Symmetry analysis of a model of stochastic volatility with time-dependent parameters

摘要

We provide the solutions for the Heston model of stochastic volatility when the parameters of the model are constant and when they are functions of time. In the former case, the solution follows immediately from the determination of the Lie point symmetries of the governing 1+1 evolution partial differential equation. This is not the situation in the latter case, but we are able to infer the essential structure of the required nonlocal symmetry from that of the autonomous problem and hence can present the solution to the nonautonomous problem. As in the case of the standard Black–Scholes problem the presence of time-dependent parameters is not a hindrance to the demonstration of a solution.