Implied Volatility and the Smile

Black–Scholes needs one unobservable input: volatility. Run the formula backwards — given the market's option price, solve for the σ that produces it — and you get implied volatility. This inversion is where the model stops being a pricing tool and becomes the language options markets speak.

Implied vol is a price, not a forecast

For every option there's a unique implied vol (option prices are monotonically increasing in σ, so the inversion is clean). Traders quote options in vol points rather than dollars because vol strips out the mechanical effects of spot, strike, and expiry — leaving the interesting part. "I bought the 25-delta puts at 22 vol" communicates everything; the dollar price would communicate almost nothing.

Two related quantities to keep distinct: realized vol is what the underlying actually did (backward-looking, measured); implied vol is what the options market charges for future movement (forward-looking, a price). The gap between them — the volatility risk premium, implied persistently above subsequently-realized in equity indices — is one of the most robust facts in derivatives, and harvesting it (selling options, hedged) is a whole industry with the exact steamroller risk profile the Greeks lesson described.

The smile: the market correcting the model

If Black–Scholes were literally true, every strike on the same expiry would trade at one implied vol. Instead, plotting implied vol against strike gives a smile (or in equities, a lopsided skew): out-of-the-money puts trade at markedly higher vols than at-the-money options, OTM calls slightly higher or flat.

Why the market disagrees with flat-σ lognormality:

• Fat tails (the CLT lesson): big moves happen far more often than lognormal predicts, so options that pay off in tails are worth more — and the inversion expresses "worth more" as "higher implied vol."
• Crash asymmetry: equities gap down, not up; volatility spikes as markets fall (correlation between spot and vol is negative). Downside strikes carry both effects.
• Supply and demand: structural buyers of protection (funds hedging) and sellers of upside (covered-call programs) tilt the curve the same direction.

Historical marker worth knowing: the pronounced equity skew dates from the 1987 crash — the market learned, in one day, that lognormal tails were wrong, and the smile is the scar.

Term structure and the surface

Vol also varies by expiry: typically upward-sloping in calm markets (uncertainty accumulates), sharply inverted in panics (near-term vol explodes past long-term — the market pricing "this week is the dangerous part"). Strike × expiry together form the vol surface, the object option desks actually mark, risk-manage, and arbitrage. The VIX is one point of it, roughly: 30-day implied vol on the S&P 500, ~20 in normal times, 80+ in crises.

Also useful vocabulary: sticky strike vs sticky delta — when spot moves, does each strike keep its vol, or does the smile float with moneyness? Neither is exactly true; knowing the terms signals you've been near a vol desk.

The interview version

"OTM puts trade at higher implied vol than ATM. Is that free money?" — No: it's compensation for real crash risk plus vol-spike correlation; selling the skew is short the exact scenario in which you'll also be losing everywhere else. "Implied vol is 20, you forecast realized 16 — trade and risks?" — Sell options delta-hedged (short gamma, collect theta); risks: your forecast is wrong, gaps between hedges, and vol marks moving against you before expiry vindicates you. "Why do all strikes not trade at one vol?" — the three bullets above, in one breath. The smile is where every "is Black–Scholes wrong?" conversation should land: yes, in a specific, priced, tradable way.