Valuation of Mortgages by Using Lévy Models

In a mortgage valuation model, the early termination (i.e., prepayment and default) hazard rates and the recovery rate can be specified as multivariate affine functions that include the correlated stochastic state variables. For good capturing of the distributions for state variables, we specify that the state variables follow Lévy models. Accordingly, the early termination hazard rates and the recovery rate also follow Lévy models. Three popular Lévy models, the normal, Variance Gamma (VG), and Negative Inverse Normal (NIG), were used to obtain the closed-form pricing formula for a mortgage. We also conduct numerical applications. Our results reveal the following findings: first, VG model is better than the normal and NIG models in fitting the actual distributions of the interest rate and the housing return rate. Thus, mortgage valuation using a VG model should be better than that using the other two models. Second, the mortgage value estimated by the normal model is the lowest among the three Lévy models. Third, a prepayment hazard rate with a deterministic value (e.g., using the PSA prepayment model) could result in an unreasonable mortgage duration. Fourth, the mortgage convexity calculated by the normal model is higher than that in the other two Lévy models. Our general pricing formula for a mortgage as described in this study can help market participants accurately value mortgages and effectively manage their risks.