Changing Numeraires to Derive Hull–White Short-Rate Dynamics
Summary
The document asks how to transform short-rate dynamics in a one-factor Linear Gaussian Markov model from its associated numeraire measure to the risk-neutral measure, in order to identify the Hull–White form. The questioner derives the diffusion of the chosen numeraire and applies a change-of-numeraire Brownian-motion relation. They are unsure how to handle the money-market account, whose value is defined through the integral of the stochastic short rate, and whether its volatility contributes to the measure change.
The post highlights the role of each numeraire’s volatility relative to its value in determining the Brownian drift adjustment. It also raises a notation issue: the short rate in the money-market account is the same stochastic short-rate process used in the model. The document contains no answer or derivation resolving the question, so it serves as a focused statement of a term-structure modeling problem rather than a validated proof. Any application would require checking sign conventions and measure-change definitions against the specific toolkit notation.
Key ideas
- A change of numeraire changes the Brownian motion and therefore adjusts the drift of the short-rate process.
- The measure-change adjustment depends on the volatility of each numeraire divided by its value.
- The money-market account is defined by integrating the stochastic short rate over time.
- The question distinguishes the numeraire’s volatility from the volatility of the short rate itself.
- The post leaves the derivation unresolved, so its proposed sign and drift adjustment require verification.
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# Obtaining the dynamics of the LGM (Hull-White) short rate (Brigo and Mercurio Toolkit)
# Obtaining the dynamics of the LGM (Hull-White) short rate (Brigo and Mercurio Toolkit)
I'm struggling with the following problem: I have a diffusion of the short rate under a numeraire associated measure $Q^N$ of the form:
$dr(t) = (...)dt+ H'(t)\alpha(t)dW_t^N$
This is the short rate of the LGM model under the LGM numeraire: $N(x_t, t) = \frac{1}{D(T)}\exp\{H(t)x_t + H(t)^2\zeta(t) /2\}$ where $dx_t = \alpha(t)dW_t^N$
I want to show that this is the Hull-White model. Already did the math to get the short rate dynamics under $Q^N$. So the last step is to change the measure from $Q^N$ to the risk neutral measure $Q^B$.
Brigo and Mercurio in their "Change of Numeraire Toolkit" say that
$BrownianShocks^{Num2}_{Corr} = BrownianShocks^{Num1}_{Corr} - Corr \Big( \frac{Vol_{Num2}}{Num2} -\frac{Vol_{Num1}}{Num1}\Big)'$
I know what I need to get. The formula from Brigo and Mercurio should yield $dW_t^B = dW_t^N + H(t)\alpha(t)$. So when substituing I get that
$dr(t) = [(...) - H'(t)H(t) \alpha^2(t)]dt + H'(t)\alpha(t)dW_t^B$
I'm working with a 1-factor model so I think I can drop $Corr$. Which means I should get:
$\frac{Vol_{Num2}}{Num2} -\frac{Vol_{Num1}}{Num1} = -H(t)\alpha(t)$.
I did the math and I know that $dN(x_t,t) = (...)dt + H(t) N(x_t,t) \alpha(t) dW_t^N$. So I can conclude that $Vol_{Num1} = H(t)N(x_t,t)\alpha(t)$ and that yields $\frac{Vol_{Num1}}{Num1} = H(t)\alpha(t)$. Which is exactly what I need.
The risk-neutral numeraire is the money-market account:
$B(t) = \exp\Big\{\int_0^t r_sds\Big\}$
so how do I continue? $r_s$ is a stochastic process isn't it? In this case, given that $r(r)$ ir normally distributed it means $\int_0^t r(s)ds$ is normally distributed and therefore $dB(t) = (...)dt + B(t) \sigma_B dW_t^B$ with some $\sigma_B\neq 0$ which will ruin the result.
Brigo and Mercurio uses the notation $r_t$ for the integrand in the risk-neutral numeraire and uses $r(t)$ when dealing with the short rate models. As far as I understand they're the same isn't it? Could someone help me?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.