What is Risk Neutral Volatility?
Stephen Figlewski, New York University Stern School of Business

A security’s expected payoff under the real world distribution for stock returns includes risk premia to compensate investors for bearing different types of stock market risk. But Black-Scholes and the great majority of derivatives valuation models developed from it produce the same option prices as would be seen under modified probabilities in a world of investors who were indifferent to risk. Implied volatility and other parameters extracted from options market prices embed these modified “risk neutral” probabilities that combine investors’ objective predictions of the real world returns distribution with their risk preferences. Under Black-Scholes assumptions, real world volatility and risk neutral volatility are equal. But Black-Scholes pricing does not hold in the real world because of unhedgeable risks that bear nonzero risk premia, and the risk neutral volatility that goes into option prices is not the market’s best estimate of the volatility that will actually occur.

This project explores what factors relating to both forecasting the empirical distribution of future returns and the risk neutralization process go into the market’s risk neutral volatility parameter. Daily risk neutral densities are extracted from S&P 500 index options from 1996-2011 using a model-free procedure. Both risk neutral volatility and realized volatility from the observation date through option expiration are computed to compare the sensitivity of the two volatility measures to a wide range of variables relating to different manifestations of volatility, such as tail risk, and to the risk neutralization process, such as the general level of consumer confidence and the size of recent volatility forecast errors.

Understanding option pricing and the information that goes into it is of obvious interest to practitioners who use financial engineering methods to characterize risk and volatility dynamics. Risk neutral volatilities have the potential to provide model-free market-based information about volatility.