn 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
2.2.1

Examples of random number generators
Linear congruential generators

The linear congruential method was one of the first RNGs. Linear congruential generators (LCGs) were first introduced by Lehmer (1949) and were
very popular for many years.


Generating Random Numbers

9

Algorithm 2.1 LCG
sn+1 = (asn + c)

mod m,

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,
• s0 ∈ N is called the seed, s0 < m.
Numbers in [0, 1) are obtained via
un =

sn
.
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 s0 = 0. Then 0
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 Zm . Preferred moduli are Mersenne Primes, primes of the
form 2k − 1, e.g. 231 − 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 sn is either constant or strictly alternates between 0 and 1. This
means that all our integer RNs sn 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 (in the case of c = 0) we call
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 253 are represented
exactly. So we just take care that a (m − 1) ≤ 253 .
• As we work in the ring Zm , we could consider multiplying with −¯ instead
a
of multiplying with a, where a = m − a is the additive inverse of a.
¯
• Another technique is called approximate factoring. If m 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
(asn )

mod m =

a (sn
a (sn

mod q) − r z/q

if it is ≥ 0
mod q) − r z/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 = 23 + 28 .
Then the modulus is decomposed as m = 2k − h with integers k, h ∈ N. Now
(asn ) mod m 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, then very small RNs are always 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 231 − 1 and this is
simply too short nowadays.
• We take a closer look at the set of all possible t-dimensional vectors filled
t
with consecutive RNs, the finite set Ψt := {(u1 , u2 , . . . , ut ) |s0 ∈ S} ⊆ [0, 1)