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Positive numerical solution for a nonarbitrage liquidity model using nonstandard finite difference schemes

Authors

  • Gilberto González-Parra,

    Corresponding author
    1. Grupo de Matemática Multidisciplinar (GMM), Fac. de Ingeniería, Universidad de los Andes, Mérida, Venezuela
    2. Centro de Investigaciones en Matemática Aplicada (CIMA), Universidad de Los Andes, Mérida, Venezuela
    3. Department of Mathematics, University of Texas at Arlington, Arlington, Texas
    • Correspondence to: Gilberto González-Parra, Grupo de Matemática Multidisciplinar (GMM), Fac. de Ingeniería, Universidad de los Andes, Mérida, Venezuela (e-mail: gcarlos@ula.ve)

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  • Abraham J. Arenasm,

    1. Departamento de Matemáticas y Estadística, Universidad de Córdoba, Montería, Colombia
    2. Grupo Teseeo, Universidad del Sinú, Montería, Colombia
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  • Benito M. Chen-Charpentier

    1. Department of Mathematics, University of Texas at Arlington, Arlington, Texas
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Abstract

In this article, we construct a numerical method based on a nonstandard finite difference scheme to solve numerically a nonarbitrage liquidity model with observable parameters for derivatives. This nonlinear model considers that the parameters involved are observable from order book data. The proposed numerical method use a exact difference scheme in the linear convection-reaction term, and the spatial derivative is approximated using a nonstandard finite difference scheme. It is shown that the proposed numerical scheme preserves the positivity as well as stability and consistence. To illustrate the accuracy of the method, the numerical results are compared with those produced by other methods. © 2013 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 30: 210-221, 2014

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