Analysis of the optimal exercise boundary of American put options with delivery lags

TL;DR

This paper investigates the optimal exercise boundary of American put options with delivery lags, decomposing the option into a European put and a novel American-style derivative, and analyzing its properties using free boundary methods.

Contribution

It introduces a decomposition of American options with delivery lags into a European put and a new American-style derivative, and studies the boundary's existence, smoothness, and asymptotic behavior.

Findings

Optimal exercise boundary exists and is strictly increasing.

The boundary is smooth and well-behaved.

Asymptotic analysis for large maturity and small time lag.

Abstract

A make-your-mind-up option is an American derivative with delivery lags. We show that its put option can be decomposed as a European put and a new type of American-style derivative. The latter is an option for which the investor receives the Greek Theta of the corresponding European option as the running payoff, and decides an optimal stopping time to terminate the contract. Based on this decomposition and using free boundary techniques, we show that the associated optimal exercise boundary exists and is a strictly increasing and smooth curve, and analyze the asymptotic behavior of the value function and the optimal exercise boundary for both large maturity and small time lag.