  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.
