Real-World Call Exercise Probability Depends on the Asset Drift

Summary

The document asks whether the Black–Scholes risk-neutral probability that a call expires in the money, expressed using N(d2), can be adapted to estimate the real-world probability. The reply says this is possible if the asset's real-world drift is known. The key distinction is between a probability under the pricing measure, which uses the risk-free rate, and a forecast under the real-world measure, which requires an estimate of expected asset growth.

The exchange provides no derivation, numerical example, or method for estimating drift. That omission matters: drift is uncertain and difficult to infer, so replacing the risk-free rate does not by itself make the result a reliable forecast. The answer is a concise conceptual pointer rather than a complete procedure, and it does not discuss how volatility assumptions, dividends, or estimation error affect the probability. It is useful as a reminder that option-implied risk-neutral probabilities and real-world forecasts answer different questions.

Key ideas

• N(d2) represents a risk-neutral probability of a call finishing in the money in the Black–Scholes setting.
• A real-world probability requires an estimate of the underlying asset’s real-world drift.
• Changing the drift input does not resolve the challenge of estimating that drift reliably.
• The exchange gives no derivation or treatment of other assumptions and estimation uncertainty.

Tags

# For a call option, what is the real-world probability of expiring in-the-money?


# For a call option, what is the real-world probability of expiring in-the-money?












In the Black-Scholes world, the risk-neutral probability of expiring in-the-money is given by N(d2). Can I just replace the risk-free rate by the drift rate to obtain real world probabilities? Thank you for your help.

## Answer by Kiwiakos (score 1)

https://quant.stackexchange.com/a/23043

Yes.

If you know what the drift is.

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