Mortgage valuation: a quasi-closed-form solution

Abstract

The main objective of this study consists in developing a quasi-analytical solution for the valuation of commercial mortgages. We consider the existence of a single source of riskthe risk of defaulting on a mortgage -and therefore, the existence of a single state variable -the value of the mortgaged property. The value of the mortgage corresponds to the present value of the future payments on the loan, minus the value of the embedded American default option. The major difficulty in designing such a model consists in calculating the value of this option, since for that purpose it is necessary to determine the lowest property price below which it must be immediately exercised, i.e. the critical value of the property.Under such a framework, the partial differential equation for the value of the default option is that presented in Black and Scholes (1973). The boundary conditions applicable to the value of the default option are naturally different, given the possibility of early default. To obtain a quasi-closed form solution, instead of continuous time framework, a discrete time setting is considered for the differential equation -the "Method of Lines". The partial differential equation is transformed into an ordinary non-homogenous differential equation, replacing the derivative of the function [price of the option in relation to time] with a finite difference, and keeping the derivatives of the mortgaged property unchanged. In order to improve the performance of the model, Richardson's extrapolation is applied, allowing for the increase in the speed of convergence of the model results towards reasonable values.As compared to the alternative numerical valuation techniques, the proposed quasi-closed form solution provides much quicker mortgage valuation results and enhances the ability to perform the corresponding sensitivity analysis. The current work constitutes one of the first attempts to address the development of analytical solutions to mortgage valuation, using contingent claims analysis.