Extracting risk-neutral densities from option prices using mixture binomial trees

Since the stock market crash in October of 1987, prices of index options deviate significantly from Black-Scholes theory. This fact is prominently documented in the literature as the volatility smile (M. Rubinstein 1994). The pricing error is a sign that the assumptions of the model do not capture all relevant information embedded in option prices. As response to this problem, previous research has relaxed some of the underlying assumptions in order to arrive at more realistic prices. Examples are changes in the data generating process of the underlying security, e.g., jump diffusion or constant elasticity of variance models. We reverse this direction of thought: we take recorded option prices as given and estimate the implied pricing kernel that is consistent with current market valuations. To be flexible in the shape of the risk-neutral density and to allow probabilities to be non-Gaussian, we use the concept of mixture distributions. We apply our methodology to a recent dataset of options on the S&P 500 index future traded on the Chicago Mercantile Exchange. The data spans the year 1998 and contains settlement prices for 22497 American-style futures options.