The Volatility Smile and Skew

Take a single underlying — say the S&P 500 — on a single expiry date, and look up the implied volatility of every option trading on it: the 90-strike put, the 100-strike (at-the-money) call, the 110-strike call, and everything in between. If the Black–Scholes lognormal story were literally true, every one of those numbers would come out identical — the model has exactly one \sigma, and it does not know or care what strike you happened to pick.

Pull the real numbers and that flat line is nowhere to be found. Implied vol depends on strike — often sharply. Plotted against strike (or against moneyness, m = \ln(K/S_0), which lines options up fairly across different spot levels), the curve bends. In some markets it bends up on both sides like a smile; in others it tilts, higher on one side than the other, like a smirk. This page is about that shape — what it looks like in different markets, and what it is telling you about what the market really believes.

Two shapes, two markets

The bend has a different personality depending on what is being optioned:

FX options (say EUR/USD) tend to show a roughly symmetric smile: implied vol is lowest near the money and rises by a similar amount on both the low-strike and high-strike wings. That symmetry makes sense for a currency pair — a big move up (dollar collapses) and a big move down (dollar spikes) are both plausible "crisis" scenarios, so both tails get priced up together.
Equity index options (S&P 500, FTSE, …) show something lopsided instead: a skew, sometimes called a smirk. Low-strike puts trade at much higher implied vol than high-strike calls. Stocks are far more prone to sudden crashes than sudden melt-ups, and the options market — dominated by investors buying downside insurance — prices that asymmetry directly into the smirk.

Here is a stylised (but realistic-shaped) snapshot of both, one month to expiry, moneyness running from a 10%-in-the-money put to a 10%-in-the-money call:

Strike (% of spot)	90%	95%	100%	105%	110%
EUR/USD implied vol (smile)	11.5%	10.2%	9.8%	10.3%	11.8%
S&P 500 implied vol (skew)	26.0%	22.0%	19.0%	17.0%	16.0%

Read across the FX row and the number dips then climbs — a genuine smile. Read across the index row and the number just falls, strike after strike — a one-sided smirk with no low-side turn-up in sight. Same underlying idea (implied vol varies with strike), two very different signatures.

See both curves move

Both stylised curves are quadratics in moneyness — a smile is dominated by a m^2 term (symmetric), a skew adds a sizeable -m term on top (asymmetric). Drag the stress slider and watch what happens in a market sell-off: both curves get more pronounced. Demand for crash protection intensifies, the equity skew steepens further, and even the "calm" FX smile widens as both of its wings get bid up.

Pull implied vol data from before October 1987 and the smile is barely there — equity index options priced almost exactly the flat line Black–Scholes predicts. Then came Black Monday: the S&P 500 fell over 20% in a single session, an event that a lognormal model with any remotely sane volatility says should happen roughly once per the age of the universe. Traders who had been pricing crash insurance too cheaply learned their lesson in the worst possible way, and never went back. Ever since, low-strike puts have carried a permanent premium over what Black–Scholes says they're "worth" — a phenomenon some quants only half-jokingly call "crashophobia." The skew is, in a very real sense, the option market's scar tissue from a single Monday in 1987.

The name is misleading. It is tempting to picture the "volatility smile" as describing something that happens over the life of the option — volatility starts one place, dips, then smiles its way back up as time passes. That is not what the chart shows at all.

The smile is a cross-sectional snapshot: at one instant, for one expiry, it plots implied vol against strike — a different axis entirely from time. It says nothing about the path any single option's volatility will take between now and expiry; it only says that the market currently prices different strikes on the same stock at different volatilities. (The separate question of how the smile itself varies with time to expiry — the volatility term structure — is a genuinely different axis, covered next.)