# Derive a short rate model from HJM

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.

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## 1 Answer

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).

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Thanks for your answer. I had a look at the chapter. I just skimmed through,but as far as I see, you do exactly this two approaches ($\sigma$ constant and $\sigma(t,T)=\sigma\exp{(-a(T-t)))}$. However, I do not see, how you should choose $\sigma(t,T)$ in general to obtain the right short rate dynamics. At least for me, it is not an obvious choice for $\sigma(t,T)$ to get the Hull-White model. Assuming HJM, I can write down the dynamics of the short rate. But in this formula just $\sigma(t,t)$ appears. And it seem rather difficult to get from $\sigma(t,t)$ to $\sigma(t,T)$, or am I wrong? – hulik Jan 16 at 8:16

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The Chapter in the book looks at "Given an HJM volatility function, what does the short rate process look like". You can always do this, but in general you will obtain a path dependent drift for the short rate and hence the model is no longer Markovian. If "short rate model" is short for "Markovian short rate model", we wouldn't call this a short rate model. - As I mentioned in my answer, you are looking at the reverse: Given a short rate model, what is HJM process. This can be calculated as described via Ito's lemma. – Christian Fries Jan 16 at 11:59
thats some cool use of MathJax Christian :) Do you know if its any good to use in Google Docs vs the Google Docs equation editor? – Nikos Jan 19 at 22:14
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