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

Andronikos Paliathanasis, K. Krishnakumar, K.M. Tamizhmani and Peter G.L. Leach
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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
K.M. Tamizhmani: Department of Mathematics, Pondicherry University, Kalapet 605014, India
Peter G.L. Leach: Institute of Systems Science, Department of Mathematics, Durban University of Technology, Durban 4000, South Africa

Mathematics, 2016, vol. 4, issue 2, 1-14

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.

Keywords: lie point symmetries; financial mathematics; stochastic volatility; Black-Scholes-Merton equation (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2016
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