Using Forward Rates in Risk-Neutral GBM Simulations

Summary

The document asks which risk-free rate to use when simulating a stock under a risk-neutral geometric Brownian motion, given a term structure derived from a Vasicek interest-rate model. It contrasts using one constant rate with using a rate that changes over the simulation horizon, and suggests using forward rates for each time step.

The answer says to use the instantaneous forward rate implied by the discount curve in continuous time. For a discrete simulation, it gives the corresponding interval rate as the negative change in log discount factors divided by the interval length. This ties the simulated drift to the term structure rather than applying a single maturity-independent rate. The guidance assumes the discount curve is the relevant risk-free curve and describes how to translate it into simulation-step rates; it does not discuss other modeling choices, such as stochastic rates jointly simulated with the stock or the calibration and limitations of the Vasicek model.

Key ideas

• A risk-neutral GBM simulation can use a risk-free rate that varies over time.
• The instantaneous forward rate is obtained from the slope of the log discount function.
• For discrete steps, use the rate implied by discount factors at the interval endpoints.
• The method links the simulation drift to the supplied term structure.
• The answer does not address joint simulation of stochastic rates and stock prices.

Tags

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


# 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.

## Answer by Antoine Conze (score 2)

https://quant.stackexchange.com/a/38654

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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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.