The Generalized Hyperbolic Model: Estimation, Financial Derivatives, and Risk Measures

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 Lévy 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 Lévy 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 Lévy model by replacing the mixing GIG distribution by a volatility process of the Ornstein-Uhlenbeck type. We investigate 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 distributions 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 …