# 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%?

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## 2 Answers

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.

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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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Thanks Alexey. May be a basic question, what does GBM stand for? – bytefire Mar 17 '13 at 7:56
@bytefire Geometric Brownian motion – Alexey Kalmykov Mar 17 '13 at 9:40