  Stochastic volatility (SV) and local stochastic volatility (LSV) processes
can be used to model the evolution of various financial variables such as FX
rates, stock prices, and so on. Considerable efforts have been devoted to
pricing derivatives written on underliers governed by such processes. Many
issues remain, though, including the efficacy of the standard alternating
direction implicit (ADI) numerical methods for solving SV and LSV pricing
problems. In general, the amount of required computations for these methods is
very substantial. In this paper we address some of these issues and propose a
viable alternative to the standard ADI methods based on Galerkin-Ritz ideas. We
also discuss various approaches to solving the corresponding pricing problems
in a semi-analytical fashion. We use the fact that in the zero correlation case
some of the pricing problems can be solved analytically, and develop a
closed-form series expansion in powers of correlation. We perform a thorough
benchmarking of various numerical solutions by using analytical and
semi-analytical solutions derived in the paper.
