Ho and Lee have proposed a family of interest rate models. Perhaps, the most general form of the normal model is the n-factor model. The model allows for n orthogonal yield curve movements such that the model remains a normal process. This generalization allows the yield curve the flexibility to take on different shapes and can fit the volatility surface observed in the market.
The model is provided in the discrete time formulation. This paper provides the continuous time formulation of the models. In the continuous time model, we have the advantage of studying the model analytically more effectively and can better compare the model with other standard models.
Specifically, we show that for a given set of orthogonal movements, we can derive the bond price process, the instantaneous forward price process and the short term rate process.
One important application of this continuous time model is to better understand the specifications of these orthogonal yield curve movements. Specifically, we investigate a particular set of the movement functional form. We assume that the first function is a constant function, the second function is an exponential decay function, and the third is a u – shaped function. Under this set of function, we then analyze the implied volatility surface (the instantaneous volatility of the forward rate expiring t delivering a T* maturity bond). Further, we can study the instantaneous correlations of the x year bond and the y year bond. Understanding the specifications of this set of functions enable us to decide on the optimal functional forms for the n-factor Ho-Lee model.
