Parametric Properties of Semi-Nonparametric Distributions, with Applications to Option Valuation

Abstract

We derive the statistical properties of the semi-nonparametric (SNP) densities of Gallant and Nychka (1987). We show that these densities, which are always positive, are more flexible than truncated Gram–Charlier expansions with positivity restrictions. We use the SNP densities for financial derivatives valuation. We relate real and risk-neutral measures, obtain closed-form prices for European options, and analyze the semiparametric properties of our pricing model. In an empirical application to S&P500 index options, we compare our model to the standard and Practitioner's Black–Scholes formulas, truncated expansions, and the Generalized Beta and Variance Gamma models.

KEY WORDS::

• Density expansions
• Gram–Charlier
• Kurtosis
• Skewness