# 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?

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