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The Generalized Hyperbolic Model: Estimation, Financial Derivatives, and Risk Measures The Generalized Hyperbolic Model: Estimation, Financial Derivatives, and Risk Measures Dissertation zur Erlangung des Doktorgrades der Mathematischen Fakult¨at der Albert-Ludwigs-Universit¨at Freiburg i. Br. vorgelegt von Karsten Prause Oktober 1999 Dekan: Prof. Dr. Wolfgang Soergel Referenten: Prof. Dr. Ernst Eberlein Prof. Ph.D. Bent Jesper Christensen Datum der Promotion: 15. Dezember 1999 Institut f¨ur Mathematische Stochastik Albert-Ludwigs-Universit¨at Freiburg Eckerstraße 1 D–79104 Freiburg im Breisgau To Ulrike Preface The aim of this dissertation is to describe more realistic models for financial assets based on generalized hyperbolic (GH) distributions and their subclasses. Generalized hyperbolic distributions were introduced by Barndorff-Nielsen (1977), and stochastic processes based on these distributions were first applied by Eberlein and Keller (1995) in Finance. Being a normal variance-mean mixture, GH distributions possess semi- heavy tails and allow for a natural definiton of volatility models by replacing the mixing generalized inverse Gaussian (GIG) distribution by appropriate volatility processes. In the first Chapter we introduce univariate GH distributions, construct an estimation algorithm and examine statistically the fit of generalized hyperbolic distributions to log- return distributions of financial assets. We extend the hyperbolic model for the pricing of derivatives to generalized hyperbolic L´evy motions and discuss the calculation of prices by fast Fourier methods and saddle-point approximations. Chapter 2 contains on the one hand a general recipe for the evaluation of option pricing models; on the other hand the derivative pricing based on GH L´evy motions is studied from various points of view: The accordance with observed asset price processes is investigated statistically, and by simulation studies the sensitivity to relevant variables; finally, theoretical prices are compared with quoted option prices. Furthermore, we examine two approaches to martingale modelling and discuss alternative ways to test option pricing models. Barndorff-Nielsen (1998) proposed a refinement of the GH L´evy model by replacing the mixing GIG distribution by a volatility process of the Ornstein-Uhlenbeck type. We investi- gate this model in Chapter 3 with a view towards derivative pricing. After a review of this model we derive the minimal martingale measure to compute option prices and investigate the behaviour of this model thoroughly from a numerical and econometric point of view. We start in Chapter 4 with a description of some “stylized features” observed in multi- variate return distributions of financial assets. Then we introduce multivariate GH distribu- tions and discuss the efficiency of estimation procedures. Finally, the multivariate Esscher approach to option pricing and, in particular, basket options are examined. In the final chapter we define more realistic risk measures based on generalized hyperbolic distributions and evaluate them following the procedure required by the Basel Committee on Banking Supervision. The proposed risk-measurement methods apply to linear and nonlinear portfolios; moreover they are computationally not demanding. Forecasting values of financial assets is not a major objective of this dissertation. However, the second and the final chapter presents some results on volatility forecasts for option prices and on tests of risk measures based on distributional forecasts. The usual procedure will always be to start with the definition of the new model and the calibration of the model. Then the properties of the model are examined theoretically and by simulation studies. Finally, we will put the model into practice and compare the performance V with benchmark models. This procedure yields a complete picture and highlightens features relevant for practical applications. The final two chapters are self-contained. Therefore, a reader only interested in multivariate modelling could skip the preceeding three chapters. I take this opportunity to thank my advisor Prof. Dr. Ernst Eberlein for his encourage- ment, confidence and always reliable support. I enjoyed the time at the Institut f¨ur Mathe- matische Stochastik, the Department of Theoretical Statistics in Arhus,˚ and the Freiburg Center for Data Analysis and Modelling (FDM). I thank their members (and guests) for the stimulating discussions; namely, I am indebted to Prof. Dr. Ole E. Barndorff-Nielsen, Jos´e Luis Bat´un, Annette Ehret, Dr. Ulrich Halekoh, Dr. Jan Kallsen, Dr. Ulrich Keller, Oliver L¨ubke, Elisa Nicolato, Fehmi Ozkan¨ and Sebastian Raible. Moreover, I gratefully acknowl- edge financial support by the Graduate College (DFG) ,,Nichtlineare Differentialgleichungen: Modellierung, Theorie, Numerik, Visualisierung.” Last but not least I thank my parents and Ulrike Vorh¨olter. VI Contents 1 Generalized Hyperbolic L´evy Motions 1 1.1GeneralizedHyperbolicDistributions....................... 2 1.2Maximum-LikelihoodEstimation......................... 7 1.3DerivativesoftheLog-LikelihoodFunction................... 10 1.4ComparisonoftheEstimatesandTests..................... 12 1.5High-FrequencyData................................ 14 1.6Finite-SampleProperties.............................. 15 1.7Minimum-DistanceEstimation.......................... 16 1.8MeasuresofRisk.................................. 18 1.9 Subordination, L´evy-KhintchineRepresentation................. 19 1.10 The Generalized Hyperbolic L´evyMotion.................... 24 1.11 The Exponential L´evyModel........................... 26 1.12DerivativePricingbyEsscherTransforms.................... 26 1.13ComputationofEsscherPricesbyFFT..................... 29 1.14 Saddle-Point Approximation to Esscher Prices . .............. 29 1.15Conclusion..................................... 32 1.16Tables........................................ 33 2 Application and Testing of Derivative Pricing Models 37 2.1Preliminaries.................................... 38 2.2ModelversusUnderlyingProcess......................... 40 2.3OptionsSensitivities................................ 42 2.4 Implicit Volatilities . ............................... 47 2.5PricingPerformance................................ 53 2.6StatisticalMartingaleMeasures.......................... 60 2.7AlternativeTestingApproaches.......................... 66 2.8Conclusion..................................... 67 3 Stochastic Volatility Models of the Ornstein-Uhlenbeck type 69 3.1EmpiricalMotivation................................ 69 3.2Ornstein-UhlenbecktypeProcesses........................ 71 3.3TheModel..................................... 75 3.4TheMinimalMartingaleMeasure......................... 75 3.5Risk-MinimizingHedgingStrategies....................... 77 3.6ConditionalLog-Normality............................ 81 3.7ComputationofPricesbyRosinskiExpansion.................. 82 VII 3.8 Saddle-Point Approximation to the Option Price . .............. 83 3.9CalibrationoftheIGOUModel.......................... 85 3.10SuperpositionofIGOUProcesses......................... 86 3.11NumericalResults................................. 88 3.12PricingPerformance................................ 94 3.13StatisticalMartingaleMeasures.......................... 96 3.14 Volatility Models driven by L´evyProcesses................... 97 4 Multivariate Models in Finance 99 4.1CorrelationStructure................................ 99 4.2SkewnessandKurtosis............................... 102 4.3MultivariateGeneralizedHyperbolicLaws.................... 103 4.4EstimationofSymmetricGHDistributions................... 107 4.5EstimationofSkewedGHDistributions..................... 112 4.6GeneratingRandomVariates........................... 113 4.7PricingofDerivatives............................... 115 4.8BasketOptions................................... 118 5 Market Risk Controlling 119 5.1ConceptsofRiskMeasurement.......................... 119 5.2StatisticalRealization............................... 123 5.3GH-IGARCH.................................... 127 5.4BacktestingExperiments.............................. 128 5.5StatisticalTests................................... 131 5.6High-DimensionalData.............................. 134 5.7NonlinearPortfolios,GHSimulationMethod.................. 140 5.8Conclusion..................................... 142 A Data Sets and Computational Aspects 143 A.1DataSets...................................... 143 A.2ComputationalAspects.............................. 144 B Modified Bessel Functions 147 C Fourth Parametrization of GH Distributions 150 References 152 VIII Chapter 1 Generalized Hyperbolic L´evy Motions The modelling of financial assets as stochastic processes is determined by distributional as- sumptions on the increments and the dependence structure. It is well known that the returns of most financial assets have semi-heavy tails, i.e. the actual kurtosis is higher than the kurtosis of the normal distribution (Mandelbrot 1963). Generalized hyperbolic distributions (GH) possess these semi-heavy tails. On the contrary, the use of models with nonexisting moments, e.g. stable distributions, does not admit the pricing of derivative contracts by mar- tingale methods where we aim at. Note, that in contrast to stable distributions the density of generalized hyperbolic distributions is known explicitly. Ole E. Barndorff-Nielsen (1977) introduced the generalized hyperbolic distributions and at first applied them to model grain size distributions of wind blown sands. An important aspect is, that GH distributions embrace many special cases, respectively
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