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


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.


  • 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.


1 Answer 1


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.

  • $\begingroup$ What do you mean by "building the distribution"? $\endgroup$
    – g g
    Dec 11, 2019 at 13:42
  • $\begingroup$ Assuming $S(r)$ is an increasing function of $r$, $P(S(r_t) < s) = P(r_t < S^{-1}(s))$ $\endgroup$ Dec 11, 2019 at 13:45
  • 1
    $\begingroup$ You claim $S(r)$ is always an increasing function of $r$ in the Hull-White model? How would you show that? $\endgroup$
    – g g
    Dec 11, 2019 at 15:06
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    $\begingroup$ I haven't made any such claim. And usually $S(r)$ is not monotonic on $]-\infty, +\infty[$. However it generally is on $+/-$ 5 stdev around $E[r_t]$ which is generally sufficient for e.g. pricing options on $S$. You should look into european swaptions pricing with the HW model as it is related to your question. $\endgroup$ Dec 12, 2019 at 7:50

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