A class of risk neutral densities with heavy tails

From observed bid and ask prices of European call and put options we estimate the risk neutral density of a stock at some future time $t>0$. We restrict attention to a class of densities with heavy tails and use a Bayesian formulation in order to study the variation in the distributions fitting the data. Heavy tails are here meant in the intuitive sense of being heavier than the tails of a normal distribution. From the fitted risk neutral density we also consider the inverse problem of finding the volatility in a diffusion model for the price process. Finally, we apply our methods to data on the S&P 500 index.