Pricing perpetual timer options under Heston Model by finite difference method: Theory and implementation

In this paper a finite difference method (FDM) is provided for pricing perpetual timer options under the Heston volatility model. Considering the degeneracy of the pricing equation, we first prove the existence and uniqueness of the solution of the pricing problem with a new notion of boundary conditions at degenerate boundary and the infinity. Then we discuss the choice of artificial boundary value conditions and obtain a prior estimate of the internal error caused by the boundary value error. This estimate helps to choose appropriate artificial boundary values and solution domain to reduce internal error of the numerical solution. Furthermore, We build a FDM with second-order convergence for the pricing problem. Finally, we implement our method and show the visualization results.