A Jumping Smile
36 Pages Posted: 9 Sep 1996
Date Written: March 1997
This paper examines systematically the consequences of jump processes in the returns of the underlying asset, for the existence of the volatility smile. The key insight in this paper is the theoretical (but little-noticed) result that the presence of jumps brings major qualitative changes to option pricing. Specifically, when there are jumps it is not possible to have a single option price derived by arbitrage methods alone even at the limit of continuous trading. In a universe of investors who are risk-averse but otherwise unspecified the absence of arbitrage possibilities generates two bounds on option prices, within which the "true" option price lies. If, as is highly plausible, each observed option price at any given time is drawn from a different cohort of risk-averse investors then there is no reason for the existence of a single distribution able to price all options. The method developed in this paper is thus suited to the pricing of contingent claims when the smile is due to the presence of jumps. Two boundary risk-neutral distributions are identified, bracketing the observed option values. These disttributions are the tightest possible distributions supportable by our data, in the sense of producing the minimum spread between possible option values. In our sample the economic difference between these two distributions in the case of S&P 500 options is minor, in the sense that the percentage error in using one rather than the other is small for the key portion of the range of exercise prices. A major advantage of our approach is that it produces results almost always. In a sample of 1991 simultaneous observations on both options and the underlying index value spread over the years 1991, 1993, and 1996 our method produced results in every single instance. By contrast, a single risk-neutral distribution satisfying all observed option values did not exist for 41% of our observations. This failure of the data to support a single option-pricing distribution cannot be explained by models based on systematically-varying volatility of the underlying asset, or by transactions costs. By contrast, it is fully consistent with the existence of jumps in the price of the underlying asset. Further, while a detailed screening of the data may eliminate some of these failures as data errors or mispricings, the fact remains that such eliminations can only take place ex post, thus reducing the usefulness of the single distribution approaches for option practitioners.
JEL Classification: G13
Suggested Citation: Suggested Citation