| Abstract: |
| We analyze the Black-Scholes model with time-dependent parameters and it is governed by a parabolic partial differential equation (PDE). First, we compute the Lie symmetries of the Black-Scholes model with time-dependent parameters. It admits six plus infinite many Lie symmetries and thus it can be reduced to the classical heat equation. We utilize the invariant criteria for a scalar linear (1+1) parabolic PDE and obtain two sets of equivalence
transformations. With the aid of these equivalence transformations, the Black-Scholes model with time-dependent parameters transforms to the classical heat equation. Moreover, the functional forms of the time-dependent parameters in the PDE are determined via this method. Then we utilize the equivalence transformations and known solutions of the heat equation to establish a number of exact solutions for the Black-Scholes model with time-dependent parameters. |
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