Volatility Smiles: Where the Market Disagrees With the Model

Black-Scholes-Merton assumes the price of the underlying is lognormally distributed at expiry, with a single volatility that does not depend on the strike. If that were true, every option on the same underlying with the same expiry would imply the same volatility.

None of them do. Plot implied volatility against strike and you get a curve, not a line, and the shape of that curve is the market telling you exactly where it disagrees with the model. The volatility smile is not a defect to be corrected. It is the most direct statement available of what the market believes the real distribution of returns looks like, and reading it is a genuine analytical skill rather than a curiosity.

Key Takeaways

• Implied volatility is the input that makes Black-Scholes reproduce the market price. It is a price in different units, not a measurement.
• The model predicts a flat line across strikes. Every liquid market shows a curve, and equity indices have since 1987.
• Equity indices skew downward from low strikes to high. Currency pairs smile, with volatility rising in both directions.
• Elevated implied volatility at a strike means the market assigns more probability to that region than a lognormal distribution does.
• Pricing the wings at the at-the-money volatility was wrong by about 40% in both directions in the worked example.
• Equity skew comes from the leverage effect and persistent demand for protection. The currency smile comes from jump risk with no structural asymmetry.

What implied volatility actually is

Implied volatility is the volatility input that makes the Black-Scholes formula return the price the option is actually trading at. It is found by solving backwards: every other input is observable, so the formula is inverted for the one that is not.

That definition carries a consequence people skip past. Implied volatility is not a measurement of anything. It is a restatement of the option’s price in different units, expressed through a specific model. Change the model and you change the number, because the quantity being reported is defined by the formula used to extract it.

Quoting in volatility terms is nevertheless useful, and for a practical reason. A price in currency depends on the strike, the spot, the rate and the time remaining, so two option prices are hard to compare directly. A volatility strips all of that out, which is why traders quote in vol and why an options market is really a market in volatility with prices attached afterwards.

The prediction the model makes, and how it fails

The model’s assumption is that the underlying follows a lognormal process with constant volatility. If that were an accurate description, implied volatility would be a property of the underlying rather than of the contract, and it would come out identical at every strike and every expiry.

The test is easy and it fails immediately. Take the options on any liquid index, all with the same expiry, and back out the implied volatility at each strike. The result is not flat. For an equity index it slopes downward from low strikes to high ones, often steeply. That single observation refutes the assumption directly, and it has done so consistently since October 1987, before which equity index implied volatilities were much closer to flat.

Figure 1. Two shapes, and the line the model predicts

Two shapes: the equity skew and the currency smile

The two canonical patterns are different and the difference is informative.

Equity indices show a skew, sometimes called a smirk. Implied volatility is highest at low strikes, falls through the money, and continues falling into high strikes. An out of the money put therefore trades at a materially higher implied volatility than an out of the money call the same distance away.

Currency pairs show something closer to a genuine smile. Implied volatility is lowest around the money and rises in both directions, so options struck well away from the current rate are expensive in volatility terms on both sides.

The convention for quoting matters here and trips people up. Currency options are usually quoted by delta rather than by strike, so a 25 delta risk reversal is the difference in implied volatility between a 25 delta call and a 25 delta put, and it is the standard measure of how skewed a currency smile currently is. Equity options are more often quoted by strike or by moneyness. The underlying object is the same curve; only the horizontal axis changes, and a question that mixes the two conventions is usually testing whether you noticed.

Single stocks sit between the two, usually skewed like an index but less steeply, and the skew steepens around events such as results announcements. Commodities vary by market and often skew upward, since the feared move in an energy or agricultural market is frequently a price spike rather than a collapse.

What the shape says about the distribution

Here is the interpretive step, and it is the one worth being able to make in both directions.

A higher implied volatility at a strike means the market is paying more for that option than the flat-volatility model says it is worth. Paying more for a payoff that only occurs in a particular region means assigning more probability to that region than the lognormal assumption does. So the implied distribution has more weight wherever the implied volatility is elevated.

The equity case deserves emphasis because it is counter-intuitive on first meeting. The skew is not saying that falls are more likely than rises. It is saying that the far left of the distribution, the severe fall, carries more probability than a lognormal with the at the money volatility would give it, and the far right carries less. That is a statement about the tails rather than about direction.

What it costs to ignore it

The practical consequence is that using one volatility across strikes misprices everything away from the money, and the errors are large relative to the option premiums involved.

The same error propagates into the greeks. Delta, gamma and vega are all computed from the model, so a delta calculated at the wrong volatility produces a hedge ratio that is wrong at exactly the strikes where the hedging matters most. A desk that hedges its downside book using at the money volatility is under-hedged in precisely the scenario the book exists to protect against.

There is a second-order consequence worth knowing about, because it decides how a desk actually behaves day to day. When the underlying moves, does the smile move with it or stay where it is? Under a sticky strike assumption each strike keeps its own implied volatility as the spot moves, so an option that was at the money and is now out of the money keeps the volatility it had. Under sticky delta, or sticky moneyness, the whole curve shifts along with the spot, so the volatility attaches to the distance from the money rather than to the strike. The two assumptions produce different deltas for the same option, and choosing between them is a modelling decision that a hedging book lives or dies by. Equity markets in calm conditions tend to behave closer to sticky strike; in stressed conditions the curve moves.

Why the two shapes differ

Two explanations are usually given for the equity skew, and they are complementary rather than competing.

The first is the leverage effect. As a company’s equity value falls, its debt does not fall with it, so the firm’s leverage rises and the equity becomes riskier. Volatility and price therefore move inversely, which is a mechanism rather than a sentiment and which produces exactly the negatively skewed distribution observed.

The second is demand. Institutional investors are structurally long equities and buy puts for protection, while the natural sellers of that protection require compensation for taking on a position that loses most when everything else they hold is also losing. That persistent one-way demand raises the price, and therefore the implied volatility, of low strike options. The observation that the skew appeared after October 1987 rather than before supports this reading, since what changed then was the market’s memory rather than any company’s balance sheet.

The currency case has neither mechanism. An exchange rate has no leverage in this sense, and there is no natural asymmetry of demand, because a dollar-rupee option is a rupee-dollar option seen from the other side and the two directions are the same event described twice. What currencies do have is jump risk in both directions, from policy decisions and interventions, and jumps fatten both tails. Symmetry of structure plus fat tails on both sides produces a smile rather than a skew.

The surface, and what to do with it

Implied volatility also varies with time to expiry, which is the volatility term structure. Short-dated implied volatility moves around far more than long-dated, since a short option’s value is dominated by conditions now while a long one averages over a longer horizon. The term structure typically slopes upward when current volatility is low and inverts when markets are stressed, which makes its shape a readable indicator in itself.

Read that grid in two directions and both tell you something. Across any row the at the money level is stable while the wings move, which is the term structure. Down any column the spread between the wings is wide at one month and narrower at a year, which is the skew flattening with maturity. That flattening is a general feature and it has an intuitive reason: a single shock dominates a one month distribution and is averaged away across a year, so short-dated options price the possibility of a jump far more aggressively than long-dated ones do.

Put the two dimensions together, strike and expiry, and the result is the volatility surface: a volatility for every combination. This is what a real options desk actually maintains, and it is what “the model” means in practice. Black-Scholes has not been discarded. It has been demoted from a model of the world to a quoting convention, a translator between prices and volatilities, with the surface carrying all the information the original assumption threw away.

The wider point survives the exam. A model that fails in a systematic and stable pattern is far more useful than one that fails randomly, because the pattern itself becomes information. The smile is the residual of Black-Scholes plotted against strike, and the market has been reading that residual as data for nearly forty years.

The short version

• Implied volatility is the input that makes Black-Scholes reproduce the market price. It is a price in different units, not a measurement.
• The model predicts a flat line across strikes. Every liquid market shows a curve, and has since 1987 for equity indices.
• Equity indices skew downward from low strikes to high. Currency pairs smile, with volatility rising in both directions.
• Elevated implied volatility at a strike means the market assigns more probability to that region than a lognormal distribution does.
• Pricing the wings at the at the money volatility was wrong by about 40% in both directions in the worked example.
• Equity skew comes from the leverage effect and persistent demand for protection. The currency smile comes from jump risk with no structural asymmetry.

Frequently Asked Questions