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.