Closed Form Optimal Exercise Boundary of the American Put Option

16 Pages Posted: 4 May 2020 Last revised: 8 May 2020

Multiple version iconThere are 2 versions of this paper

Date Written: April 8, 2020


We present three models of stock price with time-dependent interest rate, dividend yield, and volatility, respectively, that allow for explicit forms of the optimal exercise boundary of the American put option. The optimal exercise boundary satisfies nonlinear integral equation of Volterra type. We choose time-dependent parameters of the model so that the integral equation for the exercise boundary can be solved in the closed form. We also define the contracts of put type with time-dependent strike price that support the explicit optimal exercise boundary. All these results can be used as approximations to the standard model with constant parameters, i.e., geometric Brownian motion process.

Keywords: American put option, geometric Brownian motion, optimal stopping, free-boundary problem, integral equation

JEL Classification: G13, C61

Suggested Citation

Kitapbayev, Yerkin, Closed Form Optimal Exercise Boundary of the American Put Option (April 8, 2020). Available at SSRN: or

Yerkin Kitapbayev (Contact Author)

North Carolina State University ( email )

Hillsborough Street
Raleigh, NC 27695
United States

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