Physical and Risk-Neutral Expected Stock Returns in Black–Scholes

Summary

The document raises a conceptual question about the expected stock price under the Black–Scholes–Merton framework. It contrasts the risk-neutral expectation, which grows at the risk-free rate in the standard model, with informal explanations of volatility that describe a future price distribution around that growth rate as though it were the real-world expectation. The author asks what determines the physical-measure expectation and whether treating it as the risk-free expectation is a reasonable approximation.

The text provides no answer or empirical evidence; it is a statement of the issue rather than a resolved explanation. Its useful distinction is between a pricing measure used to value derivatives and the physical measure used to describe actual investor outcomes. Any comparison depends on assumptions about expected returns and risk premia, which the document does not specify. It also flags that a simple “mean plus or minus volatility” intuition can obscure the compounding and distributional details of stock prices.

Key ideas

• The risk-neutral expected stock price grows at the risk-free rate under the standard model.
• The physical-measure expectation concerns real-world returns and need not match the risk-neutral expectation.
• Volatility describes dispersion, not the expected return or drift by itself.
• Whether the two expectations are close depends on assumptions not supplied in the document.
• The document poses these questions but does not provide a resolution.

Tags

# expected value of a stock under BSM


# expected value of a stock under BSM












> Roughly speaking, if a stock is \$100 and has a volatility of 30% and the risk-free rate is 10% then the distribution of stock prices in a year is a mean of $110 give or take 30%. [This is not mathematically true because of compounding, but the details are a hindrance for this post]

This is from an article explaining the BSM model.

From my understanding, it seems that under the BSM model, the expected value of a stock under the risk-free measure is $S_0e^{r \cdot t}$, where $r$ is the risk-free rate. However, when I see people explain what volatility means intuitively, like the article I've cited, they ignore the risk-free measure part and just say that the expected value of the stock is $S_0e^{r \cdot t}$ in the real world.

I have two questions. One, what is the expected value of the stock under the physical measure? And second, are they close enough that this approximation is reasonable?

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