The Volatility Smile and Skew

If Black-Scholes were true, implied volatility would be a single number, the same for every strike. It is not. Plot implied vol against strike and you get a curve: a smile in FX, a downward-sloping skew in equity indices. That curve is the market shouting that returns are not lognormal. Reading the smile, its level, slope, and curvature, is the core skill of a volatility trader, and understanding why it exists is understanding exactly how Black-Scholes fails.

The contradiction with Black-Scholes

Black-Scholes assumes a single constant $\sigma$ generating a lognormal terminal distribution. Under that assumption, all options on the same underlying and expiry must imply the same volatility, because they are all bets on the same distribution. Empirically, inverting market prices strike-by-strike yields a non-flat curve. There is no way to reconcile this with one lognormal: the smile is proof that the risk-neutral distribution has fatter tails (and often asymmetry) than lognormal. Equivalently, via Breeden-Litzenberger, option prices encode the entire risk-neutral density, and that density is not lognormal.

Equity skew vs FX smile

The shape of the curve encodes the shape of the risk-neutral distribution:

• Equity indices: a downward skew. Out-of-the-money puts (low strikes) trade at higher implied vol than OTM calls. The market pays up for crash protection, so the left tail is fat and the risk-neutral distribution is negatively skewed. Two stories: (1) the leverage effect, as equity falls, a firm's leverage rises, raising its volatility, creating a negative spot-vol correlation; and (2) crashophobia, post-1987, the market permanently prices a jump-down risk that lognormal ignores. The skew is steep and persistent for indices.
• FX: a more symmetric smile. Currency pairs are roughly symmetric (a move up in EUR/USD is a move down in USD/EUR), so both wings are bid, a smile rather than a one-sided skew, tilted by interest-rate and risk-reversal demand.
• Commodities: often a reverse/forward skew. Supply shocks push the right tail (upside) fat for e.g. energy, giving an upward skew.

How desks parametrize the smile

Rather than an implied vol per strike, FX and increasingly equity desks quote the smile through three intuitive coordinates at a fixed delta (usually 25-delta):

• ATM vol, the level, the vol of the at-the-forward option.
• Risk reversal (RR), the slope: $\text{RR}_{25} = \sigma_{25\text{-call}} - \sigma_{25\text{-put}}$. Negative for equities (puts richer), it measures skew and prices the difference between up and down tail risk. Long a risk reversal is a bet on the skew.
• Butterfly (BF), the curvature: $\text{BF}_{25} = \tfrac12(\sigma_{25\text{-call}} + \sigma_{25\text{-put}}) - \sigma_{\text{ATM}}$. It measures how fat both tails are (kurtosis) and prices convexity (volga).

From (ATM, RR, BF) you reconstruct the wing vols:

$\sigma_{25\text{-call}} \approx \sigma_{\text{ATM}} + \text{BF} + \tfrac12\text{RR}, \qquad \sigma_{25\text{-put}} \approx \sigma_{\text{ATM}} + \text{BF} - \tfrac12\text{RR}.$

This decomposition maps cleanly onto distribution moments: ATM $\leftrightarrow$ variance, RR $\leftrightarrow$ skewness, BF $\leftrightarrow$ kurtosis.

What generates a smile

Any mechanism that fattens or skews the risk-neutral tails produces a smile:

• Jumps. A jump component (Merton) puts mass in the tails that diffusion cannot, bidding up wing options, especially OTM puts if the jump is downward.
• Stochastic volatility. If vol is random, the terminal distribution is a mixture of lognormals, which is leptokurtic (fat-tailed), a symmetric smile. Add negative spot-vol correlation $\rho$ and the smile tilts into a skew. This is the Heston/SABR story: vol-of-vol drives the smile's curvature, correlation drives its slope.
• Leverage / spot-vol correlation. Empirically vol rises when equities fall, mechanically steepening the put wing.

Smile dynamics: sticky strike vs sticky delta

A trader must know how the smile moves when spot moves, because it changes the effective delta:

• Sticky strike: each strike keeps its implied vol as spot moves. Then BS delta is (approximately) correct.
• Sticky delta (sticky moneyness): the smile shape stays fixed in moneyness and rides with spot. Then a strike's vol changes as spot moves, adding a vanna term to delta, the "true" delta of a skewed book differs from BS delta.

The wrong assumption systematically mis-hedges; index desks typically hedge somewhere between the two, and empirical smile dynamics ("the skew is sticky-strike in the short run") are an active research question.

Worked example

A three-month index option surface quotes ATM vol 18%, 25-delta risk reversal $-3$ vol points (puts over calls), and butterfly $+1$. The wings are then $\sigma_{25\text{-put}} \approx 18 + 1 + 1.5 = 20.5\%$ and $\sigma_{25\text{-call}}\approx 18 + 1 - 1.5 = 17.5\%$. The OTM put is 3 points richer than the OTM call, the crash premium, and both wings sit above ATM by the butterfly, reflecting fat tails on both sides.

What breaks in practice

• The smile is not static. Its level, slope, and curvature all move, and often more than spot, vol-of-vol and skew are themselves traded. Modeling the smile as fixed is a classic hedging error.
• Local vol gets the dynamics backward. A local-vol model fit to today's smile predicts the smile flattens as spot moves, whereas real equity skew is sticky/steepens, which is why local vol misprices forward-starting and barrier products.
• Wings are illiquid and noisy. Deep OTM quotes have huge vega-to-price leverage; the extreme wings are extrapolated (via SVI or SABR), not observed, and arbitrage-free extrapolation is delicate.
• Calendar and butterfly arbitrage. A fitted surface must be free of static arbitrage (monotone total variance in $T$, convex in $K$); careless interpolation introduces negative densities.

In interviews

The core question is "why does the volatility smile exist?", because returns are not lognormal: fat tails give a smile, negative skew (leverage/crashophobia) gives the equity skew. Distinguish equity skew (downward, OTM puts bid, crash premium) from the FX smile (symmetric). Know the parametrization: ATM = level, risk reversal = slope/skew, butterfly = curvature/kurtosis, and their link to distribution moments. Strong follow-ups: "what generates the slope vs the curvature in a stochastic-vol model?" (correlation $\rho$ vs vol-of-vol), and "sticky-strike vs sticky-delta, why does it matter?" (it changes your hedge via vanna). This motivates local and stochastic volatility models.