Pricing S&P 500 index options using a Hilbert space basis

This paper tests the approach of Madan and Milne (1994) and its extension in Abken, Madan, and Ramamurtie (1996) for pricing contingent claims as elements of a separable Hilbert space. We specialize the Hilbert space basis to the family of Hermite polynomials and test the model on S&P 500 index options. Restrictions on the prices of Hermite polynomial risk are imposed that allow all option maturity classes to be used in estimation. These restrictions are rejected by our empirical tests of a four-parameter specification of the model. Nevertheless, the unrestricted four-parameter model, based on a single maturity class, demonstrates better out-of-sample performance than that of the Black-Scholes version of the Hermite model. The unrestricted four-parameter model results indicate skewness and excess kurtosis in the implied risk-neutral density. The skewness of the risk-neutral density contrasts with the symmetry of the statistical density estimated using the Hermite model on the S&P 500 index returns