The Generalized Hyperbolic Model: Financial Derivatives and Risk Measures

Statistical analysis of data from the nancial markets shows that generalized hyper-bolic (GH) distributions allow a more realistic description of asset returns than the classical normal distribution. GH distributions contain as sub-classes hyperbolic as well as normal inverse Gaussian (NIG) distributions which have recently been proposed as basic ingredients for the modelling of price processes. We derive an option pricing formula for GH driven models using the Esscher transform as one possibility to determine prices in an incomplete market. The GH option pricing model is a generalization of the hyper-bolic model developed by Eberlein and Keller (1995), Keller (1997). We compare this model with the classical Black-Scholes (1973) model. We also propose a general recipe for the comparison of models for the pricing of derivatives with the market reality. The objectives of this study are to examine the consistency of our model assumptions with the empirically observed price processes and the performance of option pricing models from a practitioner's point of view. Due to the diiculty, if not impossibility, of constructing a test in a strict statistical sense for pricing models, we compare the models focussing on criteria which are relevant for practical purposes. Finally, we present a simpliied approach to the estimation of high-dimensional generalized hyperbolic distributions. We also consider models with stochastic volatility and investigate the application of GH distributions to measure risk in nancial markets.