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We examine the small expiry behaviour of European call options in stock pricemodels of exponential L\'evy type. In most cases of interest, we are able toidentify the exact small expiry asymptotics. In "complete generality" we areable 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 calloptions close to expiry we show that implied volatility explodes as$\tau\to0^+$ in "most" exponential L\'evy models. Attention is restricted tocalls and implied volatilities that are not at-the-money.

Implied volatility explosions: European calls and implied volatilities close to expiry in exponential L\'evy models