# How does the 2-factor Hull White model propagate the forward rates curve?

I've been trying to get a grasp on some of the basics of interest rate modeling, and am looking to simulate rates using the 2 factor Hull White model, which I am aware offers a more realistic model of rates which allows for imperfect correlation between instantaneous forward rates.

I've found resources on the web which reduce the model to an additive Gaussian one, where one has for the short rate $r(t)$:

$r(t) = \varphi(t) + x_1(t) + x_2(t)$

where $x_1, x_2$ are mean reverting processes governed by:

$dx_1(t) = -a_1x_1(t)\cdot dx + \sigma_1\cdot dW_1(t)$

$dx_2(t) = -a_1x_2(t)\cdot dx + \sigma_2\cdot dW_2(t)$

with $dW_1(t)dW_2(t) = \rho$, and $\varphi(t)$ is deterministic and chosen to fit the initial forward rate curve $f(0,t)$:

$\varphi(t) = f(0,t) + \frac{\sigma_1^2}{2a_1^2}\left(1-e^{-a_1t}\right)^2+\frac{\sigma_2^2}{2a_2^2}\left(1-e^{-a_2t}\right)^2+\rho\frac{\sigma_1\sigma_2}{a_1a_2}\left(1-e^{-a_1t}\right)\left(1-e^{-a_2t}\right)$

This tells me how to simulate the short rate (by updating $x_1,x_2$ at each time increment and adding to $\varphi$), but my question is, how could one simulate the evolution of the whole forward curve? I have also found (unwieldy) closed-form expressions for $P(t,T)$ the price of a term $T$ zero coupon bond at time $t$, from which you can obtain the forward curve, but is there a way to generate the forward curve at time $t+\Delta t$ by updating the curve at time $t$, akin to the way we can do it for the short rate $r(t)$?

-
add comment

## 1 Answer

The Heath-Jarrow-Morton representations of short interest rate models (such as Hull-White) will give you an expression for the evolution of the entire forward curve, but it doesn't make the problem any easier. The closed form ZC formulae you mention above are probably your best bet.

-
I was worried this would be the case. By the way, what does ZC stand for? –  Eddie E. Oct 12 '13 at 2:51
By ZC I mean zero coupone. –  experquisite Oct 14 '13 at 16:29
add comment