Mathematical analysis and numerical methods for a PDE model of a stock loan pricing problem

In this paper the mathematical analysis of a model for pricing stock loan contracts, when the accumulative dividend yield associated to the stock is returned by the lender to the borrower on redemption, is carried out. More precisely, the model is formulated in terms of an obstacle problem associated to a Kolmogorov equation and the existence and uniqueness in the set of solutions with polynomial growth are obtained. Also some regularity properties of the solution are analyzed. Next, for the numerical solution of the problem the combination of Crank–Nicolson Lagrange–Galerkin with the augmented Lagrangian active set method is described. Finally, some numerical examples illustrate the theoretical properties of the optimal redeeming boundary previously stated in the literature.