Monte Carlo methods and models in finance and insurance:
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| Main Authors: | , , |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Boca Raton [u.a.]
CRC Press
2010
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| Series: | Chapman & Hall/CRC financial mathematics series
A Chapman & Hall book |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XIII, 470 S. graph. Darst. |
| ISBN: | 9781420076189 |
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MARC
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Record in the Search Index
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| adam_text | Titel: Monte Carlo methods and models in finance and insurance
Autor: Korn, Ralf
Jahr: 2010
Contents
List of Algorithms xi
1 Introduction and User Guide 1
1.1 Introduction and concept .................... 1
1.2 Contents ............................. 2
1.3 How to use this book ...................... 3
1.4 Further literature ........................ 3
1.5 Acknowledgments ........................ 4
2 Generating Random Numbers 5
2.1 Introduction ........................... 5
2.1.1 How do we get random numbers?............ 5
2.1.2 Quality criteria for RNGs................ 6
2.1.3 Technical terms...................... 8
2.2 Examples of random number generators............ 8
2.2.1 Linear congruential generators............. 8
2.2.2 Multiple recursive generators.............. 12
2.2.3 Combined generators................... 15
2.2.4 Lagged Fibonacci generators .............. 16
2.2.5 F2-linear generators................... 17
2.2.6 Nonlinear RNGs..................... 22
2.2.7 More random number generators............ 24
2.2.8 Improving RNGs..................... 24
2.3 Testing and analyzing RNGs .................. 25
2.3.1 Analyzing the lattice structure............. 25
2.3.2 Equidistribution..................... 26
2.3.3 Diffusion capacity .................... 27
2.3.4 Statistical tests...................... 27
2.4 Generating random numbers with general distributions ... 31
2.4.1 Inversion method..................... 31
2.4.2 Acceptance-rejection method.............. 33
2.5 Selected distributions ...................... 36
2.5.1 Generating normally distributed random numbers . . 36
2.5.2 Generating beta-distributed RNs............ 38
2.5.3 Generating Weibull-distributed RNs.......... 38
2.5.4 Generating gamma-distributed RNs.......... 39
2.5.5 Generating chi-square-distributed RNs......... 42
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 Halton sequences..................... 47
2.7.2 Sobol sequences...................... 48
2.7.3 Randomized quasi-Monte Carlo methods........ 49
2.7.4 Hybrid Monte Carlo methods.............. 50
2.7.5 Quasirandom sequences and transformations into
other random distributions............... 50
2.8 Parallelization techniques.................... 51
2.8.1 Leap-frog method .................... 51
2.8.2 Sequence splitting.................... 52
2.8.3 Several RNGs....................... 53
2.8.4 Independent sequences.................. 53
2.8.5 Testing parallel RNGs.................. 53
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
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 .......... Ill
4.4.1 Properties of Brownian motion............. 113
4.4.2 Weak convergence and Donsker s theorem....... 116
4.4.3 Brownian bridge..................... 120
4.5 Basics of ltd calculus ...................... 126
4.5.1 The ltd integral...................... 126
4.5.2 The Ito formula...................... 133
4.5.3 Martingale representation and change of measure . . . 135
4.6 Stochastic differential equations ................ 137
4.6.1 Basic results on stochastic differential equations .... 137
4.6.2 Linear stochastic differential equations......... 139
4.6.3 The square-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 differentia] 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
Simulating Financial Models: Continuous Paths 175
5.1 Introduction ........................... 175
5.2 Basics of stock price modelling................. 176
5.3 A Black-Scholes type stock price framework ......... 177
5.3.1 An important special case: The Black-Scholes model . 180
5.3.2 Completeness of the market model........... 183
5.4 Basic facts of options ...................... 184
5.5 An introduction to option pricing ............... 187
5.5.1 A short history of option pricing............ 187
5.5.2 Option pricing via the replication principle...... 187
5.5.3 Dividends in the Black-Scholes setting......... 195
5.6 Option pricing and the Monte Carlo method in the Black-
Scholes setting .......................... 196
5.6.1 Path-independent European options.......... 197
5.6.2 Path-dependent European options........... 199
5.6.3 More exotic options ................... 210
5.6.4 Data preprocessing by moment matching methods . . 211
5.7 Weaknesses of the Black-Scholes model ............ 213
5.8 Local volatility models and the CEV model.......... 216
5.8.1 CEV option pricing with Monte Carlo methods .... 219
5.9 An excursion: Calibrating a model............... 221
5.10 Aspects of option pricing in incomplete markets ....... 222
5.11 Stochastic volatility and option pricing in the Heston model 224
5.11.1 The Andersen algorithm for the Heston model .... 227
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 Hull-
White approach ..................... 283
5.17.6 Log-normal models and further short rate models . . . 287
5.18 The forward rate approach to interest rate modelling .... 288
5.18.1 The continuous-time Ho-Lee model........... 289
5.18.2 The Cheyette model................... 290
5.19 LIBOR market models ..................... 293
5.19.1 Log-normal forward-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
Continuous-Time Stochastic Processes: Discontinuous Paths 309
6.1 Introduction ........................... 309
6.2 Poisson processes and Poisson random measures: Definition
and simulation.......................... 310
6.2.1 Stochastic integrals with respect to Poisson processes 312
6.3 Jump-diffusions: Basics, properties, and simulation ..... 315
6.3.1 Simulating Gauss-Poisson jump-diffusions....... 317
6.3.2 Euler-Maruyama scheme for jump-diffusions...... 319
6.4 Levy processes: Properties and examples ........... 320
6.4.1 Definition and properties of Levy processes...... 320
6.4.2 Examples of Levy processes............... 324
6.5 Simulation of Levy processes .................. 329
6.5.1 Exact simulation and time discretization........ 329
6.5.2 The Euler-Maruyama scheme for Levy processes . . . 330
6.5.3 Small jump approximation ............... 331
6.5.4 Simulation via series representation........... 333
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 Levy models and their simulation........... 340
7.3.1 The Esscher transform.................. 341
7.3.2 The hyperbolic Levy model............... 342
7.3.3 The variance gamma model............... 344
7.3.4 Normal inverse Gaussian processes........... 352
7.3.5 Further aspects of Levy type models.......... 354
Simulating Actuarial Models 357
8.1 Introduction ........................... 357
8.2 Premium principles and risk measures............. 357
8.2.1 Properties and examples of premium principles .... 358
8.2.2 Monte Carlo simulation of premium principles .... 362
8.2.3 Properties and examples of risk measures....... 362
8.2.4 Connection between premium principles and risk
measures ......................... 365
8.2.5 Monte Carlo simulation of risk measures........ 366
8.3 Some applications of Monte Carlo methods in life insurance . 377
8.3.1 Mortality: Definitions and classical models ...... 378
8.3.2 Dynamic mortality models................ 379
8.3.3 Life insurance contracts and premium calculation . . . 383
8.3.4 Pricing longevity products by Monte Carlo simulation 385
8.3.5 Premium reserves and Thiele s differential equation . . 387
8.4 Simulating dependent risks with copulas ........... 390
8.4.1 Definition and basic properties............. 390
8.4.2 Examples and simulation of copulas.......... 393
8.4.3 Application in actuarial models............. 402
8.5 Nonlife insurance ........................ 403
8.5.1 The individual model .................. 404
8.5.2 The collective model................... 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
in actuarial mathematics ................ 429
8.7 Asset-liability management and Solvency II.......... 433
8.7.1 Solvency II........................ 433
8.7.2 Asset-liability management (ALM)........... 435
References 441
Index 459
|
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| language | English |
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| series2 | Chapman & Hall/CRC financial mathematics series A Chapman & Hall book |
| spelling | Korn, Ralf 1963- Verfasser (DE-588)171321642 aut Monte Carlo methods and models in finance and insurance Ralf Korn ; Elke Korn ; Gerald Kroisandt Boca Raton [u.a.] CRC Press 2010 XIII, 470 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Chapman & Hall/CRC financial mathematics series A Chapman & Hall book Mathematik Business mathematics Insurance Mathematics Monte Carlo method Finanzmathematik (DE-588)4017195-4 gnd rswk-swf Versicherungsmathematik (DE-588)4063194-1 gnd rswk-swf Monte-Carlo-Simulation (DE-588)4240945-7 gnd rswk-swf Finanzmathematik (DE-588)4017195-4 s Versicherungsmathematik (DE-588)4063194-1 s Monte-Carlo-Simulation (DE-588)4240945-7 s b DE-604 Korn, Elke Verfasser aut Kroisandt, Gerald Verfasser aut HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=018975555&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Korn, Ralf 1963- Korn, Elke Kroisandt, Gerald Monte Carlo methods and models in finance and insurance Mathematik Business mathematics Insurance Mathematics Monte Carlo method Finanzmathematik (DE-588)4017195-4 gnd Versicherungsmathematik (DE-588)4063194-1 gnd Monte-Carlo-Simulation (DE-588)4240945-7 gnd |
| subject_GND | (DE-588)4017195-4 (DE-588)4063194-1 (DE-588)4240945-7 |
| title | Monte Carlo methods and models in finance and insurance |
| title_auth | Monte Carlo methods and models in finance and insurance |
| title_exact_search | Monte Carlo methods and models in finance and insurance |
| title_full | Monte Carlo methods and models in finance and insurance Ralf Korn ; Elke Korn ; Gerald Kroisandt |
| title_fullStr | Monte Carlo methods and models in finance and insurance Ralf Korn ; Elke Korn ; Gerald Kroisandt |
| title_full_unstemmed | Monte Carlo methods and models in finance and insurance Ralf Korn ; Elke Korn ; Gerald Kroisandt |
| title_short | Monte Carlo methods and models in finance and insurance |
| title_sort | monte carlo methods and models in finance and insurance |
| topic | Mathematik Business mathematics Insurance Mathematics Monte Carlo method Finanzmathematik (DE-588)4017195-4 gnd Versicherungsmathematik (DE-588)4063194-1 gnd Monte-Carlo-Simulation (DE-588)4240945-7 gnd |
| topic_facet | Mathematik Business mathematics Insurance Mathematics Monte Carlo method Finanzmathematik Versicherungsmathematik Monte-Carlo-Simulation |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=018975555&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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