Exact solutions of a Black-Scholes model with time-dependent parameters by utilizing potential symmetries

Abstract

We analyze the local conservation laws, auxiliary (potential) systems, potential symmetries and a class of new exact solutions for the Black-Scholes model time-dependent parameters (BST model). First, we utilize the computer package GeM to construct local conservation laws of the BST model for three different forms of multipliers. We obtain two conserved vectors for the second-order multipliers of form \begin{document}$ \Lambda(x,u,u_x,u_{xx}) $\end{document} . We define two potential variables \begin{document}$ v $\end{document} and \begin{document}$ w $\end{document} corresponding to the conserved vectors. We construct two singlet potential systems involving a single potential variable \begin{document}$ v $\end{document} or \begin{document}$ w $\end{document} and one couplet potential system involving both potential variables \begin{document}$ v $\end{document} and \begin{document}$ w $\end{document} . Moreover, a spectral potential system is constructed by introducing a new potential variable \begin{document}$ p_\alpha $\end{document} which is a linear combination of potential variables \begin{document}$ v $\end{document} and \begin{document}$ w $\end{document} . The potential symmetries of BST model are derived by computing the point symmetries of its potential systems. Both singlet potential systems provide three potential symmetries. The couplet potential system yields three potential symmetries and no potential symmetries exist for the spectral potential system. We utilize the potential symmetries of singlet potential systems to construct three new solutions of BST model.