Of Smiles and Smirks: A Term Structure Perspective

Abstract

Empirical anamolies in the Black-Scholes model have been widely documented in the Finance literature. Patterns in these anamolies (for instance, the behavior of the volatility smile or of unconditional returns at different maturities) have also been widely documented. Theoretical efforts in the literature at addressing these anamolies have largely centered around extensions of the basic Black-Scholes model. Two approaches have become especially popular in this context -- introducing jumps into the return process, and allowing volatility to be stochastic. This paper employs commonly-used versions of these two classes of models to examine the extent to which the models are theoretically capable of resolving the observed anamolies. We focus especially on the possible "term-structures": of skewness, kurtosis, and the implied volatility smile that can arise under each model. Our central finding is that each model exhibits moment patterns and implied volatility smiles that are consistent with some of the observed anamolies, but not with others. In sum, neither class of models constitutes an adequate explanation of the empirical evidence, although stochastic volatility models fare better than jumps in this regard.