A Multivariate Variance Gamma Model for Financial Applications

Abstract

In this paper we subordinate a multivariate Brownian motion with independent components by a multivariate gamma subordinator. The resulting process is a generalization of the bivariate variance gamma process proposed by Madan and Seneta [7], mentioned in Cont and Tankov [4] and calibrated in Luciano and Schoutens [5] as a price process. Our main contribution here is to introduce a multivariate subordinator with gamma margins. We investigate the process, determine its Lévy triplet and analyze its dependence structure. At the end we propose an exponential Lévy price model.

Multivariate asset models using Lévy processes and applications

“…The previous results highlight an advantage of our model compared to the multivariate subordinator approach of Semeraro (2008) and Luciano and Semeraro (2010), and the factor copula approach of Baxter (2007) and Moosbrucker (2006a,b). All these constructions, in fact, can only accommodate strictly positive correlation values due to restrictions on the parameter controlling the correlation coefficient, which are required to ensure the existence of the characteristic function of the processes involved.…”

“…This is also documented, for example, by Wallmeir and Diethelm (2012) whose empirical analysis shows the limited potential to match observed correlations of the multivariate Variance Gamma models of Leoni and Schoutens (2006) and Semeraro (2008). We note that for the case of subordinated Brownian motions, Semeraro (2008) and Luciano and Semeraro (2010) improve the richness of the correlation structure through an alternative construction which uses correlated Brownian motions.…”

“…This is also documented, for example, by Wallmeir and Diethelm (2012) whose empirical analysis shows the limited potential to match observed correlations of the multivariate Variance Gamma models of Leoni and Schoutens (2006) and Semeraro (2008). We note that for the case of subordinated Brownian motions, Semeraro (2008) and Luciano and Semeraro (2010) improve the richness of the correlation structure through an alternative construction which uses correlated Brownian motions. However, as pointed out by the the same authors, this is achieved at the cost of increasing the number of parameters required for calibration: the presence of a correlation matrix for the Brownian motion part of the components, in fact, implies that the number of parameters grows with the square of the number of assets included in the basket, whilst market data available for calibration is usually linear in the number of instruments.…”

“…We cite, amongst others, the approaches put forward by Baxter (2007), Moosbrucker (2006a,b), Lindskog and McNeil (2003), Brigo et al (2007), Semeraro (2008) and Luciano and Semeraro (2010). In more details, Baxter (2007) and Moosbrucker (2006a,b) use a factor copula approach for both subordinated Brownian motions and Jump Diffusion (JD) processes, whilst Lindskog and McNeil (2003) make use of linear combinations to develop a common Poisson shock process framework for dependent events frequencies in the context of insurance loss modelling and credit risk modelling.…”

“…This approach is then extended in Brigo et al (2007) to a formulation which avoids repeated defaults at both cluster level and single name level. Semeraro (2008) and Luciano and Semeraro (2010) instead apply the factor approach to build multivariate subordinators from which they derive the multivariate version of several families of subordinated Brownian motions, such as the Variance Gamma process, in this way generalizing the approaches of Schoutens (2006), Cont andTankov (2004), Leoni and Schoutens (2006) and Eberlein and Madan (2009).…”