American Options, the Method of Images and Closed Form Solutions

29 Pages Posted: 1 Aug 2018

See all articles by Gabriela Baumgartner

Gabriela Baumgartner

Deloitte Consulting

Andrew D. Smith

University College Dublin (UCD); Deloitte Touche Tohmatsu

Date Written: July 20, 2018

Abstract

This paper considers solutions for a class of optimal stopping problems, maximising the expected product of a Wiener process and a positive decreasing scale function. The general approach to such problems involves a partial differential equation with movable boundary. The method of images is a useful tool for solving fixed-boundary PDE problems. We adapt this method to a sub-class of movable boundary problems.
When the scale function in the original problem is a survival function of Generalised Pareto (GPD) type, we use a self-similarity property to reduce the PDE to an ODE. This ODE was widely studied in the 19th century, and the solution involves confluent hypergeometric functions. In cases of integer parameters, we give simpler closed form solutions involving the normal distribution function. The same approach also works when the Wiener process is reflected at zero.

Keywords: American Option, Closed Form, Movable Boundary, Optimal Stopping, Confluent Hypergeometric Function

Suggested Citation

Baumgartner, Gabriela and Smith, Andrew D., American Options, the Method of Images and Closed Form Solutions (July 20, 2018). Available at SSRN: https://ssrn.com/abstract=3213781 or http://dx.doi.org/10.2139/ssrn.3213781

Gabriela Baumgartner

Deloitte Consulting ( email )

United Kingdom

Andrew D. Smith (Contact Author)

University College Dublin (UCD) ( email )

Belfield
Belfield, Dublin 4 4
Ireland

Deloitte Touche Tohmatsu ( email )

Paramount Plaza
Midtown Manhattan
New York, NY AB
United States

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