An Arbitrage-Free Estimate of Prepayment Option Prices in Fixed-Rate GNMA Mortgage-Backed Securities

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Standard arbitrage arguments imply that any two assets, with identical payoffs in all future states of the world, must have the same present value. This logic has been successfully used to model contingent claims on a broad range of financial assets. This argument also implies that in an efficient market, the difference in prices between assets must equal the market value of all differences in their future payoffs. Although cash flow differences between assets are often so disparate that this observation is of little practical value, for assets in which the future payoffs differ by only one attribute, this observation provides a means to isolate the market price of that attribute. We use this reasoning to extract prices for the prepayment option embedded in virtually every home mortgage in the United States.

A substantial literature has developed to explain prepayment because the mortgage-backed securities market is huge, and prepayment is the key obstacle to pricing mortgage-backed assets. Past research has predominantly focused on explaining the observed rates of prepayment. Studies which estimate prepayment rates in order to use the estimates in broader mortgage valuation models include a study of adjustable rate mortgage (ARM) prepayment rates by Bartholomew, Berk and Roll (1988), and studies of fixed-rate mortgage prepayments by Arak and Goodman (1985), Curley and Guttentag (1977), Richard and Roll (1989), and Schwartz and Tourous (1989). In a similar vein, Green and Shoven (1986) estimate prepayment rates, but only explore their effect on future cash flows rather than present value, while Peters, Pinkus and Askin (1984), Milonas and Lacey (1988) and Navratil (1985) present estimates of prepayment rates without further considering the impact on value. Most Wall Street firms and money center banks have also developed proprietary prepayment rate models. Heuson (1987) takes a different tack by estimating Government National Mortgage Association (GNMA) prices as a function of variables thought to be important to prepayment. Estimating prepayment rates has been pursued because once prepayment rates are projected, they can be used to forecast future cash flows, and once the cash flows are estimated, a value can be calculated. Many studies, such as Berk and Roll (1988), Kau (1986), Davidson and Lim (1987, 1988) and Waldman and Modzelewski (1986) implement this approach through Monte Carlo simulation.

A second line of prepayment research has evolved from combining interest rate models with theoretical models of optimal prepayment. Continuous-time models including prepayment have been developed by Brennan and Schwartz (1985), Dunn and McConnell (1981a,b), Cox, Ingersol and Ross (1980), Schwartz and Torous (1992) and Kau, Keenan, Muller and Epperson (1985). Discrete-time binomial interest rate models including prepayment have been explored by Hall (1985), Pozdena and Ben Iben (1984), Leung and Sirmans (1990) and Rubinstein (1990). The thrust of this literature has been to value mortgage-backed securities as a contingent claim on interest rates with an attached prepayment option.

A third line of thought on prepayment focuses on the yield spread between GNMAs and Treasury Bonds. In principle, an appropriately selected long term Treasury Bond can have essentially the same duration and (zero) default risk as a GNMA mortgage pool. Based on this, past researchers have either modeled the difference in yield between long term Treasury securities and GNMA pools as a function of the prepayment option, or explained prepayment rates using the difference in yields between these assets as one of the explanatory variables. For example, Hendershott, Shilling and Villani (1983), Milonas (1987) and Rothberg, Nothaft and Gabriel (1989) explain the GNMA-Treasury Bond spread, while Brennan and Schwartz (1985) explore the impact of different model assumptions on the yield differences between Treasuries and GNMAs.

All of the above approaches can yield fruitful results, limited primarily by the assumptions needed to support the models. …