# Term structure used in Geometric Brownian Motions under Risk Neutral Measure?

When using a GBM under a risk-neutral measure to simulate stock prices, we have to use the risk-free interest rate, but how exactly do you determine what interest rate to use?

I have used the Vasicek model to price ZCB and create the term structure. So when simulating the stock prices should I use a constant risk-free rate or should it be time-dependent and follow the term structure? I was thinking it would be more correct to use the forward rate as the risk-free rate for each time period.

• I agree with your suggestion to use the correct forward rate for each period. – dm63 Mar 6 '18 at 11:22
• You need to use the short rate as described by the Vasicek model. Just simulate GBM and the Vasicek model jointly. – Calculon Mar 7 '18 at 14:25

It should be time dependent and set to the spot forward rate $= -\frac{\partial}{\partial t} \ln(\text{discount}(t))$ when simulating in continuous time. When discretizing the simulation use the forward rate $= -\frac{\ln(\text{discount}(t_{i+1})) - \ln(\text{discount}(t_{i}))}{t_{i+1} - t_{i}}$ from one time pillar $t_i$ to the next time pillar $t_{i+1}$.