Title: Mortgage valuation: a quasi-closed-form solution
Abstract: Click to increase image sizeClick to decrease image size Acknowledgements The authors thank the referees for their valuable suggestions and remarks which improved the paper. Notes §It could also be seen, probably even more aptly, as a European compound option, i.e. as a series of European put options with payments made on each of the due dates inherent in the loan (see, for example, Kau et al. (Citation1995) and Azevedo-Pereira et al. (2002, 2003)). †Two exceptions are Collin-Dufresne and Harding (Citation1999), who propose a closed-form solution for fixed rate residential mortgages using a single variable, the short-term interest rate, and Sharp et al. (Citation2008), who use singular perturbation theory to develop a closed-form solution for the value of a mortgage when both default and prepayment are included. †The value of the European option is obtained according to the formula given by Black and Scholes (Citation1973). ‡Some use linear extrapolation, while others utilize exponential extrapolation. †Considering the period to maturity of the contract as being divided into four time intervals gives a most acceptable estimate for the value of an American put option. In mortgage valuation, the same rationale is applied, since the valuation of a default option is conceptually similar to the valuation of an American put option. ‡For an introduction to the use of this type of extrapolation, see Carr (Citation1998). §k is equal to the number of sub-periods in relation to the total number of time intervals, n = 4, considered in the model. In this way, the expression of the mortgage value for time periods is calculated with . † corresponds to the critical value of the property over a period of possession of the contract equal to . ‡Corresponds to the default range defined in the contract. †It is possible to rapidly solve equation (Equation25) using Maple software with ‘fsolve’. ‡See, for example, works by Titman and Torous (Citation1989) and Riddiough and Thompson (Citation1993). †See equation (Equation3).