Normal Distribution

The normal distribution (also called the bell curve or Gaussian distribution) is a probability distribution where most outcomes cluster around the average, with progressively fewer outcomes occurring further from the center. In options pricing, stock returns are often modeled as normally distributed (or more precisely, log-normally distributed), forming the mathematical basis for most pricing models, probability calculations, and risk estimates.

Why It Matters

Nearly every probability number you see on an options trading platform — probability of profit, expected move, probability of touch — is calculated using the normal distribution as a foundation. When your broker shows that a trade has a 70% probability of profit, that number comes from fitting stock returns to a normal (or log-normal) distribution and calculating the area under the curve.

Understanding the normal distribution also helps you understand its limitations. Real stock returns have fatter tails than the bell curve predicts, which means extreme moves (crashes, short squeezes) occur more often than the model suggests. This "tail risk" is the reason why selling far OTM options based purely on normal distribution probabilities can be dangerously misleading.

How It Works

The normal distribution is defined by two parameters: the mean (center) and the standard deviation (width). For stock returns:

• The mean is the expected return (often assumed to be near zero for short periods)
• The standard deviation is derived from implied volatility

Key properties:

• 68.2% of outcomes fall within 1 standard deviation of the mean
• 95.4% fall within 2 standard deviations
• 99.7% fall within 3 standard deviations
• The distribution is symmetric — moves up and down are equally likely

Normal vs. log-normal distribution: Stock prices cannot go below zero, so the Black-Scholes model uses the log-normal distribution, where the logarithm of returns is normally distributed. This introduces a slight upward skew and prevents negative prices. For short time periods and moderate IV levels, the difference is small, but it becomes significant for long-dated options or very high IV.

Where the model works well:

• Short-term expected moves (days to weeks)
• Options on liquid, large-cap stocks during normal conditions
• Relative pricing (comparing one option to another)

Where the model fails:

• Tail events (crashes, squeezes) happen 5-10x more often than the normal distribution predicts
• Volatility is not constant — it clusters and spikes
• Returns are often skewed (stocks tend to crash faster than they rally)
• Earnings and binary events create bimodal distributions, not bell curves

Fat tails in practice: A 3 SD move should happen roughly once every 333 trading days. In reality, the S&P 500 has experienced 3 SD daily moves dozens of times since 2000. This discrepancy is why professional risk managers add buffer beyond what the normal distribution suggests.

Quick Example

Stock DEF trades at $100 with 30% IV and 30 days to expiration. The expected 1 SD move is $8.60. Under the normal distribution:

• 68% chance the stock lands between $91.40 and $108.60
• 95% chance it lands between $82.80 and $117.20
• 99.7% chance it lands between $74.20 and $125.80

You sell a $91 put (roughly 1 SD out) and the model says there is about an 84% chance it expires worthless. This is useful for setting expectations, but remember that the actual frequency of large moves is higher than the model predicts — your 84% trade might really be closer to 80% in the real world.