Pricing options with stochastic volatilities by the local differential quadrature method

A local differential quadrature (LDQ) method to solve the optionpricing models with stochastic volatilities is proposed. The present LDQ method is a newly developed numerical method which preserves the advantage of highorder numerical solution from the classic differential quadrature (DQ) method. The scheme also overcomes the negative effect of the ill-condition for the resultant full matrix and the sensitivity to the grid distribution. It offers a much better approach for finding the optimal order of polynomial approximation when compared to the conventional DQ method. The option-pricing problem under the stochastic volatilities is an important financial engineering topic governed by the Black-Scholes equation, a two-dimensional partial differential equation. For option-pricing problems, it would be helpful to improve the computational efficiency if we adopt the non-uniform grids to reflect the high gradient areas. Besides the requirement of non-uniform grids, a high-order solution is also easy to solve several important parameters such as delta and gamma values in the option-pricing modeling. Based on the advantages of the accuracy of the solution, the efficiency for nonuniform grids, and the appropriation for the regular-domain computation (because of its orthogonal grids), the LDQ method is proved to be very powerful to solve option-pricing problems with the stochastic volatilities. This work will consider two types of option-pricing problems under the stochastic volatilities, such as the standard options and lookback options. For standard options, we will test the effects of the final conditions, while for the lookback options we show the good capability for evaluating the exotic options. The comparisons of the numerical results for three case studies, namely the European standard call, the cash-or-nothing call and the lookback put, all indicate that the LDQ method is a very effective, stable and flexible numerical algorithm for solving the option-pricing models with stochastic volatilities. © 2009 Tech Science Press.