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I have two questions regarding the drift term in the geometric Brownian motion that I cannot find any clear answers to online.

  1. When would we use risk-free rate as drift and when would we use the expected rate of return of the stock as drift rate?

  2. If we are in the risk-neutral framework, what is the appropriate risk-free rate to use in the drift term? Is it the risk-free rate in the country of the stock or is it the risk-free rate used in the discount rate (assuming that these differ)?

Please share your insights or articles on the topic.

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These are fairly basic concepts, although I do acknowledge that it's often not discussed explicitly in text books in great depth.

1. When to use risk-free rate?

You use the risk-free rate only when you want to value derivatives (forwards, futures, options... on the stock under consideration).

On the other hand, if you (for example) want to estimate Potential Future Exposure (PFE) on a derivative portfolio against a counterparty, you need to run the Monte-Carlo simulation that computes the PFE under the real-world historical measure, where you would set the stock drift to the "expected rate of return" (often calibrated to historical data). In other words, in the PFE simulation, you first evolve the underlying stock under the real-world measure, and then you'd value the derivatives at discrete future points in time under the risk-neutral measure (using the risk-free rate).

The above might sound complicated, but the crux of it is:

  • risk-free rate for valuation of derivatives
  • expected rate of return for evolving the underlying to get a distribution of "potential" future prices

The key concept here is that derivative prices are independent of the underlying's future "potential" price distribution (because every market participant has a different, subjective view of these); rather, the derivative prices are only dependent on the underlying's volatility & the cost of borrowing money (where this cost is reflected in the risk-free rate).

2. What risk-free rate to use?

In most cases, we would use the local currency OIS curve to get the corresponding risk-free rate. So for example in USD currency, to value an option that expires in 1 year on some stock, you'd want to get the 1-year SOFR OIS rate.

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  • $\begingroup$ Thank you for a comprehensive answer, although I still have some questions. 1) I am valuing a structured product, would you treat this as a derivative no matter what? The structured product is an autocall that pays fixed coupons depending on the value of the underlying assets. I find it hard to choose between risk-free rate and expected return as drift for the underlying assets. I have read papers on such products, but one paper use risk-free rate and the other use expected returns for drift. 2) The issuer is located in US, whereas the 2 underlying assets in UK and Sweden. What is the rf? $\endgroup$
    – Chell
    Nov 22, 2021 at 11:43
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    $\begingroup$ @Chell: pricing an autocall is a derivative valuation problem, therefore the classical risk-neutral framework should be used. The risk-free rate is related to being able to replicate the pay-off, therefore it doesn't matter where the issuer is located, rather what matters are the underlying assets. I'd need to know the exact structure to be able to comment on the appropriate risk-free rate. $\endgroup$ Nov 22, 2021 at 13:12

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