Markov Chains for Modeling and Pricing Installment Options in Financial Markets

Abstract

In this paper, we apply Markov-modulated models to value continuous-installment options of the European style with a partial differential equation approach. Under regime-switching models and the opportunity for continuing or stopping to pay installments, the valuation problem can be formulated as coupled partial differential equations (CPDE) with free boundary features, which in many ways is similar to the free boundary problem for vanilla American options due to the possibility of early exercise. In this paper, to value the continuous-installment options under the proposed model with a numerical approach, we first express the truncated CPDE as a linear complementarity problem (LCP), and then a finite element method is applied to solve the resulting variational inequality. We studied the existence and uniqueness of the solution and analyzed the stability of the proposed method under some appropriate assumptions, then we illustrated the error estimates on the appropriate spaces. We presented some numerical results to examine the rate of convergence and accuracy of the proposed method for the pricing problem under the regime-switching model.