Implied volatility: small time-to-expiry asymptotics in exponential Lévy models

Michael Roper

Abstract

In this paper, we examine the small time-to-expiry behaviour of implied volatility in models of exponential Lévy type. In the at-the-money case, it turns out that the implied volatility converges, as time-to-expiry goes to zero, to the square root of the Gaussian member of the driving Lévy process� characteristic triplet. In particular, the limit is zero if the Lévy process has no Gaussian part. In the not at-the-money case, there are a number of possible behaviours. In most cases of interest, however, the implied volatility goes to infinity as time-to-expiry goes to zero. It is also shown that there are exponential Lévy models in which the implied volatility converges to zero as time-to-expiry goes to zero.

Keywords: Implied volatility; Levy processes.