Pricing American options under proportional transactioncosts using a penalty approach and a finite difference scheme

Wen Li; Song Wang

doi:10.3934/jimo.2013.9.365

Article Contents

2013, Volume 9, Issue 2: 365-389. Doi: 10.3934/jimo.2013.9.365

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Pricing American options under proportional transaction costs using a penalty approach and a finite difference scheme

Wen Li^1, and
Song Wang^1,

1.
School of Mathematics & Statistics, The University of Western Australia, 35 Stirling Highway, Crawley, WA 6009

Received: December 31, 2011

Revised: April 30, 2012

Published: March 2013

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Abstract

In this paper we propose a penalty method combined with a finite difference scheme for the Hamilton-Jacobi-Bellman (HJB) equation arising in pricing American options under proportional transaction costs. In this method, the HJB equation is approximated by a nonlinear partial differential equation with penalty terms. We prove that the viscosity solution to the penalty equation converges to that of the original HJB equation when the penalty parameter tends to positive infinity. We then present an upwind finite difference scheme for solving the penalty equation and show that the approximate solution from the scheme converges to the viscosity solution of the penalty equation. A numerical algorithm for solving the discretized nonlinear system is proposed and analyzed. Numerical results are presented to demonstrate the accuracy of the method.

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Mathematics Subject Classification: Primary: 49L20, 65M06; Secondary: 91G60.

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