INTERNATIONAL JOURNAL OF c  2012 Institute for Scientific NUMERICAL ANALYSIS AND MODELING, SERIES B Computing and Information Volume 3, Number 3, Pages 345–370 FINITE ELEMENT METHOD FOR AMERICAN OPTION PRICING: A PENALTY APPROACH SAJID MEMON Abstract. The model for pricing of American option gives rise to a parabolic variational inequal- ity. We first use penalty function approach to reformulate it as an equality problem. Since the problem is defined on an unbounded domain, we truncate it to a bounded domain and discuss error due to truncation and penalization. Finite element method is then applied to the penalized problem on the truncated domain. By coupling the penalty parameter and the discretization parameters, error estimates are established when the initial data in H 1 0 . Finally, some numerical experiments are conducted to confirm the theoretical findings. Key words. Penalty method, American options, variational inequality, finite element method, error analysis. 1. Introduction In a financial market, an option is a contract which gives to its owner the right to buy (call option) or to sale (put option) a fixed quantity of assets of a specified stock at a fixed price (exercise or strike price) on (European option) or before (American option) a given date called expiry date. It is known that price of an American option is governed by a linear complementarity problem [5, 7, 16] involving the Black-Scholes differential operator and a constraint (or obstacle) on the value of the option. In literature, there are several methods for the valuation of European and Amer- ican options. The first numerical approach to Black-Scholes equation is a lattice method proposed in [2]. Since, it is a linear complementarity problem, finite dif- ference methods need to combine with other techniques for solving discrete linear complementarity problem using methods like PSOR algorithm [18], operator split- ting [6]. Approaching linear complementarity problem using penalty method is not quite new, for example, penalty method and front fixing method together with finite difference method are discussed in [17, 11, 12]. Finite difference methods are by far the simplest and been favourite in computational finance but it is not suitable for mesh adaptivity. Finite volume methods are also used for pricing American/European option with constant or time-dependent volatility. Wang et al. [14, 15] have proposed a fitted finite volume method for spatial discretization and an implicit time stepping tech- nique for temporal discretization which is combined with power penalty method for option pricing. The analysis is performed within the framework of the vertical method of lines, where the spatial discretization is formulated as a Petrov-Galerkin finite element method with each basis function of the trial space being determined by a set of two-point boundary value problems. Finite element methods seem at first glance unnecessarily more complex than fi- nite difference scheme for finance, where a large class of problems is one dimensional in space. However, these methods are very flexible for mesh adaptivity. Earlier, Received by the editors July 2011 and, in revised form, March 2012. 2000 Mathematics Subject Classification. 35R35, 49J40, 60G40. 345