Asia-Pacific Financial Markets

, Volume 23, Issue 2, pp 153–174 | Cite as

Analysis of the Nonlinear Option Pricing Model Under Variable Transaction Costs

  • Daniel Ševčovič
  • Magdaléna Žitňanská


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.


Black–Scholes equation with nonlinear volatility Quasilinear parabolic equation Variable transaction costs 

Mathematics Subject Classification

35K15 35K55 90A09 91B28 


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© Springer Japan 2016

Authors and Affiliations

  1. 1.Department of Applied Mathematics and StatisticsComenius UniversityBratislavaSlovakia
  2. 2.Department of Mathematics and Actuarial ScienceUniversity of Economics in BratislavaBratislavaSlovakia

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