The large-time smile and skew for exponential Lévy models

We derive a full asymptotic expansion for call option prices and a third order approximation for implied volatility in the large-time, large log-moneyness regime for a general exponential Lévy model, by extending the saddlepoint argument used in Forde et al. (2010) [Proc. R. Soc. A, 466(2124), 3593-3620] for the Heston model. As for the Heston model, there are two special log-moneyness values where the call option asymptotics are qualitatively different, and we use a novel Edgeworth expansion to deal with these cases. We also characterize the behaviour of the implied volatility skew at large-maturities; in particular we show that the derivative of the dimensionless implied variance with respect to log-moneyness exists and is less than or equal to 4 in the large-maturity limit, which is consistent with the bound on the right and left-side derivative given in Rogers and Tehranchi (2010)[Finance and Stochastics, 14(2), 235-248].