No country for old distributions? On the comparison of implied option parameters between the Brownian motion and variance gamma process

Abstract

Advanced stochastic approaches are often suggested as a solution to real-world derivative pricing inconsistencies like the non-linearity of the implied volatility smile. Using a novel high-frequency data set with over one million option trades and corresponding order books from the German market, we compare the normal distribution approach with a variance gamma process, which is – as a pure jump process – especially suitable for tick-by-tick data. We are able to report a flattened implied volatility smile with the variance gamma process. Other low-frequency results like time, information, and underlying moment dependencies for both stochastic processes are unchanged. All in-sample residuals of the normal distribution have a smaller variance below the 1% significance level compared to the variance gamma process. Additionally, we reveal a mean-reversion process. We show that the normal distribution is superior to the variance gamma process in an out-of-sample context and conclude that – even in a high-frequency environment – it is still “a country for old distributions”.