Recovering Probability Distributions from Contemporaneous Security Prices

Probability distributions of stock market returns have typically been estimated from historical time series. The possibility of extreme events such as the stock market crash of 1987 makes this a perilous enterprise. Alternative parametric and nonparametric approaches use contemporaneously observed option prices to recover their underlying risk-neutral probability distribution. Parametric methods assume an option pricing formula which is inverted to obtain parameters of the distribution. The nonparametric methods pursued here choose probabilities to minimize an objective function subject to requiring that the chosen probabilities are consistent with observed option and underlying asset prices. This paper examines alternative specifications of the minimization criterion using historically observed S&P 500 index option prices over an eight-year period. With the exception of the lower left-hand tail of the distribution, alternative optimization specifications typically produce approximately the same implied distributions. Most prominently, the paper introduces a new optimization technique for estimating expirationdate risk-neutral probability distributions based on maximizing the smoothness of the resulting probability distribution. Since an "almost closed-form" solution for this case is available, the smoothness method is computationally orders of magnitude faster than the alternatives. 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 and to use time aggregation to correct for unrealistic persistent jaggedness of implied volatility smiles. 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 three (four) standard deviation decline in the S&P index (about -36% (-46%) over a year) is about 10 (100) times more likely than under the assumption of lognormality, and about 10 (10) times more likely than apparent in the implied distribution prior to the crash. __________________________________________ * Jens Carsten Jackwerth is a post-doctoral visiting scholar and Mark Rubinstein a professor of finance, both at the Haas School of Business, University of California at Berkeley. For helpful discussions, we would like to thank Ron Lagnado, Hayne Leland, Dennis Klapacz, Steve Manaster, Stewart Mayhew, William Redfearn, and Robert Whaley.