Symmetry reduction and exact solutions of the non-linear Black--Scholes equation

Oleksii Patsiuk; Sergii Kovalenko

doi:10.1016/j.cnsns.2018.02.028

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arXiv · 1512.06151

Symmetry reduction and exact solutions of the non-linear Black--Scholes equation

Oleksii Patsiuk·Sergii Kovalenko

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Abstract

In this paper, we investigate the non-linear Black--Scholes equation: $$u_t+ax^2u_{xx}+bx^3u_{xx}^2+c(xu_x-u)=0,\quad a,b>0,\ c\geq0.$$ and show that the one can be reduced to the equation $$u_t+(u_{xx}+u_x)^2=0$$ by an appropriate point transformation of variables. For the resulting equation, we study the group-theoretic properties, namely, we find the maximal algebra of invariance of its in Lie sense, carry out the symmetry reduction and seek for a number of exact group-invariant solutions of the equation. Using the results obtained, we get a number of exact solutions of the Black--Scholes equation under study and apply the ones to resolving several boundary value problems with appropriate from the economic point of view terminal and boundary conditions.

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Oleksii Patsiuk, Sergii Kovalenko. 2015-11-30. Symmetry reduction and exact solutions of the non-linear Black--Scholes equation. https://doi.org/10.1016/j.cnsns.2018.02.028

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