Pricing of vanilla and first generation exotic options in the local stochastic volatility framework: survey and new results

摘要

Stochastic volatility (SV) and local stochastic volatility (LSV) processescan be used to model the evolution of various financial variables such as FXrates, stock prices, and so on. Considerable efforts have been devoted topricing derivatives written on underliers governed by such processes. Manyissues remain, though, including the efficacy of the standard alternatingdirection implicit (ADI) numerical methods for solving SV and LSV pricingproblems. In general, the amount of required computations for these methods isvery substantial. In this paper we address some of these issues and propose aviable alternative to the standard ADI methods based on Galerkin-Ritz ideas. Wealso discuss various approaches to solving the corresponding pricing problemsin a semi-analytical fashion. We use the fact that in the zero correlation casesome of the pricing problems can be solved analytically, and develop aclosed-form series expansion in powers of correlation. We perform a thoroughbenchmarking of various numerical solutions by using analytical andsemi-analytical solutions derived in the paper.