Is Volatility Risk Priced in the Option Market?

Abstract

Rubinstein (1994) shows evidence of a significant time pattern in the shape of the volatility smile after the crash of 1987 and proposes an implied binomial tree approach to overcome the empirical limitations of the Black and Scholes model. This approach, and more generally the class of generalized deterministic volatility models, is based on the assumption that the local volatility of the underlying asset is a known function of time and of the path and level of the underlying asset price. In these economies, options are redundant assets. We use this observation as a testable restriction and ask three questions. First, is the observed dynamics of the smile consistent with deterministic volatility models? Second, if volatility is stochastic, so that two assets cannot dynamically complete the market, is volatility also priced and if so how important is to model explicitly the price of volatility in the design of risk management strategies? We address this question by testing if the returns on the underlying and on at-the-money options span the asset prices in the economy or if we need additional information from returns on other options or the riskfree rate. Third, are there any differences in the spanning properties of the option market before and after the 1987 market crash? We cast these questions in terms of martingale restrictions on the pricing kernel and conduct tests based on daily S&P500 index options data from April 1986--December 1995. All our tests suggest that both in- and out-of-the-money options are needed for spanning purposes. This finding is even stronger in the postcrash period and suggests that returns on away-from-the-money options are driven by at least one additional economic factor compared to returns on at-the-money options. This finding is inconsistent with the implications of deterministic volatility models based on generalized deterministic volatility. The finding is consistent with explanations of the smile in which volatility is stochastic and priced in equilibrium and with models in which away-from-the-money options are used in equilibrium by a different specialized clientele, such as portfolio insurers, subject to different budget constraints and/or portfolio objectives than the typical investor in at-the-money options.