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Mathematics 2016, 4(2), 34; doi:10.3390/math4020034

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

1
Instituto de Ciencias Físicas y Matemáticas, Universidad Austral de Chile, Valdivia 5090000, Chile
2
Department of Mathematics, Institute of Systems Science, Durban University of Technology, PO Box 1334, Durban 4000, South Africa
3
School of Mathematics, Statistics and Computer Science, University of KwaZulu-Natal, Private Bag X54001, Durban 4000, South Africa
4
Department of Mathematics and Statistics, University of Cyprus, Lefkosia 1678, Cyprus
These authors contributed equally to this work.
*
Author to whom correspondence should be addressed.
Academic Editor: Indranil SenGupta
Received: 30 January 2016 / Revised: 20 April 2016 / Accepted: 3 May 2016 / Published: 13 May 2016
(This article belongs to the Special Issue Mathematical Finance)
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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. View Full-Text
Keywords: lie point symmetries; financial mathematics; prices of commodities; black-scholes equation lie point symmetries; financial mathematics; prices of commodities; black-scholes equation
This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. (CC BY 4.0).

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MDPI and ACS Style

Paliathanasis, A.; Morris, R.M.; Leach, P.G.L. Lie Symmetries of (1+2) Nonautonomous Evolution Equations in Financial Mathematics. Mathematics 2016, 4, 34.

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