Comparison of the analytical approximation formula and Newton's method for solving a class of nonlinear Black-Scholes parabolic equations

摘要

Market illiquidity, feedback effects, presence of transaction costs, riskfrom unprotected portfolio and other nonlinear effects in PDE based optionpricing models can be described by solutions to the generalized Black-Scholesparabolic equation with a diffusion term nonlinearly depending on the optionprice itself. Different linearization techniques such as Newton's method andanalytic asymptotic approximation formula are adopted and compared for a wideclass of nonlinear Black-Scholes equations including, in particular, the marketilliquidity model and the risk-adjusted pricing model. Accuracy and timecomplexity of both numerical methods are compared. Furthermore, market quotesdata was used to calibrate model parameters.