Numer. Math. Theor. Meth. Appl., 2 (2009), pp. 202-223. A Power Penalty Approach to Numerical Solutions of Two-Asset American Options K. Zhang 1*, S. Wang 2, X. Q. Yang 3, K. L. Teo 41 Department of Finance, Business School, Shenzhen University, Shenzhen, China; and School of Mathematics and Statistics, University of Western Australia, Australia. 2 School of Mathematics and Statistics, University of Western Australia, Australia. 3 Department of Applied Mathematics, The Hong Kong Polytechnic University, Hong Kong. 4 Department of Mathematics and Statistics, Curtin University of Technology, Australia. Received 1 August 2008; Accepted (in revised version) 9 January 2009 Abstract This paper aims to develop a power penalty method for a linear parabolic variational inequality (VI) in two spatial dimensions governing the two-asset American option valuation. This method yields a two-dimensional nonlinear parabolic PDE containing a power penalty term with penalty constant $\lambda > 1$ and a power parameter $k>0$. We show that the nonlinear PDE is uniquely solvable and the solution of the PDE converges to that of the VI at the rate of order $O( \lambda^{-k/2})$. A fitted finite volume method is designed to solve the nonlinear PDE, and some numerical experiments are performed to illustrate the usefulness of this method. AMS subject classifications: 65M12, 65M60, 91B28 Key words: Complementarity problem, option pricing, penalty method, finite volume method. *Corresponding author. Email: mazhangkai@gmail.com (K. Zhang), swang@maths.uwa.edu.au (S. Wang), mayangxq@polyu.edu.hk (X. Q. Yang), K.L.Teo@curtin.edu.au (K. L. Teo)