SHORT-TIME IMPLIED VOLATILITY IN EXPONENTIAL LEVY MODELS

We show that a necessary and sufficient condition for the explosion of implied volatility near expiry in exponential Levy models is the existence of jumps towards the strike price in the underlying process. When such jumps do not exist, the implied volatility converges to the volatility of the Gaussian component of the underlying Levy process as the time to maturity tends to zero. These results are proved by comparing the short-time asymptotics of the Black-Scholes price with explicit formulas for upper and lower bounds of option prices in exponential Levy models.