Derive a short rate model from HJM

Question

Suppose we are assuming the HJM framework. My question is, if it is possible to derive for different choices of the volatility function $\sigma$ (and hence of the drift function) the most common short rate models, i.e. Vasicek, CIR,Dothan, Ho-Lee,Hull-White? I know if we choose the volatility function $\sigma$ as constant then we would end up with the Ho-Lee model. I guess that it is possible for exogenous models, but could be rather hard for endogenous such as Vasicek and CIR for example. However I'm particular interested in finding a condition on $\sigma$ such that we can derive from HJM framework the Vasicek model.

Answer 1

It looks as if you are actually asking the following: given a short rate model, how does the HJM volatility function look like. If your short rate model has an analytic bond price formula (many do have this, because this makes them "pratical") then you get the instanteous forward rate from the bonds and via Ito the HJM process and the HJM volatility.

Examples for HJM volatility functions $\sigma(t,T)$:

• Hull-White Model with mean reversion parameter a and volatility $\sigma_{r}$.: Set $\sigma(t,T) = \sigma_{r} * \exp(-a (T-t))$
• Vasicek Model: Same as Hull-White Model (the difference of the two is the initial data of the forward rate curve, not the volatiltiy).
• Ho-Lee Model with volatility $\sigma_{r}$: Set $\sigma(t,T) = \sigma_{r} = constant$ (Ho-Lee is Hull-Whilte with a=0).

For the derivation of the short rate model from the HJM model see Chapter 24 in http://www.amazon.com/Mathematical-Finance-Theory-Modeling-Implementation/dp/0470047224 (you can preview the page with Amazon LOOK INSIDE).