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!