Risk-Neutral Monte Carlo Drift Equals the Risk-Free Rate

Summary

The document clarifies the drift used when simulating stock prices under a risk-neutral probability measure for option pricing. It presents a discrete geometric Brownian motion step with drift proportional to the risk-free rate, volatility scaled by the square root of the time step, and a standard normal random shock. The question asks whether a stated risk-neutral drift of five percent represents a five percent interest rate.

The answers affirm that, in the basic risk-neutral GBM setup, the stock’s expected growth rate under the pricing measure is the risk-free rate, while volatility is retained from the real-world process. Monte Carlo pricing then approximates an expectation under the risk-neutral measure, with payoffs discounted consistently. This statement concerns the stated framework and does not cover extensions such as dividends, foreign currencies, or other carrying costs, which can alter the risk-neutral drift.

Key ideas

• Under the stated risk-neutral GBM model, the stock price drift is the risk-free rate.
• The random shock is scaled by volatility and the square root of the simulation time step.
• Monte Carlo option pricing approximates expected payoffs under the risk-neutral measure.
• The stated drift relationship is specific to a basic model and may change with carrying costs.

Tags

# Is drift rate the same as interest rate in risk-neutral random walk when using Monte Carlo for option pricing?


# Is drift rate the same as interest rate in risk-neutral random walk when using Monte Carlo for option pricing?












When using following risk-neutral random walk

$$\delta S = rS \delta t + \sigma S \sqrt{\delta t} \phi$$

where $\phi \sim N(0,1)$.

Now when a text mentions drift = 5% does that mean that interest rate (r) is 5%?

## Answer by zuiqo (score 4, accepted)

https://quant.stackexchange.com/a/7540

Yes. The risk neutral and the real path share the same volatility, so the difference is in the drift rate, where the risk-neutral path drifts with the risk-free rate r.

You may want to check out Paul Willmots book, esp. ch. 26, for applications.

## Answer by Alexey Kalmykov (score 3)

https://quant.stackexchange.com/a/7541

When using Monte Carlo for option pricing you numerically approximate expectation under a risk-neutral probability measure $Q$. Your undiscounted stock price process in GBM framework has as a drift equal to risk free rate under $Q$. So the answer to your question is affirmative.

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