Monte Carlo Methods and Models in Finance and Insurance

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ISBN: 9781420076189 | 1420076183
Cover: Hardcover
Copyright: 2/26/2010

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Offering a unique balance between applications and calculations, this book incorporates the application background of finance and insurance with the theory and applications of Monte Carlo methods. It presents recent methods and algorithms, including the multilevel Monte Carlo method, the statistical Romberg method, and the Heath'Platen estimator, as well as recent financial and actuarial models, such as the Cheyette and dynamic mortality models. The book enables readers to find the right algorithm for a desired application and illustrates complicated methods and algorithms with simple applications to provide an easy understanding of key properties.

List of Algorithms	p. xi
Introduction and User Guide	p. 1
Introduction and concept	p. 1
Contents	p. 2
How to use this book	p. 3
Further literature	p. 3
Acknowledgments	p. 4
Generating Random Numbers	p. 5
Introduction	p. 5
How do we get random numbers?	p. 5
Quality criteria for RNGs	p. 6
Technical terms	p. 8
Examples of random number generators	p. 8
Linear congruential generators	p. 8
Multiple recursive generators	p. 12
Combined generators	p. 15
Lagged Fibonacci generators	p. 16
F₂-linear generators	p. 17
Nonlinear RNGs	p. 22
More random number generators	p. 24
Improving RNGs	p. 24
Testing and analyzing RNGs	p. 25
Analyzing the lattice structure	p. 25
Equidistribution	p. 26
Diffusion capacity	p. 27
Statistical tests	p. 27
Generating random numbers with general distributions	p. 31
Inversion method	p. 31
Acceptance-rejection method	p. 33
Selected distributions	p. 36
Generating normally distributed random numbers	p. 36
Generating beta-distributed RNs	p. 38
Generating Weibull-distributed RNs	p. 38
Generating gamma-distributed RNs	p. 39
Generating chi-square-distributed RNs	p. 42
Multivariate random variables	p. 43
Multivariate normals	p. 43
Remark: Copulas	p. 44
Sampling from conditional distributions	p. 44
Quasirandom sequences as a substitute for random sequences	p. 45
Halton sequences	p. 47
Sobol sequences	p. 48
Randomized quasi-Monte Carlo methods	p. 49
Hybrid Monte Carlo methods	p. 50
Quasirandom sequences and transformations into other random distributions	p. 50
Parallelization techniques	p. 51
Leap-frog method	p. 51
Sequence splitting	p. 52
Several RNGs	p. 53
Independent sequences	p. 53
Testing parallel RNGs	p. 53
The Monte Carlo Method: Basic Principles	p. 55
Introduction	p. 55
The strong law of large numbers and the Monte Carlo method	p. 56
The strong law of large numbers	p. 56
The crude Monte Carlo method	p. 57
The Monte Carlo method: Some first applications	p. 60
Improving the speed of convergence of the Monte Carlo method: Variance reduction methods	p. 65
Antithetic variates	p. 66
Control variates	p. 70
Stratified sampling	p. 76
Variance reduction by conditional sampling	p. 85
Importance sampling	p. 87
Further aspects of variance reduction methods	p. 97
More methods	p. 97
Application of the variance reduction methods	p. 100
Continuous-Time Stochastic Processes: Continuous Paths	p. 103
Introduction	p. 103
Stochastic processes and their paths: Basic definitions	p. 103
The Monte Carlo method for stochastic processes	p. 107
Monte Carlo and stochastic processes	p. 107
Simulating paths of stochastic processes: Basics	p. 108
Variance reduction for stochastic processes	p. 110
Brownian motion and the Brownian bridge	p. 111
Properties of Brownian motion	p. 113
Weak convergence and Donsker's theorem	p. 116
Brownian bridge	p. 120
Basics of Itô calculus	p. 126
The Itô integral	p. 126
The Itô formula	p. 133
Martingale representation and change of measure	p. 135
Stochastic differential equations	p. 137
Basic results on stochastic differential equations	p. 137
Linear stochastic differential equations	p. 139
The square-root stochastic differential equation	p. 141
The Feynman-Kac representation theorem	p. 142
Simulating solutions of stochastic differential equations	p. 145
Introduction and basic aspects	p. 145
Numerical schemes for ordinary differential equations	p. 146
Numerical schemes for stochastic differential equations	p. 151
Convergence of numerical schemes for SDEs	p. 156
More numerical schemes for SDEs	p. 159
Efficiency of numerical schemes for SDEs	p. 162
Weak extrapolation methods	p. 163
The multilevel Monte Carlo method	p. 167
Which simulation methods for SDE should be chosen?	p. 173
Simulating Financial Models: Continuous Paths	p. 175
Introduction	p. 175
Basics of stock price modelling	p. 176
A Black-Scholes type stock price framework	p. 177
An important special case: The Black-Scholes model	p. 180
Completeness of the market model	p. 183
Basic facts of options	p. 184
An introduction to option pricing	p. 187
A short history of option pricing	p. 187
Option pricing via the replication principle	p. 187
Dividends in the Black-Scholes setting	p. 195
Option pricing and the Monte Carlo method in the Black-Scholes setting	p. 196
Path-independent European options	p. 197
Path-dependent European options	p. 199
More exotic options	p. 210
Data preprocessing by moment matching methods	p. 211
Weaknesses of the Black-Scholes model	p. 213
Local volatility models and the CEV model	p. 216
CEV option pricing with Monte Carlo methods	p. 219
An excursion: Calibrating a model	p. 221
Aspects of option pricing in incomplete markets	p. 222
Stochastic volatility and option pricing in the Heston model	p. 224
The Andersen algorithm for the Heston model	p. 227
The Heath-Platen estimator in the Heston model	p. 232
Variance reduction principles in non-Black-Scholes models	p. 238
Stochastic local volatility models	p. 239
Monte Carlo option pricing: American and Bermudan options	p. 240
The Longstaff-Schwartz algorithm and regression-based variants for pricing Bermudan options	p. 243
Upper price bounds by dual methods	p. 250
Monte Carlo calculation of option price sensitivities	p. 257
The role of the price sensitivities	p. 257
Finite difference simulation	p. 258
The pathwise differentiation method	p. 261
The likelihood ratio method	p. 264
Combining the pathwise differentiation and the likelihood ratio methods by localization	p. 265
Numerical testing in the Black-Scholes setting	p. 267
Basics of interest rate modelling	p. 269
Different notions of interest rates	p. 270
Some popular interest rate products	p. 271
The short rate approach to interest rate modelling	p. 275
Change of numeraire and option pricing: The forward measure	p. 276
The Vasicek model	p. 278
The Cox-Ingersoll-Ross (CIR) model	p. 281
Affine linear short rate models	p. 283
Perfect calibration: Deterministic shifts and the Hull-White approach	p. 283
Log-normal models and further short rate models	p. 287
The forward rate approach to interest rate modelling	p. 288
The continuous-time Ho-Lee model	p. 289
The Cheyette model	p. 290
LIBOR market models	p. 293
Log-normal forward-LIBOR modelling	p. 294
Relation between the swaptions and the cap market	p. 297
Aspects of Monte Carlo path simulations, of forward-LIBOR rates and derivative pricing	p. 299
Monte Carlo pricing of Bermudan swaptions with a parametric exercise boundary and further comments	p. 305
Alternatives to log-normal forward-LIBOR models	p. 308
Continuous-Time Stochastic Processes: Discontinuous Paths	p. 309
Introduction	p. 309
Poisson processes and Poisson random measures: Definition and simulation	p. 310
Stochastic integrals with respect to Poisson processes	p. 312
Jump-diffusions: Basics, properties, and simulation	p. 315
Simulating Gauss-Poisson jump-diffusions	p. 317
Euler-Maruyama scheme for jump-diffusions	p. 319
Lévy processes: Properties and examples	p. 320
Definition and properties of Lévy processes	p. 320
Examples of Lévy processes	p. 324
Simulation of Lévy processes	p. 329
Exact simulation and time discretization	p. 329
The Euler-Maruyama scheme for Lévy processes	p. 330
Small jump approximation	p. 331
Simulation via series representation	p. 333
Simulating Financial Models: Discontinuous Paths	p. 335
Introduction	p. 335
Merton's jump-diffusion model and stochastic volatility models with jumps	p. 335
Merton's jump-diffusion setting	p. 335
Jump-diffusion with double exponential jumps	p. 339
Stochastic volatility models with jumps	p. 340
Special Lévy models and their simulation	p. 340
The Esscher transform	p. 341
The hyperbolic Lévy model	p. 342
The variance gamma model	p. 344
Normal inverse Gaussian processes	p. 352
Further aspects of Lévy type models	p. 354
Simulating Actuarial Models	p. 357
Introduction	p. 357
Premium principles and risk measures	p. 357
Properties and examples of premium principles	p. 358
Monte Carlo simulation of premium principles	p. 362
Properties and examples of risk.measures	p. 362
Connection between premium principles and risk measures	p. 365
Monte Carlo simulation of risk measures	p. 366
Some applications of Monte Carlo methods in life insurance	p. 377
Mortality: Definitions and classical models	p. 378
Dynamic mortality models	p. 379
Life insurance contracts and premium calculation	p. 383
Pricing longevity products by Monte Carlo simulation	p. 385
Premium reserves and Thiele's differential equation	p. 387
Simulating dependent risks with copulas	p. 390
Definition and basic properties	p. 390
Examples and simulation of copulas	p. 393
Application in actuarial models	p. 402
Nonlife insurance	p. 403
The individual model	p. 404
The collective model	p. 405
Rare event simulation and heavy-tailed distributions	p. 410
Dependent claims: An example with copulas	p. 413
Markov chain Monte Carlo and Bayesian estimation	p. 415
Basic properties of Markov chains	p. 415
Simulation of Markov chains	p. 419
Markov chain Monte Carlo methods	p. 420
MCMC methods and Bayesian estimation	p. 427
Examples of MCMC methods and Bayesian estimation in actuarial mathematics	p. 429
Asset-liability management and Solvency II	p. 433
Solvency II	p. 433
Asset-liability management (ALM)	p. 435
References	p. 441
Index	p. 459
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Monte Carlo Methods and Models in Finance and Insurance

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