Converting Hull–White Short Rates into Swap Rates

Summary

The document explains how to obtain swap rates along simulated paths in a one-factor Hull–White short-rate model. The model describes the short-rate dynamics, and its analytical zero-coupon bond pricing formula relates each bond price to the current short rate through time-dependent model functions. Once those functions are determined, bond prices can be calculated at each simulation time and path.

A vanilla swap rate is then computed from the relevant zero-coupon bond prices: the floating-leg value is represented by the difference between bonds at the swap’s start and end dates, while the fixed-leg annuity is the sum of discounted accrual periods. Their ratio gives the swap rate for each path. The formula shown assumes the valuation time precedes the first swap payment date. The discussion does not provide implementation details for calibrating the model functions or handling conventions such as payment calendars and day-count rules, so those must be specified for an actual pricing setup.

Key ideas

• A Hull–White short-rate simulation can be used to price zero-coupon bonds along each path.
• The bond pricing formula depends on time-specific model functions and the simulated short rate.
• A vanilla swap rate is the floating-leg bond value divided by the fixed-leg annuity.
• Compute the required bond prices and swap rate separately for every Monte Carlo path.
• The stated swap-rate formula assumes valuation occurs before the first swap payment date.

Tags

# Convert Short rate from HW simulation into Swap rates


# 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.

## Answer by rvignolo (score 2)

https://quant.stackexchange.com/a/57836

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.

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.