Simulating Short Rates and Pricing Bonds in the Hull–White Model

# Simulating the short rate in the Hull-White model


# 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!

## Answer by Richi Wa (score 6, accepted)

https://quant.stackexchange.com/a/10198

- 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.

## Answer by wsw (score 2)

https://quant.stackexchange.com/a/10773

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 .

## Answer by Probilitator (score 1)

https://quant.stackexchange.com/a/10780

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

## Answer by ZRH (score 1)

https://quant.stackexchange.com/a/34647

Coming across the post somewhat late: I attempted the same, and had Bloomberg caplet data for calibration (6mth EURIBOR) at hand. I calibrate directly via MC simulation (Euler, as suggested by crunch): Starting off with current 6mth EURIBOR, choosing $\theta(t)$ to match curve implied forward 6mth EURIBOR, forward-stepping until the first caplet expiry and computing payoffs. I then vary sigma for this interval to match the caplet premium accurately. To match the next caplet premium, I start off the MC with the realizations of 6mth EURIBOR at the first caplet expiry. Again, varying sigma for the second interval until the second caplet premium is matched etc.

Perhaps not overly elegant (investing a bit of time, one may find an analytic way to work out the piecewise-constant $\sigma$ from Ornstein-Uhlenbeck formulae), but it is stable and matches caplet prices very well. For calibrating a ten-year cap on my laptop, and running the MC in Matlab, about 10s computation time will do.