A comparison study of ADI and operator splitting methods on option pricing models

In this paper we perform a comparison study of alternating direction implicit (ADI) and operator splitting (OS) methods on multi-dimensional Black-Scholes option pricing models. The ADI method is used extensively in mathematical finance for numerically solving multi-factor option pricing problems. However, numerical results from the ADI scheme show oscillatory solution behaviors with nonsmooth payoffs or discontinuous derivatives at the exercise price with large time steps. In the ADI scheme, there are source terms which include y-derivatives when we solve x-derivative involving equations. Then, due to the nonsmooth payoffs, source terms contain abrupt changes which are not in the range of implicit discrete operators and this leads to difficulty in solving the problem. On the other hand, the OS method does not contain the other variable's derivatives in the source terms. We provide computational results showing the performance of the methods for two-asset option pricing problems. The results show that the OS method is very efficient and gives better accuracy and robustness than the ADI method with large time steps.