On the Pricing and Optimal Exercise Boundary of American Options under the HJB Framework

In mathematical finance, the efficient valuation of American-style contracts is of great importance, as well as their exercise boundaries. This paper presents an alternative framework for the solution of the American put option under a finite element framework. An implicit time-stepping method is applied for the temporal discretization, while in space, we provide a non-uniform finite element mesh. The nonlinearity associated with early exercise is represented as an HJB PDE, which is solved by policy iteration at each time grid. Numerical experiments are validated within the benchmarking range. Moreover, optimal exercise boundaries are concurrently computed with the price of the option. We also benchmark the finite element results against finite difference and least-squares Monte Carlo methods, which demonstrates the effectiveness of the approach.