Statistical Tools for Finance and Insurance

by Cizek, Pavel; HSrdle, Wolfgang; Weron, Rafal; Hardle, Wolfgang; Weron, Rafal

• ISBN13: 9783540221890
• ISBN10: 3540221891
• Format: Paperback
• Copyright: 2005-07-20
• Publisher: Springer Verlag

Summary

Statistical Tools in Finance and Insurance presents ready-to-use solutions, theoretical developments and method construction for many practical problems in quantitative finance and insurance. Written by practitioners and leading academics in the field, this book offers a unique combination of topics from which every market analyst and risk manager will benefit. Covering topics such as heavy tailed distributions, implied trinomial trees, support vector machines, valuation of mortgage-backed securities, pricing of CAT bonds, simulation of risk processes and ruin probability approximation, the book does not only offer practitioners insight into new methods for their applications, but it also gives theoreticians insight into the applicability of the stochastic technology. Additionally, the book provides the tools, instruments and (online) algorithms for recent techniques in quantitative finance and modern treatments in insurance calculations. Written in an accessible and engaging style, this self-instructional book makes a good use of extensive examples and full explanations. The design of the text links theory and computational tools in an innovative way. All Quantlets for the calculation of examples given in the text are supported by the academic edition of XploRe and may be executed via XploRe Quantlet Server (XQS). The downloadable electronic edition of the book enables one to run, modify, and enhance all Quantlets on the spot.

Table of Contents

Contributors	13	(2)
Preface	15	(4)
I Finance	19	(268)
1 Stable Distributions	21	(24)
Szymon Borak, Wolfgang Hardie, and Rafal Weron
1.1 Introduction	21	(1)
1.2 Definitions and Basic Characteristic	22	(6)
1.2.1 Characteristic Function Representation	24	(2)
1.2.2 Stable Density and Distribution Functions	26	(2)
1.3 Simulation of α-stable Variables	28	(2)
1.4 Estimation of Parameters	30	(6)
1.4.1 Tail Exponent Estimation	31	(2)
1.4.2 Quantile Estimation	33	(1)
1.4.3 Characteristic Function Approaches	34	(1)
1.4.4 Maximum Likelihood Method	35	(1)
1.5 Financial Applications of Stable Laws	36	(9)
2 Extreme Value Analysis and Copulas	45	(20)
Krzysztof Jajuga and Daniel Papla
2.1 Introduction	45	(8)
2.1.1 Analysis of Distribution of the Extremum	46	(1)
2.1.2 Analysis of Conditional Excess Distribution	47	(1)
2.1.3 Examples	48	(5)
2.2 Multivariate Time Series	53	(12)
2.2.1 Copula Approach	53	(3)
2.2.2 Examples	56	(1)
2.2.3 Multivariate Extreme Value Approach	57	(3)
2.2.4 Examples	60	(1)
2.2.5 Copula Analysis for Multivariate Time Series	61	(1)
2.2.6 Examples	62	(3)
3 Tail Dependence	65	(28)
Rafael Schmidt
3.1 Introduction	65	(1)
3.2 What is Tail Dependence?	66	(3)
3.3 Calculation of the Tail-dependence Coefficient	69	(6)
3.3.1 Archimedean Copulae	69	(1)
3.3.2 Elliptically-contoured Distributions	70	(4)
3.3.3 Other Copulae	74	(1)
3.4 Estimating the Tail-dependence Coefficient	75	(3)
3.5 Comparison of TDC Estimators	78	(3)
3.6 Tail Dependence of Asset and FX Returns	81	(3)
3.7 Value at Risk - a Simulation Study	84	(9)
4 Pricing of Catastrophe Bonds	93	(22)
Krzysztof Burnecki, Grzegorz Kukla, and David Taylor
4.1 Introduction	93	(6)
4.1.1 The Emergence of CAT Bonds	94	(2)
4.1.2 Insurance Securitization	96	(1)
4.1.3 CAT Bond Pricing Methodology	97	(2)
4.2 Compound Doubly Stochastic Poisson Pricing Model	99	(1)
4.3 Calibration of the Pricing Model	100	(4)
4.4 Dynamics of the CAT Bond Price	104	(11)
5 Common Functional IV Analysis	115	(20)
Michal Benko and Wolfgang Härdle
5.1 Introduction	115	(1)
5.2 Implied Volatility Surface	116	(2)
5.3 Functional Data Analysis	118	(3)
5.4 Functional Principal Components	121	(4)
5.4.1 Basis Expansion	123	(2)
5.5 Smoothed Principal Components Analysis	125	(2)
5.5.1 Basis Expansion	126	(1)
5.6 Common Principal Components Model	127	(8)
6 Implied Trinomial Trees	135	(26)
Pavel Cižek and Karel Komorád
6.1 Option Pricing	136	(2)
6.2 Trees and Implied Trees	138	(2)
6.3 Implied Trinomial Trees	140	(7)
6.3.1 Basic Insight	140	(2)
6.3.2 State Space	142	(2)
6.3.3 Transition Probabilities	144	(1)
6.3.4 Possible Pitfalls	145	(2)
6.4 Examples	147	(14)
6.4.1 Pre-specified Implied Volatility	147	(5)
6.4.2 German Stock Index	152	(9)
7 Heston's Model and the Smile	161	(22)
Rafal Weron and Uwe Wystup
7.1 Introduction	161	(2)
7.2 Heston's Model	163	(3)
7.3 Option Pricing	166	(3)
7.3.1 Greeks	168	(1)
7.4 Calibration	169	(14)
7.4.1 Qualitative Effects of Changing Parameters	171	(2)
7.4.2 Calibration Results	173	(10)
8 FFT-based Option Pricing	183	(18)
Szymon Borak, Kai Detlefsen, and Wolfgang Härdle
8.1 Introduction	183	(1)
8.2 Modern Pricing Models	183	(5)
8.2.1 Merton Model	184	(1)
8.2.2 Heston Model	185	(2)
8.2.3 Bates Model	187	(1)
8.3 Option Pricing with FFT	188	(4)
8.4 Applications	192	(9)
9 Valuation of Mortgage Backed Securities	201	(24)
Nicolas Gaussel and Julien Tamine
9.1 Introduction	201	(3)
9.2 Optimally Prepaid Mortgage	204	(8)
9.2.1 Financial Characteristics and Cash Flow Analysis	204	(1)
9.2.2 Optimal Behavior and Price	204	(8)
9.3 Valuation of Mortgage Backed Securities	212	(14)
9.3.1 Generic Framework	213	(2)
9.3.2 A Parametric Specification of the Prepayment Rate	215	(3)
9.3.3 Sensitivity Analysis	218	(7)
10 Predicting Bankruptcy with Support Vector Machines	225	(28)
Wolfgang Härdle, Rouslan Moro, and Dorothea Schäfer
10.1 Bankruptcy Analysis Methodology	226	(4)
10.2 Importance of Risk Classification in Practice	230	(3)
10.3 Lagrangian Formulation of the SVM	233	(3)
10.4 Description of Data	236	(1)
10.5 Computational Results	237	(6)
10.6 Conclusions	243	(6)
11 Modelling Indonesian Money Demand	249	(1)
Noer Azam Achsani, Oliver Holtemöller, and Hizir Sofyan
11.1 Specification of Money Demand Functions	250	(3)
11.2 The Econometric Approach to Money Demand	253	(7)
11.2.1 Econometric Estimation of Money Demand Functions	253	(1)
11.2.2 Econometric Modelling of Indonesian Money Demand	254	(6)
11.3 The Fuzzy Approach to Money Demand	260	(6)
11.3.1 Fuzzy Clustering	260	(1)
11.3.2 The Takagi-Sugeno Approach	261	(1)
11.3.3 Model Identification	262	(1)
11.3.4 Fuzzy Modelling of Indonesian Money Demand	263	(3)
11.4 Conclusions	266	(5)
12 Nonparametric Productivity Analysis	271	(1)
Wolfgang Härdle and Seok-Oh Jeong
12.1 The Basic Concepts	272	(4)
12.2 Nonparametric Hull Methods	276	(3)
12.2.1 Data Envelopment Analysis	277	(1)
12.2.2 Free Disposal Hull	278	(1)
12.3 DEA in Practice: Insurance Agencies	279	(2)
12.4 FDH in Practice: Manufacturing Industry	281	(6)
II Insurance	287	(1)
13 Loss Distributions	289	(200)
Krzysztof Burnecki, Adam Misiorek, and Rafal Weron
13.1 Introduction	289	(1)
13.2 Empirical Distribution Function	290	(2)
13.3 Analytical Methods	292	(11)
13.3.1 Log-normal Distribution	292	(1)
13.3.2 Exponential Distribution	293	(2)
13.3.3 Pareto Distribution	295	(3)
13.3.4 Burr Distribution	298	(1)
13.3.5 Weibull Distribution	298	(2)
13.3.6 Gamma Distribution	300	(2)
13.3.7 Mixture of Exponential Distributions	302	(1)
13.4 Statistical Validation Techniques	303	(8)
13.4.1 Mean Excess Function	303	(2)
13.4.2 Tests Based on the Empirical Distribution Function	305	(4)
13.4.3 Limited Expected Value Function	309	(2)
13.5 Applications	311	(8)
14 Modeling of the Risk Process	319	(1)
Krzysztof Burnecki and Rafal Weron
14.1 Introduction	319	(2)
14.2 Claim Arrival Processes	321	(8)
14.2.1 Homogeneous Poisson Process	321	(2)
14.2.2 Non-homogeneous Poisson Process	323	(3)
14.2.3 Mixed Poisson Process	326	(1)
14.2.4 Cox Process	327	(1)
14.2.5 Renewal Process	328	(1)
14.3 Simulation of Risk Processes	329	(12)
14.3.1 Catastrophic Losses	329	(5)
14.3.2 Danish Fire Losses	334	(7)
15 Ruin Probabilities in Finite and Infinite Time	341	(1)
Krzysztof Burnecki, Pawel Mista, and Aleksander Weron
15.1 Introduction	341	(5)
15.1.1 Light- and Heavy-tailed Distributions	343	(3)
15.2 Exact Ruin Probabilities in Infinite Time	346	(4)
15.2.1 No Initial Capital	347	(1)
15.2.2 Exponential Claim Amounts	347	(1)
15.2.3 Gamma Claim Amounts	347	(2)
15.2.4 Mixture of Two Exponentials Claim Amounts	349	(1)
15.3 Approximations of the Ruin Probability in Infinite Time	350	(13)
15.3.1 Cramer-Lundberg Approximation	351	(1)
15.3.2 Exponential Approximation	352	(1)
15.3.3 Lundberg Approximation	352	(1)
15.3.4 Beekman-Bowers Approximation	353	(1)
15.3.5 Renyi Approximation	354	(1)
15.3.6 De Vylder Approximation	355	(1)
15.3.7 4-moment Gamma De Vylder Approximation	356	(2)
15.3.8 Heavy Traffic Approximation	358	(1)
15.3.9 Light Traffic Approximation	359	(1)
15.3.10 Heavy-light Traffic Approximation	360	(1)
15.3.11 Subexponential Approximation	360	(1)
15.3.12 Computer Approximation via the Pollaczek-Khinchin Formula	361	(1)
15.3.13 Summary of the Approximations	362	(1)
15.4 Numerical Comparison of the Infinite Time Approximations	363	(4)
15.5 Exact Ruin Probabilities in Finite Time	367	(1)
15.5.1 Exponential Claim Amounts	368	(1)
15.6 Approximations of the Ruin Probability in Finite Time	368	(6)
15.6.1 Monte Carlo Method	369	(1)
15.6.2 Segerdahl Normal Approximation	369	(2)
15.6.3 Diffusion Approximation	371	(1)
15.6.4 Corrected Diffusion Approximation	372	(1)
15.6.5 Finite Time De Vylder Approximation	373	(1)
15.6.6 Summary of the Approximations	374	(1)
15.7 Numerical Comparison of the Finite Time Approximations	374	(7)
16 Stable Diffusion Approximation of the Risk Process	381	(1)
Hansjörg Furrer, Zbigniew Michna, and Aleksander Weron
16.1 Introduction	381	(1)
16.2 Brownian Motion and the Risk Model for Small Claims	382	(4)
16.2.1 Weak Convergence of Risk Processes to Brownian Motion	383	(1)
16.2.2 Ruin Probability for the Limit Process	383	(1)
16.2.3 Examples	384	(2)
16.3 Stable Lévy Motion and the Risk Model for Large Claims	386	(9)
16.3.1 Weak Convergence of Risk Processes to α-stable Lévy Motion	387	(1)
16.3.2 Ruin Probability in Limit Risk Model for Large Claims	388	(2)
16.3.3 Examples	390	(5)
17 Risk Model of Good and Bad Periods	395	(15)
Zbigniew Michna
17.1 Introduction	395	(1)
17.2 Fractional Brownian Motion and Model of Good and Bad Periods	396	(3)
17.3 Ruin Probability in Limit Risk Model of Good and Bad Periods	399	(3)
17.4 Examples	402	(5)
18 Premiums in the Individual and Collective Risk Models	407	(1)
Jan Iwanik and Joanna Nowicka-Zagrajek
18.1 Premium Calculation Principles	408	(2)
18.2 Individual Risk Model	410	(6)
18.2.1 General Premium Formulae	411	(1)
18.2.2 Premiums in the Case of the Normal Approximation	412	(1)
18.2.3 Examples	413	(3)
18.3 Collective Risk Model	416	(11)
18.3.1 General Premium Formulae	417	(1)
18.3.2 Premiums in the Case of the Normal and Translated Gamma Approximations	418	(2)
18.3.3 Compound Poisson Distribution	420	(1)
18.3.4 Compound Negative Binomial Distribution	421	(2)
18.3.5 Examples	423	(4)
19 Pure Risk Premiums under Deductibles	427	(1)
Krzysztof Burnecki, Joanna Nowicka-Zagrajek, and Agnieszka Wylomanska
19.1 Introduction	427	(1)
19.2 General Formulae for Premiums Under Deductibles	428	(8)
19.2.1 Franchise Deductible	429	(2)
19.2.2 Fixed Amount Deductible	431	(1)
19.2.3 Proportional Deductible	432	(1)
19.2.4 Limited Proportional Deductible	432	(2)
19.2.5 Disappearing Deductible	434	(2)
19.3 Premiums Under Deductibles for Given Loss Distributions	436	(14)
19.3.1 Log-normal Loss Distribution	437	(1)
19.3.2 Pareto Loss Distribution	438	(3)
19.3.3 Burr Loss Distribution	441	(4)
19.3.4 Weibull Loss Distribution	445	(2)
19.3.5 Gamma Loss Distribution	447	(2)
19.3.6 Mixture of Two Exponentials Loss Distribution	449	(1)
19.4 Final Remarks	450	(3)
20 Premiums, Investments, and Reinsurance	453	(1)
Pawel Mista and Wojciech Otto
20.1 Introduction	453	(3)
20.2 Single-Period Criterion and the Rate of Return on Capital	456	(3)
20.2.1 Risk Based Capital Concept	456	(1)
20.2.2 How To Choose Parameter Values?	457	(2)
20.3 The Top-down Approach to Individual Risks Pricing	459	(4)
20.3.1 Approximations of Quantiles	459	(1)
20.3.2 Marginal Cost Basis for Individual Risk Pricing	460	(1)
20.3.3 Balancing Problem	461	(1)
20.3.4 A Solution for the Balancing Problem	462	(1)
20.3.5 Applications	462	(1)
20.4 Rate of Return and Reinsurance Under the Short Term Criterion	463	(6)
20.4.1 General Considerations	464	(1)
20.4.2 Illustrative Example	465	(2)
20.4.3 Interpretation of Numerical Calculations in Example 2	467	(2)
20.5 Ruin Probability Criterion when the Initial Capital is Given	469	(8)
20.5.1 Approximation Based on Lundberg Inequality	469	(2)
20.5.2 "Zero" Approximation	471	(1)
20.5.3 Cramér-Lundberg Approximation	471	(1)
20.5.4 Beekman-Bowers Approximation	472	(1)
20.5.5 Diffusion Approximation	473	(1)
20.5.6 De Vylder Approximation	474	(1)
20.5.7 Subexponential Approximation	475	(1)
20.5.8 Panjer Approximation	475	(2)
20.6 Ruin Probability Criterion and the Rate of Return	477	(4)
20.6.1 Fixed Dividends	477	(2)
20.6.2 Flexible Dividends	479	(2)
20.7 Ruin Probability, Rate of Return and Reinsurance	481	(6)
20.7.1 Fixed Dividends	481	(1)
20.7.2 Interpretation of Solutions Obtained in Example 5	482	(2)
20.7.3 Flexible Dividends	484	(1)
20.7.4 Interpretation of Solutions Obtained in Example 6	485	(2)
20.8 Final remarks	487	(2)
III General	489	(1)
21 Working with the XQC	491	(16)
Szymon Borak, Wolfgang Härdle, and Heiko Lehmann
21.1 Introduction	491	(1)
21.2 The XploRe Quantlet Client	492	(2)
21.2.1 Configuration	492	(1)
21.2.2 Getting Connected	493	(1)
21.3 Desktop	494	(13)
21.3.1 XploRe Quantlet Editor	495	(1)
21.3.2 Data Editor	496	(5)
21.3.3 Method Thee	501	(2)
21.3.4 Graphical Output	503	(4)
Index	507