Chebyshev spectral method for singular moving boundary problems with application to finance

Abstract

Accurate results for the inherently nonlinear models involving moving boundaries can only be produced by sophisticated, high-quality numerical algorithms. However the most general approaches are usually not very accurate, while those producing accurate results for certain cases are hard to generalize. In an attempt to bridge this gap, we propose a general method, based on a unified framework for arbitrary parabolic operators, which, in particular, can accurately treat singular problems. Our method consists of front-fixing and a Chebyshev series solution of the resulting nonlinear partial differential equation. An appropriate set of convergent smooth approximations is used in singular cases. For smooth problems, our method is very competitive in both speed and accuracy. At the same time, our method is able to produce accurate solutions in the most general setting, whenever existence theorems for moving boundary problems hold. We establish convergence of numerical solutions to the true solution for a large class of possibly singular initial conditions. In addition to the general method, we introduce computational techniques which enhance its performance for singular problems. These include derivative evaluations with Pad'e approximations; prior integration in time; and domain decomposition. We demonstrate the performance of our method with several regular and singular problems. A comparison with other methods shows that our algorithms produce more accurate results. The additional techniques, which do not use smoothing approximations, significantly shorten computing times while retaining reasonable accuracy. We present a systematic study of the mathematical finance problem of pricing American options on a dividend-paying asset from the point of view of partial differential equations. A symmetry result, obtained via a simple change of variables, allows to reduce any American option problem to one of the two canonical cases, depending on the relation between the interest rate and the dividend yield. Each of these cases is equivalent to a singular Stefan problem, which can be solved by our method. We present calculations for the classical problems of options written on a single stock and the more complicated examples, such as index options and foreign currency options, thus demonstrating the remarkable practical scope of the proposed approach.