Pricing bonds in an incomplete market: Linear and Dynamic Programming approach

We consider a finite horizon discrete time model for bond market where bond prices are functions of the short rate process. We use a variant of the Ito's formula to decompose the bond price process into unique drift and martingale processes. We then apply the Girsanov's Theorem for finding a change of measure under which the discounted bond price processes are martingales, thereby implying the existence of an arbitrage-free bond market. We next show that under a particular martingale measure given by a specific form of the Radon Nikodym derivative, the bond price process of exponentially quadratic form reduces to the well known exponentially linear form. We further prove that the bond market is incomplete and and the set of martingale measures is not a singleton. The analytical formulation of all martingale measures is dificult to obtain. A finite discretization of the state space of the rate process and subsequent solution of a set of martingale equations generates the set of all martingale measures in an incomplete bond market. A suitable cost function is then minimized to obtain a particular martingale measure. Linear programming and Dynamic programming approaches for solving the minimization problem are discussed. Assuming compactness of the bond price process, we further prove the convergence of the optimal solution of the discretized problem to the optimal solution of the original problem