An inverse finite element method for pricing American options Song-Ping Zhu n , Wen-Ting Chen School of Mathematics and Applied Statistics, University of Wollongong NSW 2522, Australia article info Article history: Received 9 December 2010 Received in revised form 24 January 2012 Accepted 27 July 2012 Available online 8 August 2012 JEL classification: G130 Keywords: Inverse finite elements Convergence analysis American options Black–Scholes model abstract The pricing of American options has been widely acknowledged as ‘‘a much more intriguing’’ problem in financial engineering. In this paper, a ‘‘convergency-proved’’ IFE (inverse finite element) approach is introduced to the field of financial engineering to price American options for the first time. Without involving any linearization process at all, the current approach deals with the nonlinearity of the pricing problem through an ‘‘inverse’’ approach. Numerical results show that the IFE approach is quite accurate and efficient, and can be easily extended to multi-asset or stochastic volatility pricing problems. The key contribution of this paper to the literature is that we have managed to provide a comprehensive convergence analysis for the IFE approach, including not only an error estimate of the adopted discrete scheme but also the convergence of the adopted iterative scheme, which ensures that our numerical solution does indeed converge to the exact one of the original nonlinear system. & 2012 Elsevier B.V. All rights reserved. 1. Introduction As is well known, one of the major topics in today’s quantitative finance research is the valuation of financial derivatives, such as options. For quite a long time, it has been widely acknowledged that pricing American options is a ‘‘much more intriguing’’ problem (see Huang et al., 1996; Ju, 1998; Longstaff and Schwartz, 2001), whose challenge mainly stems from the nonlinearity originated from the inherent characteristic that an American option can be exercised at any time during its lifespan. This additional right of being able to exercise the option early, in comparison with a European option, casts the American option pricing problem into a free boundary problem, which is highly nonlinear and far more difficult to deal with. Since most traded stock and commodity options in today’s financial markets are of American style, it is important to ensure that American-style securities can be priced accurately as well as efficiently. Recently, there is a breakthrough in the pricing of American options as an analytical closed-form pricing formula was successfully derived by Zhu (2006a). Although this formula has a great significance on the theoretical side of option pricing, it is not computationally appealing, as the formula involves two infinite sums of infinite double integrals, which take a formidable amount of time to evaluate. Till now, approximation methods are still popular among those market practitioners as they are usually faster with acceptable accuracy. In the literature, of all the approximation methods, there are predominately two types, analytical approximations and numerical methods for the valuation of an American option contract. Typical methods in the first category include the compound-option approximation method (Geske and Johnson, 1984), the quadratic approximation method (Barone-Adesi and Whaley, 1987; MacMillan, 1986), the randomization approach (Carr, 1997), the integral-equation method (Carr et al., 1992; Contents lists available at SciVerse ScienceDirect journal homepage: www.elsevier.com/locate/jedc Journal of Economic Dynamics & Control 0165-1889/$ - see front matter & 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.jedc.2012.08.002 n Corresponding author. Tel.: þ61 2 42213807; fax: þ61 2 42214845. E-mail addresses: spz@uow.edu.au (S.-P. Zhu), wtchen@uow.edu.au (W.-T. Chen). Journal of Economic Dynamics & Control 37 (2013) 231–250