Implied Probability Distributions: Empirical Analysis

Posted: 10 Oct 1998

See all articles by Mark Rubinstein

Mark Rubinstein

University of California, Berkeley - Haas School of Business

Jens Carsten Jackwerth

University of Konstanz - Department of Economics


An earlier article, "Implied Binomial Trees," introduced a theoretical model for implying the stochastic process of an underlying asset price from the prices of associated options. This sequel provides details concerning application of the model to the full record of S&P 500 index options transactions from April 2, 1986 through December 31, 1993. Most prominently, it introduces a revised optimization technique for estimating expiration-date risk-neutral probability distributions which is probably theoretically superior and definitely orders of magnitude faster than the approaches outlined in the antecedent paper. This method maximizes the smoothness of the distribution while at the same time insuring that multimodalities are not unrealistically strong. With the exception of the lower left-hand tail of the distribution, alternative optimization specifications typically produce approximately the same implied distributions. Considerable care is taken to specify such parameters as interest rates, dividends, and synchronous index levels, as well as to filter for general arbitrage violations resulting implied probability distributions exhibit changes in skewness as time-to-expiration approaches which are consistent and to use time aggregation to correct for unrealistic persistent jaggedness of implied volatility smiles. The with theoretical predictions. While time patterns of skewness and kurtosis exhibit a discontinuity across the divide of the 1987 market crash, they remain remarkably stable on either side of the divide. Moreover, since the crash, the risk-neutral probability of a four standard deviation decline in the S&P index (-46% over a year) is 100 times more likely than would appear to be the case under the assumption of lognormality.

JEL Classification: G13

Suggested Citation

Rubinstein, Mark E. and Jackwerth, Jens Carsten, Implied Probability Distributions: Empirical Analysis. Available at SSRN:

Mark E. Rubinstein

University of California, Berkeley - Haas School of Business ( email )

545 Student Services Building, #1900
2220 Piedmont Avenue
Berkeley, CA 94720
United States
510-642-3580 (Phone)
510-643-1420 (Fax)


Jens Carsten Jackwerth (Contact Author)

University of Konstanz - Department of Economics ( email )

Universitaetsstr. 10
Konstanz, 78457
+497531882196 (Phone)
+497531883120 (Fax)


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