The β-variance gamma model

Abstract

Kuznetsov (Ann Appl Prob, 2009) introduces a 10-parameter family of Levy processes for which the Wiener-Hopf factors and the distribution of the running supremum (infimum) can be determined semi-analytically. In this text we will examine the numerical performance of this so-called β-family, both in the equity world and in the field of credit risk. In order to do this, we will calibrate a particular member of this family to a vanilla option surface (by means of the Fast Fourier Transform-technique due to Carr and Madan (J Comput Fin 2(4):61–73, 1999) and use the resulting parameters to determine the prices of a digital down-and-out barrier (DDOB) option, written on the same underlying. In a second experiment, we will try and calibrate the model to some real-life credit default swap (CDS) term structures. The parameters of the model under investigation are chosen such that its Levy density is approximately equal to that of the famous Variance Gamma (VG) process, which will serve as a benchmark. Hence, the former will be referred to as the β-VG model. The option prices will be determined both semi-analytically [using the formulas derived by Kuznetsov (Ann Appl Prob, 2009)] and through a Monte-Carlo simulation. However, the CDS spreads will only be determined semi-analytically, due to the very close relation between pricing DDOB options and determining the par spread of a CDS. Furthermore, in both cases, the results will be compared with the ones obtained using the VG model [Cf. Schoutens (Levy processes in finance: pricing financial derivatives, Wiley , Chichester, 2003) and, Cariboni and Schoutens (Levy processes in credit risk, Wiley, Chichester, 2009)]. It will turn out that, w.r.t. vanilla option prices, the β-VG model performs almost identically as the VG model, whereas the semi-analytical expressions by Kuznetsov (Ann Appl Prob, 2009) lead to a (fast and) accurate pricing of DDOB options and CDSs.