A Newton method for American option pricing

Abstract

The variational inequality formulation provides a mechanism to determine both the option value and the early exercise curve implicitly [17]. Standard finite difference approximation typically leads to linear complementarity problems with tridiagonal coefficient matrices. The second order upwind finite difference formulation gives rise to finite dimensional linear complementarity problems with nontridiagonal matrices, whereas the upstream weighting finite difference approach with the van Leer flux limiter for the convection term [19, 22] yields nonlinear complementarity problems. We propose a Newton type interior-point method for solving discretized complementarity/variational inequality problems that arise in the American option valuation. We illustrate that the proposed method on average solves a discretized problem in 2 ∼ 5 iterations to an appropriate accuracy. More importantly, the average number of iterations required does not seem to depend on the number of discretization points in the spatial dimension; the average number of iterations actually decreases as the time discretization becomes finer. The arbitrage condition for the fair value of the American option requires that the delta hedge factor be continuous. We investigate continuity of the delta factor approximation using the complementarity approach, the binomial method, and the explicit payoff method. We illustrate that, while the (implicit finite difference) complementarity approach yields continuous delta hedge factors, both the binomial method and the explicit payoff method (with the implicit finite difference) yield discontinuous delta approximations. Hence the early exercise curve computed from the binomial method and the explicit payoff method can be inaccurate. In addition, it is demonstrated that the delta factor computed using the Crank-Nicolson method with complementarity approach oscillates around the early exercise curve. ∗ Computer Science Department and Cornell Theory Center, Cornell University, Ithaca, NY 14850. Research partially supported by the Applied Mathematical Sciences Research Program (KC-04-02) of the Office of Energy Research of the U.S. Department of Energy under grant DE-FG02-90ER25013.A000, NSF through grant DMS9805602 and ONR through grant N00014-96-1-0050, and the Finance Industry Solutions Center, an SGI/Cornell joint venture in New York.