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

We examine the small expiry behaviour of European call options in stock price models of exponential L\'evy type. In most cases of interest, we are able to identify the exact small expiry asymptotics. In "complete generality" we are able to show that the time value of the call option has O(\tau) decay as \tau (time to expiry) goes to zero. Using our results on the behaviour of call options close to expiry we show that implied volatility explodes as $\tau\to0^+$ in "most" exponential L\'evy models. Attention is restricted to calls and implied volatilities that are not at-the-money.