  We derive explicit formulas for time decay, for the European call and put
options at expiry, and use them to calculate analytical approximations to the
price of the American put and early exercise boundary near expiry. We show that
for many families of non-Gaussian processes used in empirical studies of
financial markets, the early exercise boundary for the American put without
dividends is separated from the strike price by a non-vanishing margin on the
interval [0,T). As the riskless rate vanishes and the drift decreases
accordingly so that the stock remains a martingale, the optimal exercise price
goes to zero uniformly over the interval [0, T). The implications for
parameters' fitting are discussed.
