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 (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. In this paper 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. The contribution of this research is twofold: We introduce a pricing framework that combines the benefits of mixture distributions with the flexibility of implied binomial trees. We show several advantages of using this technique compared to a standard implied lattice, where each end-nodal probability is modeled seperately. The approach, by design, produces smooth probability density functions even in the tails of the distribution. There is no need for further smoothing algorithms, clamping or trimming of the tree. In addition, the method significantly reduces the number of free parameters in the optimization which dramatically increases computational speed and even allows the extraction of a risk-neutral probability density function (pdf) out of American option prices. We analyze a dataset of the year 1998 which enables us to show the effect of the Asia crisis on the valuation of traded index options. We observe the implied probability of a crash, assumed by market participants, to increase after the crash has occured. This suggests that investors tend to forget (or don't care) about risks when stock markets rally, as they have in the beginning of 1998.