Converting Hull–White Discount Factors into Overnight Rates

Summary

The document explains how to relate a short-rate model, such as Hull–White, to a simulated overnight rate. The central distinction is that the model’s one-day zero-coupon bond price is a discount factor, not itself the overnight rate. For a simply compounded daily rate, the discount factor can be converted using the day-count fraction for the period. This provides a route from simulated bond prices to rates such as EONIA.

The answer also notes that the resulting daily rate is close to the model’s instantaneous short rate. Using the short rate directly may be simpler, while the converted daily rate may suit applications that need a compounded overnight convention. The explanation gives a relationship rather than a full implementation: it does not provide software, parameter estimation guidance, or discuss calibration and model limitations. The appropriate choice depends on the intended use and rate conventions.

Key ideas

• A one-day bond price from a short-rate model represents a discount factor, not an overnight rate.
• Convert the discount factor to a simply compounded daily rate using the period’s day-count fraction.
• The converted daily rate is close to the instantaneous short rate.
• Using the short rate may be simpler, while the converted rate reflects daily compounding conventions.

Tags

# Simulating the path for Interest Rate


# Simulating the path for Interest Rate












There are many ways to short term rates like `Ho-lee` process, `HW` process. However I failed to understand how this information can be used to simulate the process for Overnight rate like `EONIA` etc.

Can you please refer to any online technical papers to simulate such Overnight process?

Just to explain more, let say I define a HW process as follows

$dr(t) = (\theta(t)-\alpha r(t))dt+\sigma dW_t$

With this process, I can estimate a discount bond as $P(t, t+1)$ based on the estimated parameters (refer to https://en.wikipedia.org/wiki/Hull–White_model)

So is it right to day that the process for OI rate is just the process for $P(t,t+1)$?

Is there any software implementation like `R/Python` that someone can please refer to?

Thank you very much.

## Answer by piterbarg (score 2, accepted)

https://quant.stackexchange.com/a/61805

You are on the right path but here $P(t,t+1)$ is not your overnight rate but a daily discount factor. To convert to a simply compounded daily rate, which would be your overnight rate, you would do something like this $$ R(t,t+1) = (1 - P(t,t+1))/(\tau P(t,t+1)) $$

Here $\tau$ is your daycount fraction for $1$ day

As I mentioned in the comment $R(t,t+1)$ is very close to the short rate $r(t)$ and, depending on the intended application, you might want to use the latter as it is a bit simpler. Otherwise use $R$

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

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