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 the financial industry. One way to extend the model to the multivariate case is to subordinate a Brownian motion through a univariate subordinator. We study this 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. Apart from the empirical analysis of the multivariate VG model, our study aims to contribute to the literature on pricing of structured financial products. The existing studies typically derive fair values from the normal model, which might seriously bias the results on the degree of overpricing. We find that smile-consistent calibration is possible in 277 out of 468 cases in our sample. The estimated multivariate VG processes significantly differ from multivariate Brownian Motion. In particular, the VG model implies a higher probability of strongly negative returns with a final stock price below the barrier. The value difference to the normal model depends primarily on the time-to-maturity and the degree of skewness in the smile patterns. We conclude that non-normality of returns cannot be ignored when valuing multi-asset options, which is important for future studies on the overpricing of exotic structured products.