# Current discount rate of Hull White One-Factor Monte Carlo Simulation

I have a question about the Hull-White One-Factor Monte Carlo Simulation. As we know under the Hull-White One-Factor Model, the short rate follows a random process. So basically, every simulation path is different than each other, which can generate different discount factor value for today. As a result, when I generate the evolution of discount factors(i.e. 1 year bond) over time, it cannot converge at time 0.

How can I solve this problem to make each path' today discount rate converge. Thank you so much!

The average of simulated discount factors from the Hull-White model and market discount factor are the same in theory but very similar in the simulation due to numerical error.

I draw one figure which compares two discount factors and shows their difference.

• red line : mean of simulated discount factors
• blue line : market discount factor
• green line : difference of two discount factor

You can find two discount factors are nearly same and their difference is nearly zero.

I think it is useful to check the process for applying the no-arbitrage condition as follows.

$$P(t,T) = \frac{P(0,T)}{P(0,t)} \exp \left( -x(t)B(t,T) + \frac{1}{2}\{V(t,T)-V(0,T)+V(0,t)\} \right)$$

I hope this helps. 