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Lie Symmetry Analysis of the Black-Scholes-Merton Model for European Options with Stochastic Volatility

Author

Listed:
  • Andronikos Paliathanasis

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

  • K. Krishnakumar

    (Department of Mathematics, Pondicherry University, Kalapet 605014, India
    These authors contributed equally to this work.)

  • K.M. Tamizhmani

    (Department of Mathematics, Pondicherry University, Kalapet 605014, India
    These authors contributed equally to this work.)

  • Peter G.L. Leach

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

Abstract

We perform a classification of the Lie point symmetries for the Black-Scholes-Merton Model for European options with stochastic volatility, ? , in which the last is defined by a stochastic differential equation with an Orstein-Uhlenbeck term. In this model, the value of the option is given by a linear (1 + 2) evolution partial differential equation in which the price of the option depends upon two independent variables, the value of the underlying asset, S , and a new variable, y . We find that for arbitrary functional form of the volatility, ? ( y ) , the (1 + 2) evolution equation always admits two Lie point symmetries in addition to the automatic linear symmetry and the infinite number of solution symmetries. However, when ? ( y ) = ? 0 and as the price of the option depends upon the second Brownian motion in which the volatility is defined, the (1 + 2) evolution is not reduced to the Black-Scholes-Merton Equation, the model admits five Lie point symmetries in addition to the linear symmetry and the infinite number of solution symmetries. We apply the zeroth-order invariants of the Lie symmetries and we reduce the (1 + 2) evolution equation to a linear second-order ordinary differential equation. Finally, we study two models of special interest, the Heston model and the Stein-Stein model.

Suggested Citation

  • Andronikos Paliathanasis & K. Krishnakumar & K.M. Tamizhmani & Peter G.L. Leach, 2016. "Lie Symmetry Analysis of the Black-Scholes-Merton Model for European Options with Stochastic Volatility," Mathematics, MDPI, vol. 4(2), pages 1-14, May.
    • Handle: RePEc:gam:jmathe:v:4:y:2016:i:2:p:28-:d:69344
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    References listed on IDEAS

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