Comparison of the Analytical Approximation Formula and Newton's Method for Solving a Class of Nonlinear Black–Scholes Parabolic Equations

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. In this paper, different linearization techniques such as Newton's method and the 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.

Técnica De Funções De Base Radial Acurada Para Soluções Competitivas E Eficientes De Equações Não Lineares De Black-Scholes

“…The main motivation to solve the nonlinear equation of BS with the volatility function of σ(SV ss ) 2 is due to the pricing of more realistic options in which one can take into account the presence of transaction costs, market feedbacks, risks from unprotected portfolio and other effects (Duris, Tan, Lai & Sevcovic, 2015).…”