Extracting Risk-Neutral Density and Its Moments from American Option Prices

Abstract

Implied volatility is obtained from an option’s observed market price by solving backward through the pricing model to find the volatility that would produce the same value from the model. Unfortunately, this typically leads to the familiar volatility smile: different implied volatilities from different options on the same underlying. But from market prices for the full set of a given stock’s options, one can extract not just volatility, but the entire risk-neutral probability distribution, from which volatility and any other moment can easily be obtained, and the result does not depend on any particular pricing model. One problem, however, is that because the risk-neutral density is over the stock price at expiration day, the options involved need to be European so that exercise can only take place on that date. But exchange-traded options are nearly all American, in the U.S. at least. In this article, Tian shows how to obtain the risk-neutral density from American option prices. The trick is to build an implied lattice model from the American options then use it to find European option prices that would be consistent with the observed American prices in the market. The density can then be extracted from these implied European prices in the standard way. TOPICS:Options, VAR and use of alternative risk measures of trading risk, accounting and ratio analysis

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