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Boundary Evolution Equations for American Options

37 Pages Posted: 2 Dec 2011 Last revised: 12 Jan 2012

Daniel Mitchell

University of Texas at Austin - Red McCombs School of Business

Jonathan Goodman

New York University (NYU) - Courant Institute of Mathematical Sciences

Kumar Muthuraman

University of Texas at Austin - McCombs School of Business

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Date Written: November 30, 2011

Abstract

We consider the problem of finding optimal exercise policies for American options, both under constant and stochastic volatility settings. Rather than work with the usual equations that characterize the price exclusively, we derive and use boundary evolution equations that characterize the evolution of the optimal exercise boundary. Using these boundary evolution equations we show how one can construct very efficient computational methods for pricing American options that avoid common sources of error. First we detail a methodology for standard static grids and then describe an improvement that defines a grid that evolves dynamically while solving the problem. When integral representations are available, as in the Black-Scholes setting, we also describe a modified integral method that leverages on the representation to solve the boundary evolution equations. Finally we compare runtime and accuracy to other popular numerical methods. The ideas and methodology presented herein can easily be extended to other optimal stopping problems.

Keywords: Optimal Stopping, American Options, Stochastic Volatility, Early Exercise Boundary, Free-Boundary Problem, Dynamic Grid

Suggested Citation

Mitchell, Daniel and Goodman, Jonathan and Muthuraman, Kumar, Boundary Evolution Equations for American Options (November 30, 2011). McCombs Research Paper Series No. IROM-02-12. Available at SSRN: https://ssrn.com/abstract=1966726 or http://dx.doi.org/10.2139/ssrn.1966726

Daniel Mitchell (Contact Author)

University of Texas at Austin - Red McCombs School of Business ( email )

Austin, TX 78712
United States

Jonathan B. Goodman

New York University (NYU) - Courant Institute of Mathematical Sciences ( email )

New York University
New York, NY 10012
United States

Kumar Muthuraman

University of Texas at Austin - McCombs School of Business ( email )

Austin, TX 78712
United States

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