Markov Properties of Short Rates in HJM and Hull–White Models

Summary

The document raises a modeling question about whether the short rate is Markov in the Heath–Jarrow–Morton framework. It presents an expression for the instantaneous forward rate and a derived spot-rate dynamics expression that includes a stochastic integral over past Brownian increments. That history-dependent term appears to make the spot rate non-Markov in the general case.

The question contrasts this with the one-factor Hull–White model, whose short rate follows a mean-reverting diffusion and is Markov, even though Hull–White can be represented within HJM using a particular volatility function. The document does not include an answer or resolve the apparent inconsistency. Its value is in highlighting that Markov behavior depends on the model’s volatility structure and state representation; the displayed equations alone do not establish that the two formulations contradict each other. Readers need a derivation or a source addressing the special Hull–White case to settle the issue.

Key ideas

• The document compares spot-rate dynamics in HJM with the one-factor Hull–White short-rate model.
• A stochastic integral over past shocks can make a rate depend on more than its current value.
• Hull–White is presented as a special HJM specification with a Markov short rate.
• The question remains unanswered in the document, so it does not demonstrate a contradiction or provide a resolution.

Tags

# Hull White and HJM model not Markov


# Hull White and HJM model not Markov












In `HJM` model we have `instaneous forward rate` $f(t,T):$

$$d f(t,T) = v(t,T)v_T(t,T)d t - v_T(t,T)d W_t,$$

is `Markov`. And the `spot rate` $r(t)$

$$d r(t) = \left\{f_t(0,t) + \int^t_0 [v(\tau,t)v_{tt}(\tau,t)+v_{t}(\tau,t)^2]d\tau - \int^t_0 v_{tt}(\tau,t)d W_\tau\right\}d t$$ $$ + v_{tt}(\tau,t)|_{\tau = t}d W_\tau.$$

Term $$\int^t_0 v_{tt}(\tau,t)d W_\tau$$ makes $r(t)$ `non-Markov`generally(Ho-Lee is not).

But one thing confused me is that, `Hull-White` (One-Factor) Model: $$d r = [\theta(t) - ar]d t + \sigma d W_t.$$ is HJM model, and $r$ is obviously Markov. But we have $v(t,T)$ in Hull-White: $$v(t,T) = \sigma\dfrac{1 - e^{-a(T-t)}}{a}$$ should make $r(t)$ non-Markov.

So what's wrong here?

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