Lie Symmetry Analysis of the Black-Scholes-Merton Model for European Options with Stochastic Volatility
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 linea...
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MDPI AG
2016-05-01
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| Series: | Mathematics |
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| author | Andronikos Paliathanasis K. Krishnakumar K.M. Tamizhmani Peter G.L. Leach |
| author_facet | Andronikos Paliathanasis K. Krishnakumar K.M. Tamizhmani Peter G.L. Leach |
| author_sort | Andronikos Paliathanasis |
| collection | DOAJ |
| description | 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. |
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| last_indexed | 2024-12-19T04:26:24Z |
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| spelling | doaj.art-5e0b03103cdd4001b6c17e347f530e022022-12-21T20:36:00ZengMDPI AGMathematics2227-73902016-05-01422810.3390/math4020028math4020028Lie Symmetry Analysis of the Black-Scholes-Merton Model for European Options with Stochastic VolatilityAndronikos Paliathanasis0K. Krishnakumar1K.M. Tamizhmani2Peter G.L. Leach3Instituto de Ciencias Físicas y Matemáticas, Universidad Austral de Chile, Valdivia 5090000, ChileDepartment of Mathematics, Pondicherry University, Kalapet 605014, IndiaDepartment of Mathematics, Pondicherry University, Kalapet 605014, IndiaInstitute of Systems Science, Department of Mathematics, Durban University of Technology, Durban 4000, South AfricaWe 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.http://www.mdpi.com/2227-7390/4/2/28lie point symmetriesfinancial mathematicsstochastic volatilityBlack-Scholes-Merton equation |
| spellingShingle | Andronikos Paliathanasis K. Krishnakumar K.M. Tamizhmani Peter G.L. Leach Lie Symmetry Analysis of the Black-Scholes-Merton Model for European Options with Stochastic Volatility Mathematics lie point symmetries financial mathematics stochastic volatility Black-Scholes-Merton equation |
| title | Lie Symmetry Analysis of the Black-Scholes-Merton Model for European Options with Stochastic Volatility |
| title_full | Lie Symmetry Analysis of the Black-Scholes-Merton Model for European Options with Stochastic Volatility |
| title_fullStr | Lie Symmetry Analysis of the Black-Scholes-Merton Model for European Options with Stochastic Volatility |
| title_full_unstemmed | Lie Symmetry Analysis of the Black-Scholes-Merton Model for European Options with Stochastic Volatility |
| title_short | Lie Symmetry Analysis of the Black-Scholes-Merton Model for European Options with Stochastic Volatility |
| title_sort | lie symmetry analysis of the black scholes merton model for european options with stochastic volatility |
| topic | lie point symmetries financial mathematics stochastic volatility Black-Scholes-Merton equation |
| url | http://www.mdpi.com/2227-7390/4/2/28 |
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