Evaluation of GNMA pass-through securities /

Abstract

The equation derived is a step towards a general theory of security valuation.

The model may be restricted in scope to derive the traditional Black-Scholes option pricing theory, the option pricing formula for equities that pay a continuous dividend, a new option pricing formula for options that are kept in existence by the payment of a continuous fee, and a generalization of the Gordon model for a security paying a growing annuity for a finite or infinite period of time.

The major advantages of the model are that it combines the following features: (1) The irrational behavior of option holders is correctly analysed. (2) The multi-dimensional interest rate process used has a functional form flexible enough to model a wide selection of processes. (3) The state vector is expressed in terms of bond prices rather than interest rates. (4) The market price of risk does not appear in the final model. (5) The drift term of the generating stochastic process does not appear in the final model. (6) The instantaneous rate of interest, which can not be observed, need not appear in the final model. (7) The parameters of the model are to be estimated directly from the differential equation.

A model is derived for the valuation of a wide range of securities using Ito calculus. The prime reason for the development of the model was to evaluate callable bonds that have the call option exercised in an irrational manner such as GNMA pass-through securities, but the model's equation may be used without modification to evaluate any default-free bond, option, or common stock. The model is expressed as a parabolic partial differential equation with n-state variables. This equation may be used to generate the term structure of interest rates.