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Interpreting the Drift Term in Black–Scholes

Article Quant Q&A · Author: 4pie0
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Summary

The document examines the Black–Scholes risk-neutral drift relation and asks whether its negative volatility adjustment conflicts with the observation that higher interest rates can coincide with falling stock prices. The replies explain that the relation belongs to an arbitrage-free pricing model, rather than serving as a claim about how stock prices move on average in the real world.

The volatility adjustment arises when translating between a normally distributed log-return and the resulting lognormal price process; it is a mathematical correction, not a standalone economic signal. The risk-neutral measure sets the drift used to price derivatives so that the model avoids arbitrage. Actual expected equity returns can differ from that model drift, and the document offers no empirical test of a rate-volatility relationship. Its discussion is conceptual and does not specify assumptions beyond the stated model or explain how to estimate real-world returns.

Key ideas

  • The Black–Scholes drift relation describes risk-neutral pricing rather than empirical stock behavior.
  • The volatility adjustment reflects the conversion between log returns and prices in a lognormal process.
  • A risk-neutral drift supports arbitrage-free derivative pricing and need not equal real-world expected equity returns.
  • The relation alone does not establish how interest-rate changes affect stock prices.

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OptionsDerivatives pricingVolatilityEquitiesStatistics
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# Black-Scholes and Fundamentals


# Black-Scholes and Fundamentals












So basically

> $dS_t=\mu S_tdt+\sigma S_tdWt$

and

> $\mu=r-\frac12\sigma^2$

I have just been thinking about this later equation. This is very interesting because it ties together risk-free rate, volatility and asset drift. I always like and try to look at equation from some simple perspective, for example assuming that something is huge or very small or 0, and trying to watch how it impacts other variables. This is good approach to remember some dependencies.

So looking at this later equation, first thing to note is the negative sign of volatility. This is OK when trying to explain why VIX is index of fear and that "investors" don't like increase in volatilities. But increasing risk-free rate in macroeconomics theory translates to increased demand for bonds and decrease in demand for stocks, so their prices drop - this assumption is quite real in today's market - when US Treasuries yields rise stocks go down and vice versa.

So this is not in agreement with this also fundamental assumption $\mu=r-\frac12\sigma^2$.

How do you interpret this fact?

## Answer by vonjd (score 10, accepted)

https://quant.stackexchange.com/a/7590

I think you are interpreting too much into the matter. The $-\frac12\sigma^2$ is just a correction term that comes from Jensen's inequality.

You need this when switching from supposedly symmetric returns (normal distribution) to the skewed price process (log-normal distribution).

I think there are no deeper truths to be found here.

## Answer by ikh (score 3)

https://quant.stackexchange.com/a/7591

One thing to keep in mind here is that the world of risk-free/arbitrage-free models is not necessarily the real world. Specifically, this equation

$$ \mu = r - \frac{1}{2}\sigma^2 $$

occurs not because this is the way stocks behave in reality (they don't! For S&P 500, long-run $\mu$ is closer to 6-9%, if I recall correctly), but because using any other number in the pricing formula would create arbitrage.

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

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