American and European Options Near Expiry, Under Markov Processes with Jumps

Abstract

We derive an explicit formula for time decay, theta, for out-of-the-money European options at expiry, in terms of the density of jumps and payoff $g$. We use this formula to show that in the presence of jumps, the limit of the no-exercise region as time to expiry tends to 0 is typically larger than in the pure Gaussian case. In particular, 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).

Keywords: European out-of-the-money options, early exercise boundary, processes with jumps

JEL Classification: D81, C61