Pricing Multi-Asset American Options: A Finite Element Method-of-Lines with Smooth Penalty

This paper studies the problem of pricing multi-asset American-style options in the Black–Scholes–Merton framework. The value function of an option contract is known to satisfy a partial differential variational inequality (PDVI) when early exercise is permitted. We develop a computational method for the valuation of multi-asset American-style options based on approximating the PDVI by a non-linear penalized PDE with a penalty term with continuous Jacobian. We convert the non-linear PDE to a variational (weak) form, discretize the weak formulation spatially by a Galerkin finite element method to obtain a system of ODEs, and integrate the resulting system of ODEs in time with an adaptive variable order and variable step size solver SUNDIALS. Numerical results demonstrate that employing a penalty term with continuous Jacobian in contrast to the penalty terms with discontinuous Jacobians in use in the literature improves computational performance of the adaptive temporal integrator. In our framework we are able to price American-style options with payoffs dependent on up to six assets on a PC. This is in contrast to the existing literature on the pricing of American options by PDE methods, that has so far been limited to at most three-dimensional problems. Our results open avenues for further applications to multi-dimensional problems, such as pricing convertible bonds in multi-factor models, that will be explored in future work. This research was supported by the National Science Foundation under grants DMI–0422937 and DMI–0422985.