Direct Semi-Parametric Estimation of the State Price Density Implied in Option Prices

Abstract

We present a model for direct semi-parametric estimation of the state price density (SPD) implied by quoted option prices. We treat the observed prices as expected values of possible pay-offs at maturity, weighted by the unknown probability density function. We model the logarithm of the latter as a smooth function, using P-splines, while matching the expected values of the potential pay-offs with the observed prices. This leads to a special case of the penalized composite link model. Our estimates do not rely on any parametric assumption on the underlying asset price dynamics and are consistent with no-arbitrage conditions. The model shows excellent performance in simulations and in applications to real data.

KEYWORDS:

• Arbitrage-free estimates
• Inverse problem
• Penalized composite link model
• State price density

Acknowledgments

We thank professor Oleg Bondarenko for sharing the code used to estimate risk-neutral densities with the positive convolution approximation. We also thank the two anonymous reviewers and the associated editor for their insightful comments and remarks.