Implied volatility explosions: European calls and implied volatilities close to expiry in exponential Lévy models

We examine the small expiry behaviour of the price of call options in models of exponential Lévy type. In most cases of interest, it turns out that E ( (Sτ −K) ) − (S0 −K) ∼ { τ ∫ R (S0e x −K)+ ν(dx), S0 < K, τ ∫ R (K − S0e) ν(dx), S0 > K, as τ → 0+, i.e. as time to expiry goes to zero. (We have written ν for the Lévy measure of the driving Lévy noise.) In “complete generality”, however, we can say only that E ( (Sτ −K) ) − (S0 −K) = O(τ) as τ → 0+. Using our results on the behaviour of call options close to expiry we show that implied volatility explodes as τ → 0+ in “most” exponential Lévy models. Attention is restricted to calls and implied volatilities that are not at-themoney, i.e. S0 6= K.