Intensity-Based Framework and Penalty Formulation of Optimal Stopping Problems

Abstract

Financial derivatives commonly contain pre-mature termination clauses, which are embedded rights held by the holder or writer. Well known examples of these stopping rights include the early exercise right in American options, callable right in callable securities and pre-payment right in mortgage loans. In this paper, we show how to model the mortgagor's prepayment in mortgage loans and issuer's call in American warrant as event risks using the intensity based approach, where the propensity of prepayment or calling is modeled by the intensity of a Poisson process. We illustrate that the corresponding pricing formulation resembles the penalty approximation approach commonly used in the solution of the linear complementarity formulation of an optimal stopping problem. We obtain several theoretical results on the prepayment strategies of mortgage loans and calling polices of American warrants. We also propose robust second order accurate numerical schemes for solving the penalty formulation of an optimal stopping problem.

Keywords: linear complementarity formulation, mortgage prepayment, optimal stopping

JEL Classification: G13