Optimal Mortgage Prepayment Under the Cox-Ingersoll-Ross ModelJones, Christopher (2013) Optimal Mortgage Prepayment Under the Cox-Ingersoll-Ross Model. Doctoral Dissertation, University of Pittsburgh. (Unpublished)
AbstractWe study a parabolic variational inequality and associated free boundary problem (FBP) with financial applications. We consider a mortgage contract having the early termination option via prepayment under the Cox-Ingersoll-Ross (CIR) model for the short rate. The problem is of significant interest from both mathematical and economic points of view. The main difficulty is that the Black-Scholes-type partial differential equation (PDE) associated with the problem is not uniformly parabolic and therefore standard approaches have to be modified to treat the degeneracy of the PDE. Our main results are the existence and uniqueness of a solution to the variational inequality and the associated FBP; the free boundary represents optimal prepayment rate for the mortgage contract at each time prior to expiry. We establish regularity of the free boundary; several other properties of the free boundary are studied as well as the infinite horizon problem. From the financial point of view, the infinite horizon problem is interpreted as the mortgage prepayment problem when there is a long time to expiration. In solving the infinite horizon problem we obtain a critical rate, such that for mortgage rates below this critical rate, prepayment is never an optimal decision for a long times to expiration. With the existence of a unique solution to the FBP established, we conclude by performing a calibration of the CIR model to constant maturity Treasury yields and investigate numerical aspects of the variational inequality. Share
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