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

Gianluca Frasso*, Paul H.C. Eilers

*Corresponding author for this work

Research output: Contribution to journalArticleAcademicpeer-review


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.

Original languageEnglish
Pages (from-to)1179-1190
Number of pages12
JournalJournal of Business and Economic Statistics
Issue number3
Early online date4 May 2021
Publication statusPublished - 2022

Bibliographical note

Funding Information:
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.

Publisher Copyright:
© 2021 American Statistical Association.


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