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Implied Volatility and the Smile: What Option Prices Say About Risk
A six-part series on how option markets price volatility, from the textbook model to the arbitrage-free surfaces used by professional desks.
- Black-Scholes and the Greeks
- Implied Volatility and the Smile (you are here)
- Local Volatility and Dupire's Formula
- Stochastic Volatility: the Heston Model
- The SABR Model
- Arbitrage-Free Surfaces: SVI and Mixtures of Lognormals
Part 1 ended with a contradiction: Black-Scholes assumes one volatility, but the market charges a different one for almost every strike. This part explains how implied volatility is extracted from prices, why equity index options show a persistent skew, and how the shape of the smile can be translated into the probabilities the market assigns to different outcomes.
From price to implied volatility
The Black-Scholes formula takes volatility as an input and produces a price. Markets work the other way around: the price is observed, and the question is which volatility would justify it. That volatility is the implied volatility.
There is no closed-form way to invert the formula, so implied volatility is found numerically. Because the call price rises steadily with volatility, the inversion always has exactly one answer for any arbitrage-free price, and simple root-finding methods work well. The most common is Newton's method, which uses vega, the derivative of price with respect to volatility, to take each step:
A worked example
Use the same at-the-money three-month call from Part 1 (S = 100, K = 100, T = 0.25, r = 4%), but suppose it trades at 5.00 instead of the 4.49 implied by 20% volatility.
| Step | Volatility guess | Model price | Vega |
|---|---|---|---|
| 0 | 20.00% | 4.4852 | 19.72 |
| 1 | 22.61% | 5.0002 | 19.74 |
| 2 | 22.61% | 5.0000 | 19.74 |
After a single step the guess is within a fraction of a cent of the market price. The implied volatility of this option is about 22.6%. Near the money, where vega is large, Newton's method converges in two or three steps. Far from the money, vega becomes tiny and the method can overshoot, so production code usually falls back to bisection, which is slower but always converges.
Implied volatility is best thought of as a quoting convention. Saying an option trades at "22.6 vol" carries exactly the same information as its price, but unlike a price, it can be compared across strikes, maturities and even different underlyings.
The smile appears
If Black-Scholes were literally true, inverting every option on the same underlying and expiration would give the same number. Plotting implied volatility against strike would produce a flat line. Instead the line curves, and its shape depends on the market:
- Equity indices show a downward-sloping skew: low strikes have much higher implied volatility than high strikes, often with a slight upturn at the far right.
- Currencies between similar economies tend to show a more symmetric smile, with both tails priced above the center.
- Commodities exposed to supply shocks and VIX options often show the reverse, with higher implied volatility on upside strikes, because the feared event is a spike upward.
Equity index skew has been a persistent feature of markets since the crash of October 1987. Before that, index smiles were much flatter. A single day in which the market fell more than 20% changed how investors priced the possibility of a crash, and that change never fully went away.
Why equity indices are skewed
Several explanations reinforce each other:
- Crash risk and fat tails. Large declines happen more often than a lognormal model allows, and they happen faster than rallies. Options that pay off in a crash should cost more than the model suggests.
- Volatility rises when prices fall. Lower equity prices raise companies' leverage and investors' uncertainty, so a fall in the index tends to come with higher realized volatility. A put that pays off after a fall will be paying off in a high-volatility world.
- Demand for protection. Pension funds, asset managers and structured product issuers buy index puts to protect portfolios. Investors also sell calls against stock holdings for income. This combination pushes put prices up and call prices down.
- Hedging costs for dealers. Dealers who sell those puts are left short downside risk that is hard to hedge in a crash, and they charge for carrying it.
What the smile says about probabilities
The smile is not just a pricing quirk. It contains the market's view of the full probability distribution of future prices. In 1978, Douglas Breeden and Robert Litzenberger showed that the risk-neutral probability density can be recovered directly from call prices across strikes:
The intuition comes from a butterfly spread: buy one call at K − h, sell two at K, buy one at K + h. It pays off only if the price ends near K, so its cost, scaled by h², approximates the probability of ending at K. Doing this at every strike traces out the whole distribution.
A flat smile corresponds to a lognormal distribution. A skewed smile, with expensive low strikes, corresponds to a distribution with a fatter left tail: the market assigns more probability to large declines than the lognormal model does. To keep the average at the forward price, some probability has to move elsewhere, which typically appears as a slightly taller, narrower peak just above the forward. In other words, the market expects modest gains to be common and large losses to be rare but much more likely than a bell curve suggests.
This link also explains why the smile cannot take any shape at all. Probabilities cannot be negative, so call prices must be a convex function of strike, and they must decrease as the strike rises. A smile that violates these rules would let someone build a butterfly or a call spread with a negative cost, a free profit. Building surfaces that never break these rules is the topic of Part 6.
Measuring the smile
Traders summarize the smile with a few standard numbers:
| Measure | Definition | What it captures |
|---|---|---|
| ATM volatility | Implied volatility at the forward or spot | The overall level of the smile |
| 25-delta risk reversal | IV of the 25-delta call minus IV of the 25-delta put | Skew; strongly negative for equity indices |
| 25-delta butterfly | Average IV of the 25-delta call and put minus ATM IV | Curvature, or how expensive both tails are |
| Moneyness skew | IV at 90% moneyness minus IV at 110% | Skew measured in price terms; the 30-day version is shown on this site's Vol Surface page |
Measuring by delta rather than by fixed strike adjusts for time: a 10% drop is a tail event over one week but an ordinary move over one year. Delta-based measures put options of different maturities on a comparable footing.
How the smile changes with maturity
Short-dated smiles are much steeper than long-dated ones. Over one week, a strike 5% below spot is far in the tail, and the cost of crash protection dominates its price. Over two years, the same strike is barely out of the money. As a rough rule of thumb, skew measured in fixed moneyness flattens roughly in proportion to one over the square root of time, which is another reason why delta-based measures are used to compare maturities.
The level of the smile changes with maturity too, giving the term structure of implied volatility discussed in the volatility surface guide. Stacking smiles for every maturity produces the full surface.
How the smile moves with the market
Knowing today's smile is not enough to hedge. A trader also needs to know how it will move when the underlying moves, and Black-Scholes is silent on this. Two simple rules of thumb bracket common practice:
- Sticky strike: each strike keeps its implied volatility when spot moves. If the index falls, the at-the-money option is now a lower strike with a higher implied volatility, so ATM volatility rises.
- Sticky moneyness (or sticky delta): the smile moves with spot, so an option that stays at the same moneyness keeps the same implied volatility.
Real markets follow neither rule exactly. In equity index sell-offs, at-the-money implied volatility typically rises, often by as much as or more than the sticky-strike rule would predict. The choice matters in practice: it changes the effective delta of an option, because part of the option's price change comes from the volatility move that accompanies the price move. That extra effect is exactly the vanna exposure discussed in the GEX guide.
Modeling these dynamics consistently is the job of the models covered in the rest of this series. Local volatility, the subject of Part 3, takes the smile as given and asks what volatility process could have produced it.
Key takeaways
- Implied volatility is the volatility that makes the Black-Scholes price match the market price. It is found numerically and serves as a universal quoting convention.
- Equity index options show a persistent downward skew, driven by crash risk, the tendency of volatility to rise as prices fall, and steady demand for protection.
- The second derivative of call prices with respect to strike gives the market's risk-neutral distribution; a skewed smile means a fatter left tail than a lognormal model allows.
- Arbitrage constraints on prices mean the smile cannot take an arbitrary shape.
- Skew is steepest at short maturities, and how the smile moves with spot affects hedging through vanna.
See today's smiles in live data on the Vol Surface page: hover over the surface to project each expiration's smile onto the back wall.