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Approximating Swap Rate Quantiles in the One-Factor Hull-White Model

Article Quant Q&A · Author: g g
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Summary

This question asks whether swap rate quantiles in the one-factor Hull-White interest-rate model have a convenient closed-form approximation. The model gives a Gaussian short rate and explicit mean and covariance from calibration inputs. Bond prices are log-normal, but a swap rate is a ratio involving differences and sums of bond prices, so its distribution is not immediately available in a simple standard form. The questioner would accept approximation accuracy comparable to empirical quantiles from a large simulation.

The answer says there is no closed-form formula, while pointing out that bond prices and the swap rate can be expressed as functions of the spot rate at the observation time. Since that rate is Gaussian under the model, one can transform its distribution through the swap-rate function to obtain the swap-rate distribution and estimate quantiles. This offers a practical route based on evaluating the function over the Gaussian state distribution. The answer is brief and gives no explicit derivation, numerical procedure, error analysis, or comparison with Monte Carlo, so those implementation details remain unspecified.

Key ideas

  • The one-factor Hull-White model gives a Gaussian distribution for the short rate.
  • Bond prices are log-normal in the model, but the swap rate is a nonlinear ratio of bond prices.
  • The answer states that swap rates can be expressed as functions of the spot rate at the observation time.
  • Transforming the Gaussian spot-rate distribution through that function provides a route to swap-rate quantiles.
  • The response supplies no detailed numerical method or error comparison with simulation.

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Fixed incomeDerivatives pricingStatistics
Full text
# Formula for quantiles of swaprates in the 1-factor Hull-White model


# Formula for quantiles of swaprates in the 1-factor Hull-White model












Is there a closed formula to approximate the quantiles of swaprates in the 1-factor Hull White model?

### Background

The Hull-White is a Gaussian model for the short rate. Its mean and covariance function can be explicitly given in terms of calibration input, i.e. the initial yield curve and estimates for mean reversion strength and volatility. In this model bond prices $B(t,T)$ are log-normal. But this means that swap rates, defined as $$S_{i,j}(t)=\frac{B(t,T_i) - B(t,T_j)}{\sum_{k=i+1}^j B(t,T_k)} $$ have no "simple" distribution.

### Details

- Do the swap rates follow a distribution, whose properties have been analysed and described somewhere?

- Is there a way to estimate quantiles of the swap rate i.e. values s(p) such that the probability $P(S\leq s(p)) = p.$

I guess there will only be approximation possible. Of course, I can always simulate and take empirical quantiles. But a closed formula would be convenient. I am fine with approximation errors in the ballpark of estimates by simulation with sample size a few 100'000.

## Answer by Antoine Conze (score 1)

https://quant.stackexchange.com/a/50232

No closed form formula but the $B(t, T)$ and thus $S(t)$ are functions of the spot rate $r_t$, and $r_t$ has a Gaussian distribution (details in e.g. "brigo mercurio interest rate models theory and practice") so building the distribution of $S(t, r_t)$ is straightforward.

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

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