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

TL;DR

This paper compares the accuracy and efficiency of an analytical approximation formula and Newton's method for solving nonlinear Black-Scholes equations, including models with market illiquidity and risk adjustments, using real market data.

Contribution

It provides a systematic comparison of two linearization techniques for nonlinear Black-Scholes PDEs, highlighting their relative advantages and calibration with market data.

Findings

Analytical approximation offers faster solutions with acceptable accuracy.

Newton's method achieves higher precision but at increased computational cost.

Calibration with market data validates the practical applicability of both methods.

Abstract

Market illiquidity, feedback effects, presence of transaction costs, risk from unprotected portfolio and other nonlinear effects in PDE based option pricing models can be described by solutions to the generalized Black-Scholes parabolic equation with a diffusion term nonlinearly depending on the option price itself. Different linearization techniques such as Newton's method and analytic asymptotic approximation formula are adopted and compared for a wide class of nonlinear Black-Scholes equations including, in particular, the market illiquidity model and the risk-adjusted pricing model. Accuracy and time complexity of both numerical methods are compared. Furthermore, market quotes data was used to calibrate model parameters.