A finite element method for pricing of continuous-installment options under a Markov-modulated model: existence, uniqueness, and stability of solutions

Abstract

In this paper, we apply Markov-modulated models to value continuous-installment options of European style with 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 numerical approach, we first express the truncated CPDE as a linear complementarity problem (LCP), then a finite element method is applied to solve the resulting variational inequality. Under some appropriate assumptions, we establish the stability of the method and illustrate some numerical results to examine the rate of convergence and accuracy of the proposed method for the pricing problem under regime-switching model.