Constrained viscosity solution to the HJB equation arising in perpetual American employee stock options pricing

We consider the valuation of a block of perpetual ESOs and the optimal exercise decision for an employee endowed with them and with trading restrictions. A fluid model is proposed to characterize the exercise process. The objective is to maximize the overall discount returns for the employee through exercising the options over time. The optimal value function is defined as the grant-date fair value of the block of options, and is then shown by the dynamic programming principle to be a continuous constrained viscosity solution to the associated Hamilton-Jacobi-Bellman (HJB) equation, which is a fully nonlinear second order elliptic partial differential equation (PDE) in the plane. We prove the comparison principle and the uniqueness. The numerical simulation is discussed and the corresponding optimal decision turns out to be a threshold-style strategy. These results provide an appropriate method to estimate the cost of the ESOs for the company and also offer favorable suggestions on selecting right moments to exercise the options over time for the employee.