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-Markovgenerally(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?



Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge that you have read and understand our privacy policy and code of conduct.

Browse other questions tagged or ask your own question.