I am trying to use Hull White Model to price a zero coupon bond by Monte Carlo Simulation. The basic idea is under this equation:enter image description here

Under Hull White Model, I want to generate every short rate (r) and integrate them to get the price. Based on the HW model, the dr(t) process includes a v(t) term as follow. I don't quite understand how to simulate thee v(t) process under Monte Carlo, can anyone help? enter image description here


1 Answer 1


You do not model $v(t)$ by Monte-Carlo! As your excerpt explains $\phi(t)$ is a deterministic function of the initial yield curve and accordingly $v(t)$ is deterministic as well. Two further remarks: (i) You should not base the model on $v(t)$ but on an integrated $v(t)$, since this only involves the first derivative of the forward rates. (ii) You should not model $r(t)$ and then integrate to find $\int{r(t)}$ but model both at the same time as a joint Gaussian process.

All these issues are clearly (and in detail) explained in the standard reference Glasserman (Chapter 3.3 Gaussian short rate models).

  • $\begingroup$ Question please: why do we need to simulate $\int{r(t)}$ for discount factors? Can we not use $A*exp(-B*r(t))$, once we have simulated $r(t)$. $\endgroup$
    – InnocentR
    Oct 27, 2019 at 18:35
  • $\begingroup$ Why don't you post this as a question? $\endgroup$
    – g g
    Oct 27, 2019 at 21:05
  • $\begingroup$ I had asked earlier but sadly not much help: quant.stackexchange.com/questions/43101/… $\endgroup$
    – InnocentR
    Oct 27, 2019 at 21:08
  • $\begingroup$ "You should not model r(t) and then integrate to find ∫r(t) but model both at the same time as a joint Gaussian process." - This requires further elaboration. Linking to Glasserman is not an answer, especially in light of the fact, that this is a 50€ book. $\endgroup$ Jul 1, 2023 at 10:58

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