# Hull white model Monte Carlo simulation Zero Coupon Bond

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:

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?

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.
• 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 $r(t)$. Oct 27, 2019 at 18:35