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Statistical Tools for Finance and Insurance

Contents

I Finance 15 1 Stable distributions 17 Szymon Borak, Wolfgang H¨ardle, and Rafa
l Weron 1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 1.2 Definitions and basic characteristics . . . . . . . . . . . . . . . 18 1.2.1 Characteristic function representation . . . . . . . . . . 20 1.2.2 Stable density and distribution functions . . . . . . . . . 22 1.3 Simulation of α-stable variables . . . . . . . . . . . . . . . . . . 24 1.4 Estimation of parameters . . . . . . . . . . . . . . . . . . . . . 26 1.4.1 Tail exponent estimation . . . . . . . . . . . . . . . . . 27 1.4.2 Quantile estimation . . . . . . . . . . . . . . . . . . . . 29 1.4.3 Characteristic function approaches . . . . . . . . . . . . 30 1.4.4 Maximum likelihood method . . . . . . . . . . . . . . . 31 1.5 Financial applications of stable laws . . . . . . . . . . . . . . . 32 2 Extreme Value analysis and copulas 43 Krzysztof Jajuga and Daniel Papla 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 4 Contents 2.1.1 Analysis of distribution of the extremum . . . . . . . . . 44 2.1.2 Analysis of conditional excess distribution . . . . . . . . 45 2.1.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . 46 2.2 Multivariate time series . . . . . . . . . . . . . . . . . . . . . . 51 2.2.1 Copula approach . . . . . . . . . . . . . . . . . . . . . . 51 2.2.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . 54 2.2.3 Multivariate extreme value approach . . . . . . . . . . . 55 2.2.4 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . 58 2.2.5 Copula analysis for multivariate time series . . . . . . . 59 2.2.6 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . 60 3 Tail dependence 65 Rafael Schmidt 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 3.2 What is tail dependence? . . . . . . . . . . . . . . . . . . . . . 66 3.3 Calculation of the tail-dependence coefficient . . . . . . . . . . 69 3.3.1 Archimedean copulae . . . . . . . . . . . . . . . . . . . 69 3.3.2 Elliptically-contoured distributions . . . . . . . . . . . . 70 3.3.3 Other copulae . . . . . . . . . . . . . . . . . . . . . . . . 74 3.4 Estimating the tail-dependence coefficient . . . . . . . . . . . . 75 3.5 Comparison of TDC estimators . . . . . . . . . . . . . . . . . . 78 3.6 Tail dependence of asset and FX returns . . . . . . . . . . . . . 82 3.7 Value at Risk – a simulation study . . . . . . . . . . . . . . . . 84 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 4 Pricing of catastrophe bonds 95 Krzysztof Burnecki, Grzegorz Kukla, and David Taylor Contents 5 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 4.1.1 The emergence of CAT bonds . . . . . . . . . . . . . . . 96 4.1.2 Insurance securitization . . . . . . . . . . . . . . . . . . 98 4.1.3 CAT bond pricingmethodology . . . . . . . . . . . . . . 99 4.2 Compound doubly stochastic Poisson pricing model . . . . . . 101 4.3 Calibration of the pricing model . . . . . . . . . . . . . . . . . 102 4.4 Dynamics of the CAT bond price . . . . . . . . . . . . . . . . . 106 5 Common functional IV analysis 117 Michal Benko and Wolfgang H¨ardle 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 5.2 Implied volatility surface . . . . . . . . . . . . . . . . . . . . . . 118 5.3 Functional data analysis . . . . . . . . . . . . . . . . . . . . . . 120 5.4 Functional principal components . . . . . . . . . . . . . . . . . 124 5.4.1 Basis expansion . . . . . . . . . . . . . . . . . . . . . . . 126 5.5 Smoothed principal components analysis . . . . . . . . . . . . . 126 5.5.1 Basis expansion . . . . . . . . . . . . . . . . . . . . . . . 128 5.6 Common principal components model . . . . . . . . . . . . . . 130 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137 6 Implied Trinomial Trees 139 Pavel ˇ C´ıˇzek, Karel Komor´ad 6.1 Option pricing . . . . . . . . . . . . . . . . . . . . . . . . . . . 140 6.2 Trees and implied trees . . . . . . . . . . . . . . . . . . . . . . 142 6.3 Implied trinomial trees . . . . . . . . . . . . . . . . . . . . . . . 144 6.3.1 Basic insight . . . . . . . . . . . . . . . . . . . . . . . . 144 6.3.2 State space . . . . . . . . . . . . . . . . . . . . . . . . . 146 6 Contents 6.3.3 Transition probabilities . . . . . . . . . . . . . . . . . . 148 6.3.4 Possible pitfalls . . . . . . . . . . . . . . . . . . . . . . . 150 6.4 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151 6.4.1 Pre-specified implied volatility . . . . . . . . . . . . . . 151 6.4.2 German stock index . . . . . . . . . . . . . . . . . . . . 156 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161 7 Heston’s model and the smile 163 Rafa
l Weron and Uwe Wystup 7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163 7.2 Heston’smodel . . . . . . . . . . . . . . . . . . . . . . . . . . . 165 7.3 Option pricing . . . . . . . . . . . . . . . . . . . . . . . . . . . 168 7.3.1 Greeks . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170 7.4 Calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171 7.4.1 Qualitative effects of changing parameters . . . . . . . . 173 7.4.2 Calibration results . . . . . . . . . . . . . . . . . . . . . 175 8 FFT based option pricing 185 Szymon Borak, Kai Detlefsen and Wolfgang H¨ardle 8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185 8.2 Modern pricingmodels . . . . . . . . . . . . . . . . . . . . . . . 186 8.2.1 MertonModel . . . . . . . . . . . . . . . . . . . . . . . 186 8.2.2 HestonModel . . . . . . . . . . . . . . . . . . . . . . . . 187 8.2.3 BatesModel . . . . . . . . . . . . . . . . . . . . . . . . 189 8.3 Option Pricing with FFT . . . . . . . . . . . . . . . . . . . . . 190 8.4 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193 9 Valuation of Mortgage Backed Securities: from Optimality to Reality203 Contents 7 Nicolas Gaussel and Julien Tamine 9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203 9.2 Optimally prepaid mortgage . . . . . . . . . . . . . . . . . . . . 206 9.2.1 Financial characteristics and cash flow analysis . . . . . 206 9.2.2 Optimal behavior and price . . . . . . . . . . . . . . . . 207 9.3 Valuation of mortgage backed securities . . . . . . . . . . . . . 214 9.3.1 Generic framework . . . . . . . . . . . . . . . . . . . . . 215 9.3.2 A parametric specification of the prepayment rate . . . 217 9.3.3 Sensitivity analysis . . . . . . . . . . . . . . . . . . . . . 220 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227 10 Predicting Bankruptcy with Support Vector Machines 229 Wolfgang H¨ardle, Rouslan Moro, and Dorothea Sch¨afer 10.1 Bankruptcy Analysis Methodology . . . . . . . . . . . . . . . . 230 10.2 Importance of Risk Classification in Practice . . . . . . . . . . 234 10.3 Lagrangian Formulation of the SVM . . . . . . . . . . . . . . . 237 10.4 Description of Data . . . . . . . . . . . . . . . . . . . . . . . . . 240 10.5 Computational Results . . . . . . . . . . . . . . . . . . . . . . . 241 10.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 246 11 Modelling Indonesian Money Demand 253 Noer Azam Achsani, Oliver Holtem¨oller, and Hizir Sofyan 11.1 Specification of money demand functions . . . . . . . . . . . . . 254 11.2 The econometric approach to money demand . . . . . . . . . . 256 11.2.1 Econometric estimation of money demand functions . . 256 11.2.2 Modelling Indonesian money demand with econometric techniques . . . . . . . . . . . . . . . . . . . . . . . . . . 258 11.3 The fuzzy approach to money demand . . . . . . . . . . . . . . 264 8 Contents 11.3.1 Fuzzy clustering . . . . . . . . . . . . . . . . . . . . . . 264 11.3.2 The Takagi-Sugeno approach . . . . . . . . . . . . . . . 265 11.3.3 Model identification . . . . . . . . . . . . . . . . . . . . 266 11.3.4 Modelling Indonesian money demand with fuzzy techniques267 11.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 269 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 271 12 Nonparametric productivity analysis 275 Wolfgang H¨ardle and Seok-Oh Jeong 12.1 The basic concepts . . . . . . . . . . . . . . . . . . . . . . . . . 276 12.2 Nonparametric hull methods . . . . . . . . . . . . . . . . . . . 280 12.2.1 Data envelopment analysis . . . . . . . . . . . . . . . . 281 12.2.2 Free disposal hull . . . . . . . . . . . . . . . . . . . . . . 282 12.3 DEA in practice: insurance agencies . . . . . . . . . . . . . . . 284 12.4 FDH in practice: manufacturing industry . . . . . . . . . . . . 285 II Insurance 291 13 Loss Distributions 293 Krzysztof Burnecki, Adam Misiorek, and Rafa
l Weron 13.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293 13.2 Empirical distribution function . . . . . . . . . . . . . . . . . . 294 13.3 Analyticalmethods . . . . . . . . . . . . . . . . . . . . . . . . . 296 13.3.1 Log-normal distribution . . . . . . . . . . . . . . . . . . 296 13.3.2 Exponential distribution . . . . . . . . . . . . . . . . . . 297 13.3.3 Pareto distribution . . . . . . . . . . . . . . . . . . . . . 299 13.3.4 Burr distribution . . . . . . . . . . . . . . . . . . . . . . 302 Contents 9 13.3.5 Weibull distribution . . . . . . . . . . . . . . . . . . . . 302 13.3.6 Gamma distribution . . . . . . . . . . . . . . . . . . . . 304 13.3.7 Mixture of exponential distributions . . . . . . . . . . . 306 13.4 Statistical validation techniques . . . . . . . . . . . . . . . . . . 307 13.4.1 Mean excess function . . . . . . . . . . . . . . . . . . . . 307 13.4.2 Tests based on the empirical distribution function . . . 309 13.4.3 Limited expected value function . . . . . . . . . . . . . 313 13.5 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 315 14 Modeling of the risk process 323 Krzysztof Burnecki and Rafa
l Weron 14.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 323 14.2 Claim arrival processes . . . . . . . . . . . . . . . . . . . . . . . 325 14.2.1 Homogeneous Poisson process . . . . . . . . . . . . . . . 325 14.2.2 Non-homogeneous Poisson process . . . . . . . . . . . . 327 14.2.3 Mixed Poisson process . . . . . . . . . . . . . . . . . . . 330 14.2.4 Cox process . . . . . . . . . . . . . . . . . . . . . . . . . 331 14.2.5 Renewal process . . . . . . . . . . . . . . . . . . . . . . 332 14.3 Simulation of risk processes . . . . . . . . . . . . . . . . . . . . 333 14.3.1 Catastrophic losses . . . . . . . . . . . . . . . . . . . . . 333 14.3.2 Danish fire losses . . . . . . . . . . . . . . . . . . . . . . 338 15 Ruin probabilities in finite and infinite time 345 Krzysztof Burnecki, Pawe
l Mi´sta, and Aleksander Weron 15.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 345 15.1.1 Light- and heavy-tailed distributions . . . . . . . . . . . 347 15.2 Exact ruin probabilities in infinite time . . . . . . . . . . . . . 350 10 Contents 15.2.1 No initial capital . . . . . . . . . . . . . . . . . . . . . . 351 15.2.2 Exponential claim amounts . . . . . . . . . . . . . . . . 351 15.2.3 Gamma claimamounts . . . . . . . . . . . . . . . . . . 351 15.2.4 Mixture of two exponentials claim amounts . . . . . . . 353 15.3 Approximations of the ruin probability in infinite time . . . . . 354 15.3.1 Cram´er–Lundberg approximation . . . . . . . . . . . . . 355 15.3.2 Exponential approximation . . . . . . . . . . . . . . . . 356 15.3.3 Lundberg approximation . . . . . . . . . . . . . . . . . . 356 15.3.4 Beekman–Bowers approximation . . . . . . . . . . . . . 357 15.3.5 Renyi approximation . . . . . . . . . . . . . . . . . . . . 358 15.3.6 De Vylder approximation . . . . . . . . . . . . . . . . . 359 15.3.7 4-moment gamma De Vylder approximation . . . . . . . 360 15.3.8 Heavy traffic approximation . . . . . . . . . . . . . . . . 362 15.3.9 Light traffic approximation . . . . . . . . . . . . . . . . 363 15.3.10Heavy-light traffic approximation . . . . . . . . . . . . . 364 15.3.11Subexponential approximation . . . . . . . . . . . . . . 364 15.3.12Computer approximation via the Pollaczek–Khinchin formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 365 15.3.13Summary of the approximations . . . . . . . . . . . . . 366 15.4 Numerical comparison of the infinite time approximations . . . 367 15.5 Exact ruin probabilities in finite time . . . . . . . . . . . . . . . 371 15.5.1 Exponential claim amounts . . . . . . . . . . . . . . . . 372 15.6 Approximations of the ruin probability in finite time . . . . . . 372 15.6.1 Monte Carlomethod . . . . . . . . . . . . . . . . . . . . 373 15.6.2 Segerdahl normal approximation . . . . . . . . . . . . . 373 15.6.3 Diffusion approximation . . . . . . . . . . . . . . . . . . 375 15.6.4 Corrected diffusion approximation . . . . . . . . . . . . 375 Contents 11 15.6.5 Finite time De Vylder approximation . . . . . . . . . . 377 15.6.6 Summary of the approximations . . . . . . . . . . . . . 378 15.7 Numerical comparison of the finite time approximations . . . . 378 16 Stable diffusion approximation of the risk process 387 Hansj¨org Furrer, Zbigniew Michna, and Aleksander Weron 16.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 387 16.2 Brownian motion and the risk model for small claims . . . . . . 388 16.2.1 Weak convergence of risk processes to Brownian motion 389 16.2.2 Ruin probability for the limit process . . . . . . . . . . 390 16.2.3 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . 391 16.3 Stable L´evy motion and the risk model for large claims . . . . . 392 16.3.1 Weak convergence of risk processes to α-stable L´evy motion393 16.3.2 Ruin probability in the limit risk model for large claims 394 16.3.3 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . 397 17 Risk model of good and bad periods 403 Zbigniew Michna 17.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 403 17.2 Fractional Brownian motion and the risk model of good and bad periods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 404 17.3 Ruin probability in the limit risk model of good and bad periods 407 17.4 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 410 18 Premiums in the individual and collective risk models 417 Jan Iwanik and Joanna Nowicka-Zagrajek 18.1 Premium calculation principles . . . . . . . . . . . . . . . . . . 418 18.2 Individual riskmodel . . . . . . . . . . . . . . . . . . . . . . . . 420 12 Contents 18.2.1 General premium formulae . . . . . . . . . . . . . . . . 421 18.2.2 Premiums in the case of the normal approximation . . . 422 18.2.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . 423 18.3 Collective riskmodel . . . . . . . . . . . . . . . . . . . . . . . . 426 18.3.1 General premium formulae . . . . . . . . . . . . . . . . 427 18.3.2 Premiums in the case of the normal and translated gamma approximations . . . . . . . . . . . . . . . . . . . . . . . 428 18.3.3 Compound Poisson distribution . . . . . . . . . . . . . . 430 18.3.4 Compound negative binomial distribution . . . . . . . . 431 18.3.5 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . 433 19 Pure risk premiums under deductibles 439 Krzysztof Burnecki, Joanna Nowicka-Zagrajek, and Agnieszka Wy
loma´nska 19.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 439 19.2 General formulae for premiums under deductibles . . . . . . . . 440 19.2.1 Franchise deductible . . . . . . . . . . . . . . . . . . . . 441 19.2.2 Fixed amount deductible . . . . . . . . . . . . . . . . . 443 19.2.3 Proportional deductible . . . . . . . . . . . . . . . . . . 444 19.2.4 Limited proportional deductible . . . . . . . . . . . . . . 444 19.2.5 Disappearing deductible . . . . . . . . . . . . . . . . . . 446 19.3 Premiums under deductibles for given loss distributions . . . . 448 19.3.1 Log-normal loss distribution . . . . . . . . . . . . . . . . 449 19.3.2 Pareto loss distribution . . . . . . . . . . . . . . . . . . 451 19.3.3 Burr loss distribution . . . . . . . . . . . . . . . . . . . 454 19.3.4 Weibull loss distribution . . . . . . . . . . . . . . . . . . 456 19.3.5 Gamma loss distribution . . . . . . . . . . . . . . . . . . 460 19.3.6 Mixture of two exponentials loss distribution . . . . . . 461 Contents 13 19.4 Final remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . 462 20 Premiums, investments and reinsurance 467 Pawe
l Mi´sta and Wojciech Otto 20.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 467 20.2 Single-period criterion and the rate of return on capital (RBC concept) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 470 20.2.1 Risk Based Capital concept . . . . . . . . . . . . . . . . 470 20.2.2 How to choose parameter values . . . . . . . . . . . . . 471 20.3 The top-down approach to individual risks pricing . . . . . . . 473 20.3.1 Approximations of quantiles . . . . . . . . . . . . . . . . 473 20.3.2 Marginal cost basis for individual risk pricing . . . . . . 474 20.3.3 Balancing problem . . . . . . . . . . . . . . . . . . . . . 475 20.3.4 A solution for the balancing problem . . . . . . . . . . . 476 20.3.5 Applications . . . . . . . . . . . . . . . . . . . . . . . . 476 20.4 Rate of return and reinsurance under the short term criterion . 477 20.4.1 General considerations . . . . . . . . . . . . . . . . . . . 478 20.4.2 Illustration by the example . . . . . . . . . . . . . . . . 479 20.4.3 Interpretation of numerical calculations in Example 2 . 481 20.5 Ruin probability criterion when the initial surplus is given . . . 483 20.5.1 Approximation based on Lundberg inequalty . . . . . . 483 20.5.2 “Zero” approximation . . . . . . . . . . . . . . . . . . . 485 20.5.3 Cramer-Lundberg approximation . . . . . . . . . . . . . 485 20.5.4 The Beekman-Bowers approximation . . . . . . . . . . . 486 20.5.5 The diffusion approximation . . . . . . . . . . . . . . . . 487 20.5.6 DeVylder approximation . . . . . . . . . . . . . . . . . . 488 20.5.7 Subexponential approximation . . . . . . . . . . . . . . 489 14 Contents 20.5.8 Panjer approximation . . . . . . . . . . . . . . . . . . . 489 20.6 Ruin probability criterion and the rate of return . . . . . . . . 491 20.6.1 Fixed dividends . . . . . . . . . . . . . . . . . . . . . . . 491 20.6.2 Flexible dividends . . . . . . . . . . . . . . . . . . . . . 493 20.7 Ruin probability, rate of return and reinsurance . . . . . . . . . 495 20.7.1 Fixed dividends – an example . . . . . . . . . . . . . . . 495 20.7.2 Interpretation of solutions obtained in Example 5 . . . . 496 20.7.3 Flexible dividends . . . . . . . . . . . . . . . . . . . . . 498 20.7.4 Interpretation of solutions obtained in Example 6 . . . . 499 20.8 Final remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . 501 III General 505 21 Working with the XQC 507 Wolfgang H¨ardle, Heiko Lehmann, and Szymon Borak 21.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 507 21.2 The XploRe Quantlet Client . . . . . . . . . . . . . . . . . . . . 508 21.2.1 Configuration . . . . . . . . . . . . . . . . . . . . . . . . 508 21.2.2 Getting connected . . . . . . . . . . . . . . . . . . . . . 509 21.3 Desktop . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 510 21.3.1 XploRe Quantlet Editor . . . . . . . . . . . . . . . . . . 511 21.3.2 Data editor . . . . . . . . . . . . . . . . . . . . . . . . . 512 21.3.3 Method tree . . . . . . . . . . . . . . . . . . . . . . . . . 517 21.3.4 Graphical output . . . . . . . . . . . . . . . . . . . . . . 519

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