In the markets, not every drop signals collapse, and not every crash is the end of the world. While dramatic moments draw headlines, most of the time the market simply fluctuates—within statistical norms.
And if we’re living through one of those “normal” stretches, then we should respect the statistics. Many tough investment decisions come down to this choice:
Do you believe we’re facing a rare disaster, or just a statistically probable fluctuation?
Understanding the Historical Distribution
Let’s take a data-driven look at U.S. equity returns. The chart below shows the frequency histogram of S&P 500 returns over 20 trading days (roughly one month), using data from 1990 to the present.
- Black bars represent the actual historical distribution of one-month returns.
- Yellow bars represent a fitted normal (Gaussian) distribution, using the historical average and standard deviation.
Human Limits with Probabilities
Humans aren’t naturally good at thinking in terms of probability distributions. Most of us prefer to simplify. That’s why the normal distribution has been so influential—it reduces an entire distribution to just two numbers: mean and standard deviation.
In this case, the average one-month return of the S&P 500 since 1990 is +0.7%, with a standard deviation of 4.5%. Annualized, this corresponds to about 16% volatility—which may sound familiar, because that’s often used as a benchmark for a “reasonable” VIX level.
Comparing History vs. Normal Distribution
But here’s the issue: even though the yellow curve (normal distribution) is mathematically neat, it underestimates both calm periods and extreme events.
- The market spends more time in small-move regimes than the Gaussian model predicts.
- More importantly, large gains and large losses occur far more often than the model expects.
Chart 1: One-Month S&P 500 Return Histogram (1990–Present)
Black bars = actual historical return frequency
Yellow bars = fitted normal distribution (mean = 0.7%, std = 4.5%) → Historical data shows higher-than-normal frequency of both small changes and extreme events.
To quantify this, we can divide the actual frequency (black bars) by the fitted normal frequency (yellow bars). This gives us a distortion ratio—shown in the second chart.
For moderate moves, the discrepancy isn’t too bad.
But for extreme events—like monthly losses over 30%—the real-world probability is billions of times higher than the normal model suggests.
So yes, people do sometimes use normal distributions when pricing options or estimating risk. And yes, that can be dangerously misleading.
Chart 2: Distortion Ratio (Actual / Normal Model)
This ratio shows how much real-world frequency differs from the Gaussian model.
→ Extreme events—especially losses beyond ±10%—are dramatically more frequent in reality than in market implied distributions: and market needs a way to price that beyond normal distribution.
The Black Swan Reminder
Nassim Taleb’s The Black Swan warns exactly about this: financial systems tend to underestimate the true likelihood of tail events.
One classic example? The 1987 U.S. stock market crash. The Dow dropped more than 22% in a single day. Under a normal distribution, that’s a ten-to-the-minus-several-hundred type of event—mathematically “impossible”, yet it happened.
There are many such “impossible” events:
- An asteroid strike
- A volcano wiping out a global manufacturing hub
- A sovereign debt default from a country that’s never defaulted before
These probabilities aren’t zero—just very small. But small is not the same as negligible.
And if options are priced using a model that assumes these events are essentially zero probability… well, that’s where opportunity—and danger—live.
Why This Matters to You
If you’ve followed this far, you’ll see where this is going. Option pricing is probability estimation. If the market is pricing a put option based on a flawed distribution—one that underestimates tail risk—then it may be underpricing real-world risk.
That’s not just theory. That’s an edge.
Note: In this piece, we’ve been using simple returns (ending price minus starting price, divided by starting price), which are commonly seen in public financial charts. In academic and quantitative finance, we often use log returns instead. We’ll explore why that matters in a future post—it also affects how we think about VIX levels and boundaries.