Discrete time Ho lee model

Question

This is my first question in this forum. I am stuck with my current testing the Ho Lee model. I am having difficulty computing the perturbation factor $\Delta$.

The ho lee model should be completely determined by the initial term structure $B(0,1),B(0,2),...$ its risk neutral probability of an up jump $p$ (which is independent of time and should be the same at each node through out the tree), and the perturbation factor $\Delta=\frac{h(1;u)}{h(1;d)}$.

Now I am given the task of knowing the initial term structure $B(0,1),B(0,2),...$, $p$ and short rates $r(0)$ and $r(1;u)$, and have to compute the perturbation factor delta instead. Any help is much appreciated.

Answer 1

There is a relationship: $ \log \delta^{-1} = \frac{\sqrt{Var[r(t)]}}{\sqrt{p(1-p)}}$

Which relates the jump size to the volatility of short rate and risk neutral jump probability.

The vol of short rate is chosen to be const in basic model, could be time-varying, but makes things complicated.

To solve for $\delta$ you do need the vol of short rate given initially. What exactly are those $r(1;u)$ that you have? you have a populated interest rate tree?

Then you should be able to compute the variance of your short rate from them.