I'm struggling to understand the integration process of the Hull-White equation:

\begin{equation} dr(t)=[\nu(t)-ar(t)]dt+\sigma dW(t) \end{equation}

In the majority of the references that I have consulted, apparently is trivial to integrate between the spot time s and the valuation time t reaching the following mean and variance results:

\begin{equation} E[r(t)]=r(s)e^{-a(t-s)}+\alpha(t)-\alpha(s)e^{-a(t-s)} \end{equation}

\begin{equation} Var[r(t)]=\dfrac{\sigma^2}{2a}[1-e^{-2a(t-s)}] \end{equation}

Where $\alpha$ is related to the forward rate $f^{m}$ and is defined as:

\begin{equation} \alpha(t)=f^{m}(0,t)+\dfrac{\sigma^2}{2a}[1-e^{-at}]^{2} \end{equation}

Does anybody now a website or paper where all this integration process is explained step by step?

I have tried to look for it but in the best cases I have found just one or two intermediate steps without detailed explanation.

  • $\begingroup$ Have a look here, particularly the section titled "Why not to use Short-hand notation" where a solution to the SDE is shown step by step. $\endgroup$ Commented Apr 9 at 18:22
  • 1
    $\begingroup$ An excellent book covering a lot of useful stuff when it comes to modelling the Hull-White process, and a lot of other models, is the second volume of the trilogy "Interest Rate Modelling" by L. Andersen and V. Piterbarg. This might be a little advanced if you are very new to the topic though. $\endgroup$ Commented Apr 10 at 11:15

1 Answer 1


Start with $d(e^{(at)}r(t))=e^{at}(-ar(t)dt+vol*dW(t)+ar(t)dt+v(t)dt)$

Integrate both sides with limits. That's all there is.

Forward is the expectation of the spot, once you have the spot from here.


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