Option pricing under NIG distribution: --- The empirical analysis of Nikkei 225 option ----

It is well known that the distributions of assets returns have heavier tails than the Gaussian's. To capture such a distributional characteristic, the Generalized Hyperbolic(GH) distribution and its subclasses have been applied to assets returns as the distribution with heavier tails. GH distribution was originally introduced by Barndorff-Nielsen(1977) to depict grain size distribution of wind blown sands. Furthermore, GH distribution covers many distributions such as Normal Inverse Gaussian(NIG) as a special case. Recently these distributions have been sccessfully applied to fit to real stock returns in empirical studies. In this paper, we apply the Gaussian and the NIG distributions to the log returns of Nikkei 225. At first, we investigate the goodness of fit of these distributions to Nikkei 225. Next, we calculate two kinds of the theoretical prices for European call option with Nikkei 225 as the underlying asset. The theoretical prices are calculated by Black-Schols(BS) model as well as by NIG model which is obtained by Esscher transforms under the assumption that the log returns follow NIG distribution. Secondly we calculate the differences between option prices calculated by BS model and quoted option prices in the real Japanese market. We also calculate the similar differences between option prices obtained by NIG model and the realized option prices in the Japanese market. We use the discrepancies calculated by BS model as benchmark to examine the performance of option prices by NIG model. As a result we find that option prices obtained by NIG model are generally closer to realized option prices than those by BS model.