Black–Scholes One-Sigma Exceedances and Their Frequency

Summary

The document asks how the Black–Scholes assumption of normally distributed log returns leads to the claim that returns exceed one standard deviation about once every three days. It states that the log return over a time interval has a normal distribution centered at its stated mean, with volatility scaled by the square root of the interval, and gives the probability of a return falling within one standard deviation of that mean.

The frequency follows from the complementary probability: about 31.731% of observations lie outside the one-standard-deviation band, which corresponds to roughly one exceedance per 3.15 observations on average. Interpreting that as days requires each observation to represent one day. The discussion is theoretical and assumes the model's normal-return premise; it does not establish that real market returns follow that distribution or that exceedances arrive at regular intervals.

Key ideas

• Under the stated normal model, about 68.269% of returns fall within one standard deviation of the mean.
• The complementary probability gives the chance of an observation outside that band.
• An exceedance frequency of about once every three observations is an average, not a regular schedule.
• Calling the observations days assumes a daily sampling interval.

Tags

# Why, under Black-Scholes, do returns exceed one $\sigma$ every three days, approximately?


# Why, under Black-Scholes, do returns exceed one $\sigma$ every three days, approximately?












My book claims the following:

> "The other assumption of normally distributed returns would mean that returns would be between $-\sigma$ and $+\sigma$ with 68.269 per cent probability. This means that you would have an exceedence of roughly once every three days."

Now, since $\ln (S_T/S_0)\sim N((\mu+\dfrac12\sigma^2)\Delta t , \sigma^2\Delta t)$, it is immediately true that$$ P\Big((\mu+\dfrac12\sigma^2)\Delta t-\sigma \sqrt{\Delta t}\leq\ln (S_T/S_0)\leq (\mu+\dfrac12\sigma^2)\Delta t+\sigma \sqrt{\Delta t}\Big)=68.269\%. $$ I'm trying to reduce this to "once every three days" but I don't see how.

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