DISCRETE AND CONTINUOUS doi:10.3934/dcdss.2020122 DYNAMICAL SYSTEMS SERIES S EXACT SOLUTIONS OF A BLACK-SCHOLES MODEL WITH TIME-DEPENDENT PARAMETERS BY UTILIZING POTENTIAL SYMMETRIES Rehana Naz a and Imran Naeem *b a Centre for Mathematics and Statistical Sciences Lahore School of Economics Lahore, 53200, Pakistan b Department of Mathematics School of Science and Engineering Lahore University of Management Sciences LUMS, Lahore Cantt 54792, Pakistan Abstract. We analyze the local conservation laws, auxiliary (potential) sys- tems, 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 Λ(x, u, ux,uxx). We define two potential vari- ables v and w corresponding to the conserved vectors. We construct two singlet potential systems involving a single potential variable v or w and one couplet potential system involving both potential variables v and w. Moreover, a spec- tral potential system is constructed by introducing a new potential variable pα which is a linear combination of potential variables v and w. 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. 1. Introduction. The Black-Scholes model is one of fundamental model of math- ematical finance [21, 22, 3]. This model is expressed as a linear evolutionary partial differential equation (PDE) having variable coefficients. Gazizov and Ibragimov [15] studied this model in Lie symmetry classification perspective and converted it to the heat equation. Edelstein and Govinder [12] established the conservation laws. They also computed potential symmetries and then constructed exact solutions by utilizing the potential symmetries. Tamizhmani et al. [35] analyzed different forms of the Black-Scholes models subject to the terminal condition u(x, T )= U in Lie symmetry perspective. The BST model [33] for the value of an option satisfies the following PDE u t + 1 2 θ(t) 2 x 2 u xx + r(t)  xu x - u  =0. (1) 2010 Mathematics Subject Classification. 76M60, 83C15, 35L65. Key words and phrases. Conservation laws, potential systems, potential symmetries, black- Scholes model, exact solutions. * Corresponding author: Imran Naeem. 1