Recovering probability distributions from option prices

Article Abstract:

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 to requiring that the probabilities are consistent with observed option and underlying asset prices. Alternative optimization specifications produce approximately the same implied distributions. A new and fast optimization technique for estimating probability distributions based on maximizing the smoothness of the resulting distribution is proposed. Since the crash, the risk-neutral probability of a three (four) standard deviation decline in the index (about -36 percent (-46 percent) over a year) is about 10 (100) times more likely than under the assumption of lognormality. (Reprinted by permission of the publisher.)

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A simple formula for the expected rate of return of an option over a finite holding period

Article Abstract:

A simple formula is developed for the expected rate of return of an option over a finite holding period, perhaps shorter than the option's expiration time. Under conditions consistent with the Black-Scholes formula, the expected future value of a European option-even prior to expiration - is shown equal to the current Black-Scholes value of the option. However, the expected future value of the stock at the end of the holding period replaces the current stock price in the Black-Scholes formula and the future value of a riskless investment of the striking price replaces the striking price. Moments of the distribution of returns from an option portfolio are approximated using an extension of this result.

Comparative analysis, Criticism and interpretation, Investment analysis, Securities analysis, Black, Fischer, Scholes, Myron

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Path dependent options: the case of lookback options

Article Abstract:

Lookback options are path dependent contingent claims whose payoffs depend on the extrema of a given security's price over a period of time. Using probabilistic tools, we derive formulas for various European lookback options, and provide some results about their American countrerparts. (Reprinted by permission of the publisher.)

Securities, Securities prices

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