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Analysis of the Nonlinear Option Pricing Model Under Variable Transaction Costs

Daniel Ševčovič () and Magdaléna Žitňanská ()
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Daniel Ševčovič: Comenius University
Magdaléna Žitňanská: University of Economics in Bratislava

Asia-Pacific Financial Markets, 2016, vol. 23, issue 2, 153-174

Abstract: Abstract 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.

Keywords: Black–Scholes equation with nonlinear volatility; Quasilinear parabolic equation; Variable transaction costs; 35K15; 35K55; 90A09; 91B28 (search for similar items in EconPapers)
Date: 2016
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