Abstract
After the latest world-wide financial crisis, Residential Mortgage-Backed Securities (RMBS) become more and more important in financial instruments for long-term investment. Since RMBS have longer maturities and more frequent payoffs than the ordinary bonds, it is desired to evaluate their values quickly in business transactions, and especially in risk managements where valuations are repeated under various kinds of economic conditions. In this article, we derive semi-analytical pricing formulas of RMBS where prepayment rates have a term structure and a dependence on the interest rates. As an stochastic interest rate model, we adopt CIR++ model or Quadratic Gaussian++ model because their future interest rates become positive and these models can perfectly fit the initial term structure of the market rates. Moreover, we consider the time-dependence of the interest rate sensitivities of prepayment rates. The technical key points are the forward-neutral pricing method and the Feynman-Kac theorem. The obtained pricing formulas have three different expressions, and the interest-rate sensitivity of prepayment rates determines which one is used among the three expressions. All formulas can be calculated so quickly that they would be useful in practice.
