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Lognormal assumption

The pricing model's premise that returns are normally distributed and prices lognormally, which bounds prices at zero but understates real-world tails.

Assuming lognormal prices is convenient and partly sensible: it prevents negative prices and makes percentage moves symmetric. It also produces thin tails, continuous paths and constant volatility, none of which describe an equity market on a bad morning.

Every visible distortion of the volatility-surface is the market correcting for this. volatility-skew exists because traders know downside jumps are more likely than the model allows, and they pay up for the strikes the model considers nearly impossible.

Example: with XYZ at $50 and 25% implied volatility, a lognormal model puts the 30-day chance of a close below $40 at well under 1%. Actual equity history says gaps of that size happen often enough that the $40 put will never trade at the model's price.

Related: black-scholes-assumptions, volatility-skew, tail-risk, standard-deviation-move

See it drawn

Original diagrams for the ideas on this page. Illustrative, not real market data.

The volatility smile across strikesImplied volatility plotted against strike, dipping near the money and turning up at both ends, more steeply on the downside.Implied volatility32%28%24%20%8090110120Puts below the money cost moreFar calls cost more tooLowest IV near the moneyATM 100Strike price
The volatility smile. Options on the same stock and the same expiry are not priced off one volatility. Strikes near the money carry the lowest implied volatility, and it rises towards both ends — usually faster on the downside, which tilts the smile into a skew.

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