# Simulating the short rate in the Hull-White model

What is the best way to simulate the short rate $r(t)$ in a simple one factor Hull White process? Suppose I have

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

where $\theta(t)$ is calibrated to swap curve, constants $\alpha$ and $\sigma$ are calibrated to caps using closed form solution for zero-coupon bond options. The best way I can think to do it is an Euler discretisation, that is:

$$r(t+\Delta t) = r(t) + \theta(t)\Delta t - \alpha r(t) \Delta t + \sigma \sqrt {\Delta t} Z$$ where $Z \sim N(0,1)$. In this case, I need $t$ to go from 0 to 10 years, ideally in 0.25 increments. But with Euler, I'd need to use small $\Delta t$, so perhaps 0.025 or less? Once I have string of $r(t)$, I can easily calculate $P(t,T)$ zero coupon bonds.

Appreciate any other ideas or if someone could point me in the right direction. I'm quite new to rates modelling!

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• In fact you can calibrate $\theta(t)$ piecewise constant and $\alpha$ and $\sigma$ to bond prices only. You don't need the swaption prices in mM. If you let $\sigma(t)$ depend on $t$ (this is called the generalized Hull-White model) then you need information about the options market.
• For the model as you write it you don't necessarily need MC to calculate zero-coupon bond prices and thus discount factors. It is not that easy but following procedures as described here could help.
• if you stick to MC: For $\Delta t$ I would use it small like $1/250 \approx 0.004$. This is one time-step per banking-day. You should be able to simulate many paths with all the $10*250=2500$ random variables. Without digging into the theoretical aspects of choosing the step-size. If this takes too long then double the step size $\approx 0.008$. Maybe much bigger steps work too. But this looks natural to me.

A remark: if there were no mean-reversion then I would use much bigger step-sizes. You could take steps up to the coupon dates. But in order to feel mean reversion I would keep the step size small. Another thing is negative rates. In HW you can have them and they exist in reality these days. Again: mean reversion to a non-negative $\theta(t)$ will keep rates positive most of the time if $\Delta t$ is small.

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Thanks, this is somewhat helpful. I ended up purchasing Brigo & Mercurio's book. It gives closed form solution for the conditional distribution of $r(t)$ given its filtration. –  crunch Feb 12 at 15:36
But this is not what you need. You need $E[\exp(-\int_t^T r_u du)|F_t]$ - this is the discount factor. You should find something like this there. –  Richard Feb 13 at 8:03
What about approximating $\int_t^T r_u du$ with $r_t \Delta_t$ as long as $\Delta_t$ remains reasonable? –  crunch Feb 13 at 12:55
Yes ... this should work. I have never implemented this, but it's worth trying. –  Richard Feb 13 at 14:12
thanks for all your help –  crunch Feb 13 at 14:51

Once the single-factor Hull-White model is calibrated, you can compute zero-coupon bond prices in closed form (i.e., without running simulations). See http://en.wikipedia.org/wiki/Hull%E2%80%93White_model#Analysis_of_the_one-factor_model .

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Note that you can also use trees instead of running monte carlo (if a closed form solution is not avaliable)

As far as I know it is even an industry standard to work with the Hull-White tree instead of monte-carlo.

For mote informiation you can have a look at the paper: USING HULL-WHITE INTEREST-RATE TREES

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