The American put and European options near expiry, under Levy processes

TL;DR

This paper derives explicit formulas for option time decay near expiry under Levy processes, revealing how American put early exercise boundaries behave and their implications for financial modeling.

Contribution

It provides analytical approximations for American put options near expiry under Levy processes, highlighting boundary behavior and parameter fitting implications.

Findings

Early exercise boundary remains separated from strike for many Levy processes.

As riskless rate approaches zero, optimal exercise price tends to zero.

Explicit formulas for time decay at expiry are derived.

Abstract

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.