The standard model assumes a stock drifts around in small, bell-shaped percentage moves with the same volatility every day. That is a fair description of an ordinary Tuesday. It is a poor description of the days that decide whether a trader survives, and the market has known that for a long time. This lesson is about the gap between the model and the world, because that gap is where the rest of the course lives.
Fat tails
Under a bell curve a fall of four standard deviations should happen about once in 30,000 trading days, well over a century. Real markets manage it every few years. Big moves are far more common than the model expects, in both directions but especially downward. Traders say real returns have fat tails: the far ends of the distribution carry more weight than the bell curve gives them. An option seller who trusts the model is being paid for a risk the model says is nearly impossible and the world says is routine.
Jumps
The model also assumes prices move smoothly, so a hedger can always trade on the way from one level to another. Earnings, a takeover, a central bank decision, a pandemic: the stock reopens somewhere else and the prices in between never trade. A jump is a move you cannot hedge through. Stocks with earnings ahead, and the whole market on some days, carry jump risk the smooth model does not see, and options on them are priced higher than the model alone would say.
Volatility that changes
The model also fixes volatility for the option's whole life. In practice it wanders, and it tends to rise when stocks fall, which makes the downside tail fatter still.
The day the model broke
Before October 1987, index options were largely priced with one volatility across all strikes. A put 10% below the market cost the same volatility as a call 10% above it. On October 19, 1987, US stocks fell more than 20% in a single day, a move of well over ten standard deviations that the model rated as effectively impossible. Traders who had sold cheap downside puts were wiped out. From that day on, the market has charged a higher volatility for downside protection than a flat model would, on indexes above all, and it has never gone back. The pattern is called skew: implied vol rising as the strike falls. It is not a flaw in the chain. It is the market correcting the model.
An index-style curve: 18 vol at the money, rising 4 points for every 10% the strike falls. Set skew to 0 and curvature up for a single-stock smile; set both to 0 and you are back to the flat model.
Plotted against strike, implied vol is a curve, not a line. Index curves slope down from left to right, steep on the downside. Single stocks often curl up at both ends, a smile, because a takeover or a blowout quarter can send them sharply higher too. The dashed line is what the model would use if it believed its own assumptions.
What skew does to prices
A few vol points matter most where the tail lives: a far out-of-the-money put is almost all tail, so pricing it a few points above the at-the-money vol can multiply its value, while the upside calls get cheaper. Module 11 will price that properly, strike by strike.
One number is not enough
So a stock does not have a volatility. It has one at every strike, and the shape across strikes is information: what the market fears, and how much it is willing to pay to cover it. The standard yardstick compares two options an equal distance from the money in probability terms, the put with a delta near 0.25 and the call with a delta near 0.25. Put vol minus call vol is the 25-delta skew, in vol points. Positive means the downside costs more. On an index it is almost always positive, often by several points. A $500 index at 15 vol at the money might show 19 on the 25-delta put and 13 on the 25-delta call: a skew of 6 points, and a market paying half again as much vol for protection as for upside.
The model is still used every day, because it is the common language: the dictionary between vol and price. The correction is to let every strike carry its own number. Module 11 maps the whole shape.
The maths: fat tails in numbers
Under the normal distribution, the chance of a single day falling more than z standard deviations is 1 - N(z). With 252 trading days a year:
| Down move | Chance on a given day | Once every |
|---|---|---|
| 3 standard deviations | 0.135% | about 740 days, 3 years |
| 4 standard deviations | 0.0032% | about 31,600 days, 125 years |
| 5 standard deviations | 0.00003% | about 3.5 million days, 13,800 years |
Each extra standard deviation cuts the odds by a factor of 20 to 100, because the bell curve's tails fall off like e^(-z²/2). Real index returns fall off far more slowly. For an index with about 1% daily volatility, a 4% down day is a four-standard-deviation event, and the US market has produced dozens of them in the last few decades, clustered in 1987, 2008 and 2020. A statistician measures this with kurtosis, the weight in the tails compared with a bell curve: 3 for a normal distribution, and far higher for daily stock returns.
How does skew fix it? A put's price is, roughly, the probability-weighted value of the outcomes below its strike. Raising that put's implied vol is the chain's way of saying those outcomes are more likely than the bell curve admits. The smile is the fat-tailed distribution, written in the only unit the model understands.
Your turn
Open SPY. Find the two 25-delta strikes on the chain, compare their implied vols, and answer the questions. Then keep the number in mind: every module from here on assumes you know that downside and upside are not priced alike.