Inferring the Ho–Lee Short-Rate Jump Size from Volatility

Summary

The document discusses how to infer the perturbation factor in a discrete-time Ho–Lee interest-rate tree when the initial bond term structure, risk-neutral up probability, and short rates are known. It gives a relationship between the logarithm of the inverse perturbation factor, short-rate volatility, and the up-jump probability, linking the tree’s jump size to the dispersion of short rates.

The answer says that the perturbation factor can be calculated if the short-rate volatility is available, and suggests estimating that volatility from a populated rate tree. It does not provide a worked calculation or specify conventions for measuring variance, so users must check that the relationship matches their tree setup. The discussion also notes that the basic model uses constant short-rate volatility, while allowing it to vary over time adds complexity.

Key ideas

• The Ho–Lee perturbation factor is related to short-rate volatility and the risk-neutral probability of an up jump.
• Short-rate volatility is needed to infer the perturbation factor from the stated relationship.
• A populated short-rate tree may provide the values needed to estimate short-rate variance.
• The basic model assumes constant short-rate volatility, while time-varying volatility complicates the setup.

Tags

# Discrete time Ho lee model


# Discrete time Ho lee model












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 by Vytautas (score 1)

https://quant.stackexchange.com/a/3029

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.

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.