Recovering Probability Distributions from Option Prices

摘要: This article derives underlying asset risk-neutral probability distributions of European options on the S&P 500 index. Nonparametric methods are used to choose probabilities that minimize an objective function subject requiring consistent with observed option and prices. Alternative optimization specifications produce approximately same implied distributions. A new fast technique for estimating based maximizing smoothness resulting distribution is proposed. Since crash, a three (four) standard deviation decline in index (about -36 percent (-46 percent) over year) about 10 (100) times more likely than under assumption lognormality. RECENTLY, THE INCREASING POPULARITY derivatives some highly publicized failures control risk have led increased efforts find reasonable measure sensitivity large institutional portfolios extreme events. Merely because such events rare not sufficient ignore them, since few occasions when they do occur, significant amounts money can change hands, potentially wiping out profits accumulated long prior periods. key behind estimation joint constituent returns. has been concern financial economists, assumptions critical much their research during last quarter century. Heretofore, stock market returns typically estimated from historical time series. Unfortunately, common hypotheses may capture events, interest or be present record, even though clearly possible.