On Deterministic Shift Extensions of Short Rate Models

Abstract:     
In the present paper we show how to extend any time-homogeneous short-rate model and analytically tractable short-rate model (such as Vasicek (1977), Cox-Ingersoll-Ross (1985), Dothan (1978)) to a model which can reproduce any observed yield curve, through a procedure that preserves the possible analytical tractability of the original model. In the case of the Vasicek (1977) model, our extension is equivalent to that of Hull and White (1990), whereas in the case of the Cox-Ingersoll-Ross (1985) (CIR) model, our extension is more analytically tractable and avoids problems concerning the use of numerical solutions. Our approach can also be applied to the Dothan (1978) or Rendleman and Bartter (1980) model, thus yielding a "quasi" lognormal short-rate model which fits any given yield curve and for which there exist analytical formulae for prices of zero coupon bonds. We also consider the extension of time-homogeneous models without analytical formulae but whose tree-construction procedures are particularly appealing, such as the exponential Vasicek's. We explain why the CIR++ extended CIR model is the more interesting model obtained through our procedure. We also give explicit analytical formulae for bond options, hence swaptions, caps and floors, and we explain how the model can be used for Monte Carlo evaluation of European path-dependent interest-rate derivatives. We finally hint at the same extension for multifactor models and explain its strong points for concrete applications.

Number of Pages in PDF File: 25

Keywords: Short-rate models, Analytical tractability, Yield-Curve fitting, Vasicek's model, Dothan's model, Cox-Ingersoll-Ross' model, Longstaff and Schwartz's model, Monte Carlo evaluation

JEL Classification: G13

working papers series