Multivariate Downside Risk: Normal Versus Variance Gamma

Abstract

Although several types of options on multiple assets are popular in today's financial markets, valuing multi-asset options is still a challenge in finance. The standard framework of multivariate normality is often inappropriate, since it ignores fat tails and other stylized facts of asset returns. The Variance Gamma (VG) model appears to be a promising alternative. In the univariate case, it has become a standard tool in finance. The traditional way to extend the model to the multivariate case is to subordinate a Brownian motion through a univariate subordinator. In recent years, generalizations with multivariate subordinators have been proposed. Our objective is to study two versions of the multivariate VG model in a large-scale application with multi-asset options traded in an active market. Our database consists of 468 multivariate barrier reverse convertibles at the Swiss market for structured products. The Swiss market ranks among the largest in the world and is characterized by an exceptional popularity of multiple asset options. We find that there is a trade-off between the two VG models considered: one performs better in capturing the smile, the other is more often able to capture the correlation structure. In all, based on our calibration, only 316 out of 468 products can be evaluated with at least one of the two VG models. We conclude that there is a need for more flexible extensions.JEL classification: G13; G15; G14

Multivariate asset models using Lévy processes and applications

“…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.…”

Option pricing in time-changed Lévy models with compound Poisson jumps

“…Luciano and Schoutens [27] modeled the S&P500, the Nikkei225 and the Eurostoxx50 financial indexes by the variancegamma process. Luciano et al [29] and Wallmeier and Diethelm [43] confirmed the using of variance-gamma distribution for the modeling of the US and the Swiss stock markets, respectively.…”

A Credit-Risk Valuation under the Variance-Gamma Asset Return

“…Among others, let us mention the papers by Daal and Madan (2005), Finlay and Seneta (2006), Linders and Stassen (2016), Luciano et al (2016), Luciano and Schoutens (2006), Moosbrucker (2006), Mozumder et al (2015), Rathgeber et al (2016), and Wallmeier and Diethelm (2012), where the variance-gamma distribution is confirmed as a very good model to make out the statistics. For approximations of processes by the variance-gamma one, see Eichelsbacher and Thäle (2015).…”