Integrating the Hull–White Short-Rate SDE with an Integrating Factor

Summary

The document asks for a step-by-step derivation of the Hull–White short-rate model’s conditional mean and variance. The stochastic differential equation combines mean reversion toward a time-dependent drift with constant volatility. The response gives the key method: multiply the rate by the integrating factor exp(at), apply the stochastic differential rule, and integrate between the starting and ending times. This cancels the mean-reversion terms and leaves an expression involving the drift integral and a weighted Brownian increment.

The exchange does not show the intermediate integration steps or derive the stated mean and variance in detail. It briefly links the forward rate to the expected future short rate. The result assumes the model’s specified parameters and setup; readers seeking a full derivation must work through the stochastic integral and its expectation and variance, including the relationship between the time-dependent drift and the initial forward curve.

Key ideas

• Multiplying the short rate by exp(at) is the integrating-factor step for the Hull–White SDE.
• The product rule cancels the mean-reversion terms before integration over time.
• The integrated expression contains a drift contribution and a weighted Brownian increment.
• The response names the core technique but does not derive the mean and variance fully.

Tags

# Step by step integration of the Hull-White SDE


# Step by step integration of the Hull-White SDE












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.

## Answer by Arshdeep (score 1)

https://quant.stackexchange.com/a/78923

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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