# Convert Short rate from HW simulation into Swap rates

I am trying to price an exotic option that requires me to simulate 10 yr swap rates. I have calibrated a 1 factor HW model to swaption prices. However, my understanding is that the HW model describes the evolution of short rates. Is there any way or where can I read to find out more on how to convert my simulated short rates in every Monte Carlo path into the swap rate?

Thank you very much in advance for any advice.

## 1 Answer

The Hull-White short rate model (or any other short rate model) describes the short rate dynamics $$dr(t)$$ as well as provide the analytical solution of the zero coupon bond $$P(t, T)$$:

$$P(t, T) = E_t^Q \left[ \exp \left( - \int_t^T r(s) ds \right) \right] = \exp(A(t, T) - B(t, T) \cdot r(t))$$

Depending on the notation you are using, the zero coupon bonds can differ with the previous expression, however, the following still applies. Once you find the solution for $$A(t, T)$$ and $$B(t, T)$$ by means of a system of ordinary differential equations (Riccati System of ODEs), you can compute, for each simulation path, the zero coupon bond at any $$(t, T)$$.

Consider a vanilla swap described by the tenor structure $$T$$ such that $$0 \leq T_1 < T_2 < \dots < T_N$$, its swap rate is given by:

$$S(t) = \frac{P(t, T_1) - P(t, T_N)}{\sum_{n=1}^{N} \tau_n \cdot P(t, T_{n+1})} \quad \text{with } t < T_1,$$

then you can compute, for each path, its swap rate.