On the Early Exercise Boundary of the American Put Option

Abstract

We study the short time behavior of the early exercise boundary for American style put options in the Black--Scholes theory. We develop an asymptotic expansion which shows that the simple lower bound of Barles et al. is a more accurate approximation to the actual boundary than the more complex upper bound. Our expansion is obtained through iteration using a boundary integral equation. This integral equation is derived from the time derivative of the option value function, which closely resembles the classical Stefan free boundary value problem for melting ice. Our analytical results are supported by numerical computations designed for very short times.

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