Valuation of Real Options by the Gradient Projection Method

Most real options models are American-type options involving a free boundary problem which can be modeled in the form of variational inequalities. In this paper, we provide a viable mathematical formulation and promising computational approach for the valuation of real options. We study an equivalent optimization problem with an inequality constraint and boundary conditions, whose necessary conditions are the variational inequality articulation of an American option. The objective functional is defined in a Hilbert space and includes a partial differential operator. We propose gradient projection methods for both infinite and finite time horizon real option problems, while we approximate the partial derivatives by finite differences. We test the performance and accuracy of the proposed algorithm, and compare with an existing method, the projective successive over-relaxation.