I am looking for some insights and worked-out example on how exactly should I estimate the Mean reversion parameter of the One factor Hull White model

This link suggests to fit some regression equation ().equation number 2 : https://link.springer.com/content/pdf/10.1007/978-3-030-13463-1_2

Let say I have historical values of Zero curve for last one year. In that case what values should I chose for $r\left(t\right)$ and $r\left(s\right)$?

Any direction would be very helpful.


2 Answers 2


This is classic.

For pricing, under RN measure, MR (curve) need to be calibrated to market data. If you calibrate jointly to caps and swaptins, you want the vol curve to mostly pick up expires, and MR curve to mostly pick up maturities.

Under historical measure, you are never really calibrating HW. Historical measure equivalent of HW, or rather Vasicek is AR(1) process

$r_{n+1} = A + B r_n + \sigma \epsilon_n$

Here all coefficients are constant. Vasicek is reduced to that by discretising

$dr_t \approx r_{n+1} - r_n$, and then moving $r_n$ to the RHS, collecting and introducing the discredited coefficients.

Now this equation can be estimated by OLS, but, if you care, you need to be careful with the confidence intervals, as in OLS the standard assumption is that the explanatory variable is not stochastic. Here it is, as you regress the series on itself lagged.

This is not an issue in practice, and you want B<1 for the series to be stationary.

Normally this is the approach to build "Hostorical LMM", but if you want to find a historical estimation for a single MR number in HW, then you need to choose some very short term rate as a proxy for the short rate and do the described estimation to that curve's time series. Then you fix this parameter in your HW and calibrate vol conditioned on that MR speed.


Typically, the zero curve that is used for calibration is originally backed-out from market prices. Then it’s interpolated and used to calibrate the mean-reversion parameter. This is basically the market's interpretation of how interest rate will move in the future.

This paper is trying to use historical data to calibrate the mean-reversion parameter because they believe the reversion behaviour is quite consistent.

The process is not much different than calibrating any other model, e.g. the Heston, except we require different sets of data for the different parameters.

$s$ is just a previous time less than $t$, from the interpolated curve.

You are getting confused because you usually see $t$ on both sides for:

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

Here the LHS is the increment at time $t$, which uses the price at time $t$

Instead, equation 2 is:

$$r(t) = … r(s)…$$

The LHS isn’t the increment, and it makes no sense as to why $t(t)$ would also be a function of $t$ on the RHS.

  • $\begingroup$ Thanks. But not sure I understand your statement. I have 2 confusions 1) How exactly the HW model would translate to the equation 2) in that paper? and 2) In real calibration case, what value if s should I take? $\endgroup$
    – Bogaso
    Commented May 10 at 15:40

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