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Lie Symmetries of (1+2) Nonautonomous Evolution Equations in Financial Mathematics

Author

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  • Andronikos Paliathanasis

    (Instituto de Ciencias Físicas y Matemáticas, Universidad Austral de Chile, Valdivia 5090000, Chile)

  • Richard M. Morris

    (Department of Mathematics, Institute of Systems Science, Durban University of Technology, PO Box 1334, Durban 4000, South Africa
    These authors contributed equally to this work.)

  • Peter G. L. Leach

    (Department of Mathematics, Institute of Systems Science, Durban University of Technology, PO Box 1334, Durban 4000, South Africa
    School of Mathematics, Statistics and Computer Science, University of KwaZulu-Natal, Private Bag X54001, Durban 4000, South Africa
    Department of Mathematics and Statistics, University of Cyprus, Lefkosia 1678, Cyprus
    These authors contributed equally to this work.)

Abstract

We analyse two classes of ( 1 + 2 ) evolution equations which are of special interest in Financial Mathematics, namely the Two-dimensional Black-Scholes Equation and the equation for the Two-factor Commodities Problem. Our approach is that of Lie Symmetry Analysis. We study these equations for the case in which they are autonomous and for the case in which the parameters of the equations are unspecified functions of time. For the autonomous Black-Scholes Equation we find that the symmetry is maximal and so the equation is reducible to the ( 1 + 2 ) Classical Heat Equation. This is not the case for the nonautonomous equation for which the number of symmetries is submaximal. In the case of the two-factor equation the number of symmetries is submaximal in both autonomous and nonautonomous cases. When the solution symmetries are used to reduce each equation to a ( 1 + 1 ) equation, the resulting equation is of maximal symmetry and so equivalent to the ( 1 + 1 ) Classical Heat Equation.

Suggested Citation

  • Andronikos Paliathanasis & Richard M. Morris & Peter G. L. Leach, 2016. "Lie Symmetries of (1+2) Nonautonomous Evolution Equations in Financial Mathematics," Mathematics, MDPI, vol. 4(2), pages 1-14, May.
    • Handle: RePEc:gam:jmathe:v:4:y:2016:i:2:p:34-:d:70024
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    References listed on IDEAS

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    2. Schwartz, Eduardo S, 1997. "The Stochastic Behavior of Commodity Prices: Implications for Valuation and Hedging," Journal of Finance, American Finance Association, vol. 52(3), pages 923-973, July.
    3. Black, Fischer & Scholes, Myron S, 1972. "The Valuation of Option Contracts and a Test of Market Efficiency," Journal of Finance, American Finance Association, vol. 27(2), pages 399-417, May.
    4. Black, Fischer & Scholes, Myron S, 1973. "The Pricing of Options and Corporate Liabilities," Journal of Political Economy, University of Chicago Press, vol. 81(3), pages 637-654, May-June.
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