In this paper, we consider a model of valuing callable financial securities when the underlying asset price dynamic depends on a finite-state Markov chain. The callable securities enable both an issuer and an investor to exercise their rights to call. We formulate this problem as a coupled stochastic game for the optimal stopping problem with two stopping boundaries. Then, we show that there exists a unique optimal price of the callable contingent claim which is a unique fixed point of a contraction mapping. We derive analytical properties of optimal stopping rules of the issuer and the investor under general payoff functions by applying a contraction mapping approach. In particular, we derive specific stopping boundaries for the both players by specifying for the callable securities to be the callable American call and put options.

In this paper, we propose a primal-dual interior point method for nonlinear semidefinite programming problems and show its superlinear convergence. This method is based on generalized shifted barrier Karush-Kuhn-Tucker (KKT) conditions, which include barrier KKT conditions and shifted barrier KKT conditions as a special case. This method solves two Newton equations in a single iteration to guarantee superlinear convergence. We replace the coefficient matrix of the second Newton equation with that of the first to reduce the computational time of the single iteration. We show that the superlinear convergence of the proposed method with the replacement under the usual assumptions.