Analysis of the Nonlinear Option Pricing Model Under Variable Transaction Costs

• Published in: Asia-Pacific financial markets Vol. 23; no. 2; pp. 153 - 174
• Main Authors: Sevcovic, Daniel, Itanská, Magdaléna
• Format: Journal Article
• Language: English
• Published: Tokyo Springer Japan 01-06-2016; Springer Nature B.V
• Subjects: Applied mathematics; Costs; Econometrics; Economic models; Economic theory; Economic Theory/Quantitative Economics/Mathematical Methods; Economics and Finance; Finance; Hedging; International; International Economics; Macroeconomics/Monetary Economics//Financial Economics; Options markets; Put & call options; Securities prices; Volatility; Asia; Variable transaction costs; Quasilinear parabolic equation; 90A09; 35K15; 91B28; Black–Scholes equation with nonlinear volatility; 35K55
• ISSN: 1387-2834; 1573-6946
• DOI: 10.1007/s10690-016-9213-y

In this paper we analyze a nonlinear Black–Scholes model for option pricing under variable transaction costs. The diffusion coefficient of the nonlinear parabolic equation for the price
V
is assumed to be a function of the underlying asset price and the Gamma of the option. We show that the generalizations of the classical Black–Scholes model can be analyzed by means of transformation of the fully nonlinear parabolic equation into a quasilinear parabolic equation for the second derivative of the option price. We show existence of a classical smooth solution and prove useful bounds on the option prices. Furthermore, we construct an effective numerical scheme for approximation of the solution. The solutions are obtained by means of the efficient numerical discretization scheme of the Gamma equation. Several computational examples are presented.