Fitting Generalized Hyperbolic processes - New insights for generating initial values

Abstract

The fi tting of L évy processes is an important fi eld of interest in both option pricing and risk management. In literature a large number of fi tting methods requiring adequate initial values at the start of the optimization procedure exists. A so-called simplifi ed method of moments (SMoM) generates by assuming a symmetric distribution these initial values for the Variance Gamma process, whereby the idea behind can be easily transferred to the Normal Inverse Gaussian process. However, the characteristics of the Generalized Hyperbolic process prevent such a easy adaption. Therefore, we provide by applying a Taylor series approximation for the modifi ed Bessel function of the third kind, a Tschirnhaus transformation and a symmetric distribution assumption a SMoM for the Generalized Hyperbolic distribution. Our simulation study compares the results of our SMoM with the results of the maximum likelihood estimation. The results show that our proposed approach is an appropriate and useful way for estimating Generalized Hyperbolic process parameters and signifi cantly reduces estimation time.

Financial modelling applying multivariate Lévy processes: New insights into estimation and simulation

“…Furthermore, we observe an acceptable performance of the SMoM. Limitations and insights for this method are explained in [54] and also valid for the multivariate case.…”

“…The αρNIG model provides Barndorff-Nielsen [11], Rydberg [14] Madan and Seneta [15], Madan and Seneta [16], Finlay and Seneta [38] SMoM The calculation of the SMoM for the GH process is more difficult as for the other two processes due to the modified Bessel function within the moments. Therefore, we apply and refer to the approach of Rathgeber et al [54] δ =…”

The Semi-Hyperbolic Distribution and Its Applications

“…Baciu [35] showed that the GH model is the best fit for Bucharest Stock Exchange returns. Rathgeber et al [36] extended the simplified method of moments for the problem of the estimation of the parameters of the GH process. Balter and McNeil [37] calibrated symmetric GH distribution based on S&P500 index statistics.…”