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

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. Implied volatility at strike K and time-to-expiry τ is the unique volatility parameter that when plugged into the Black-Scholes formula recovers the quoted call option price at strike K and and time-to-expiry τ. Hence implied volatility is another way of quoting call option prices. In practice, it is more popular than quoting actual call option prices. Hence the interest in implied volatility. In this paper, we present first order asymptotics for implied volatility as time-to-expiry goes to zero. There has recently been much work on small time-to-expiry asymptotics of call options in stochastic volatility models, see, for example, Forde and Jacquier (2009) and references therein. In this paper, however, we exclusively examine small time-to-expiry asymptotics of implied volatility in models of exponential Lévy type. Despite the popularity of Lévy processes in mathematical finance, see Cont and Tankov (2004) and references therein, the literature treating the small timeto-expiry behaviour of implied volatility in this class of models is small. The method we use is to establish small time-to-expiry asymptotics for the call option price and then to use the results of Roper and Rutkowski (2009) to relate the call option price asymptotics to the implied volatility asymptotics. It appears that the not at-the-money asymptotics in exponential Lévy models were first rigorously analysed in Roper (2008) and the at-the-money case was first rigorously analysed in Roper (2009). Note though that the results of Durrleman (2008) are applicable to some exponential Lévy models and when they are they agree with the results of this paper. Since then, there have been a number of works devoted to small time-to-expiry asymptotics of implied volatility in models of exponential Lévy type. The contributions that we make here are two-fold. Firstly, in models of exponential