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Article

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

by 2,†, 2,† and 3,4,5,†
1
Instituto de Ciencias Físicas y Matemáticas, Universidad Austral de Chile, Valdivia 5090000, Chile
2
Department of Mathematics, Pondicherry University, Kalapet 605014, India
3
Institute of Systems Science, Department of Mathematics, Durban University of Technology, Durban 4000, South Africa
4
School of Mathematics, Statistics and Computer Science, University of KwaZulu-Natal, Durban 4000, South Africa
5
Department of Mathematics and Statistics, University of Cyprus, Lefkosia 1678, Cyprus
*
Author to whom correspondence should be addressed.
These authors contributed equally to this work.
Mathematics 2016, 4(2), 28; https://doi.org/10.3390/math4020028
Received: 30 January 2016 / Revised: 14 April 2016 / Accepted: 15 April 2016 / Published: 3 May 2016
We perform a classification of the Lie point symmetries for the Black-Scholes-Merton Model for European options with stochastic volatility, σ, 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, $σ ( y )$ , 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 $σ ( y ) = σ 0$ 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. View Full-Text
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MDPI and ACS Style

Paliathanasis, A.; Krishnakumar, K.; Tamizhmani, K.M.; Leach, P.G.L. Lie Symmetry Analysis of the Black-Scholes-Merton Model for European Options with Stochastic Volatility. Mathematics 2016, 4, 28. https://doi.org/10.3390/math4020028

AMA Style

Paliathanasis A, Krishnakumar K, Tamizhmani KM, Leach PGL. Lie Symmetry Analysis of the Black-Scholes-Merton Model for European Options with Stochastic Volatility. Mathematics. 2016; 4(2):28. https://doi.org/10.3390/math4020028

Chicago/Turabian Style

Paliathanasis, Andronikos, K. Krishnakumar, K.M. Tamizhmani, and Peter G.L. Leach 2016. "Lie Symmetry Analysis of the Black-Scholes-Merton Model for European Options with Stochastic Volatility" Mathematics 4, no. 2: 28. https://doi.org/10.3390/math4020028

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