Quantitative Methods & RiskProbability & Distributions for Traders
The Normal Distribution — and Why Markets Aren't
The bell curve, and the trouble with it
The normal (Gaussian) distribution is the default model for returns because it is mathematically convenient: fully described by two numbers (mean and standard deviation), closed under addition, and the limiting shape of sums of independent shocks via the Central Limit Theorem. It is also wrong in exactly the way that hurts traders most. Real returns have fat tails — extreme moves happen far more often than a normal curve predicts — and the gap between the model and reality is where blow-ups live.
Normal density and the sigma rule
f(x) = (1 / (σ√2π)) · exp(−(x−μ)² / 2σ²)
1987, 2008, 2015, 2020 say otherwise
October 19 1987 was about a −22σ event for the S&P under a normal model — astronomically impossible, yet it happened. Equity and FX markets routinely print 5σ+ days every few years. When your model says 'once per 14,000 years' and reality says 'every few years,' the model is broken, not the market.
Moments: skew and kurtosis
The mean and variance are the first two moments of a distribution. Two more matter for traders. Skewness measures asymmetry — negative skew means the left (loss) tail is longer, which is typical of carry trades and short-volatility strategies that 'pick up pennies in front of a steamroller.' Kurtosis measures tail heaviness; a normal distribution has kurtosis 3, and anything above that is leptokurtic — fat-tailed.
Skewness and excess kurtosis
Skew = E[(X−μ)³] / σ³ Kurt = E[(X−μ)⁴] / σ⁴ Excess = Kurt − 3
A
What the normal model assumes
- Symmetric — rallies and crashes equally sized
- Excess kurtosis 0 — thin tails
- Extreme moves are essentially impossible
- Volatility constant through time
- Each day independent of the last
B
What real returns actually do
- Negatively skewed — crashes are deeper than rallies
- Excess kurtosis 3–10 — fat tails
- 5σ+ days occur every few years, not every 14,000
- Volatility clusters — calm and storm regimes
- Big moves bunch together (autocorrelated |returns|)
Fat tails (leptokurtosis)
A distribution whose extreme outcomes are more probable than a normal distribution with the same variance would predict. Practically: the 'once-a-century' loss shows up once a decade. Student's t and the family of stable distributions are common fat-tailed models; even simpler, just use the empirical history and don't assume normality at all.
Underestimated tails
Parametric VaR built on a normal assumption systematically understates the loss at high confidence levels. The 99% VaR is where the fat tail bites hardest, and that is exactly the number risk committees lean on.
1import numpy as np2from scipy import stats34r = np.array(daily_log_returns) # your real history5mu, sd = r.mean(), r.std(ddof=1)67# How many >3sigma days did a normal model predict vs. how many we saw?8z = (r - mu) / sd9observed_3sig = np.mean(np.abs(z) > 3)10normal_3sig = 2 * (1 - stats.norm.cdf(3)) # ~0.00271112print(f"observed |z|>3 : {observed_3sig:.4%}")13print(f"normal |z|>3 : {normal_3sig:.4%}")14print(f"excess kurtosis: {stats.kurtosis(r):.2f}") # >0 means fat tails15print(f"skewness : {stats.skew(r):.2f}") # <0 means left tail
Practical rule
Use the normal distribution for intuition and quick back-of-envelope work. Never use it as the final word on tail risk. When the number actually matters — capital at risk, position size, the loss you must survive — use the empirical distribution or a fat-tailed model, and add a discrete stress scenario on top.
Knowledge check
A risk model assumes normal returns and reports that a −10% day is a '6-sigma, once-in-a-million-years' event. Over five years you observe two such days. The correct conclusion is: