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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.

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    $\begingroup$ I agree with your suggestion to use the correct forward rate for each period. $\endgroup$ – dm63 Mar 6 '18 at 11:22
  • $\begingroup$ You need to use the short rate as described by the Vasicek model. Just simulate GBM and the Vasicek model jointly. $\endgroup$ – Calculon Mar 7 '18 at 14:25
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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 $t_{i+1}$.

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