The Volatility Surface and Skew: Smiles, Term Structure, Sticky Strike vs Sticky Delta, and Risk Reversals

The volatility surface is the grid of implied volatilities across every strike and expiry for one underlying, and skew is the pattern along the strike axis: for equity indexes, out-of-the-money puts trade at higher implied volatility than at-the-money options, and out-of-the-money calls at lower. If the Black-Scholes model were literally true the surface would be flat; it is not, and its shape is the options market’s statement about crash risk, volatility dynamics, and demand for protection.

In short: implied volatility varies by strike (the skew or smile) and by expiry (the term structure). In S&P 500 options the skew was nearly flat before October 1987 and has been steep ever since, as Mark Rubinstein documented in his 1994 Journal of Finance paper “Implied Binomial Trees.” Desks summarise the strike shape with the 25-delta risk reversal (call vol minus put vol, negative for equity indexes) and the butterfly. How the surface moves when the index moves matters as much as its shape: the sticky-strike, sticky-delta and local-volatility rules give different hedge ratios for the same option, and in our example they disagree by about $2,400 per contract after a 5% drop.

This deep dive assumes you know what implied volatility is and how it is backed out of a price (see implied volatility explained) and what delta measures (see delta explained). It explains mechanisms and evidence. It does not recommend any trade.

A common misconception: “skew means puts are overpriced”

A steep skew is often described as puts being “expensive.” That framing assumes the Black-Scholes lognormal distribution is the right benchmark. It is not. Skew is better read as the market pricing a distribution with a fatter left tail than a lognormal one: index returns tend to fall faster than they rise, volatility tends to rise when prices fall, and large investors pay to insure against crashes. Whether puts are expensive relative to the losses they actually insure is an empirical question about the variance risk premium, not something the shape of the skew answers on its own.

Reading the surface

Every point on the surface is one option’s price, translated into the volatility that makes Black-Scholes reproduce it. Three conventions dominate:

By strike or moneyness. Implied volatility plotted against strike, or against strike divided by the forward price.
By delta. Strikes labelled by their Black-Scholes delta, so that a “25-delta put” is the put with delta of -0.25. Delta-based quoting makes surfaces comparable across levels and expiries, and it is standard on over-the-counter desks.
By expiry. The at-the-money volatility for each expiry is the term structure.

From those, desks build three summary numbers for each expiry:

ATM volatility      = implied vol at the money (forward)
25d risk reversal   = vol(25-delta call) - vol(25-delta put)
25d butterfly       = [vol(25-delta call) + vol(25-delta put)] / 2 - ATM vol

The risk reversal measures tilt: negative means puts carry higher implied volatility than calls. The butterfly measures curvature: how much both wings are priced above the centre.

Where equity skew came from: October 1987

Before the 1987 crash, index option smiles were close to flat. Mark Rubinstein’s American Finance Association presidential address, published as “Implied Binomial Trees” in the Journal of Finance (July 1994, volume 49, pages 771-818), built a method for recovering the risk-neutral distribution, and a consistent price tree, from the options themselves. Emanuel Derman’s Columbia lecture notes on the smile reproduce the paper’s representative S&P 500 implied volatilities: a nearly flat line across strikes from 95% to 105% of the index before 1987, and a clear downward slope after it.

Rubinstein and Jens Jackwerth later summarised the evidence in their research survey “Recovering Probabilities and Risk Aversion from Option Prices and Realized Returns.” For 164-day S&P 500 index options on the Chicago Board Options Exchange on July 1, 1987, they show a smile that is “remarkably flat, well within the bounds of realistic trading costs.” For the same maturity on January 2, 1990, a “very steep smile” appears, a shape they describe as typical from mid-1988 onward. Giving Black-Scholes the benefit of the doubt by choosing the single volatility that minimises the largest pricing error, one option is still mispriced by about $4.00, or one by about 15%. They conclude that the crash “marks a divide in the pricing of S&P 500 Index options”: before October 1987, the recovered risk-neutral distribution and the lognormal one were nearly indistinguishable; afterwards the recovered distribution had much more probability in the lower left tail.

Emanuel Derman’s Goldman Sachs note “Regimes of Volatility” (January 1999) adds a direct measurement from a later stress. Tracking three-month S&P 500 implied volatilities from September 1997 to October 1998, he found the skew was negative on every day. The spread between the 800-strike and 1,200-strike volatilities was “about eight volatility points” before the October 27, 1997 sell-off (when the index fell more than 7%), widened to about 16 points by mid-December, and widened further in August 1998.

Why the skew exists

No single cause explains it. Four mechanisms are well documented:

Spot and volatility move in opposite directions. Derman’s 1997-1998 data show three-month at-the-money volatility as “the reflected image of the index.” Steven Heston’s 1993 stochastic volatility paper shows that this negative correlation is what produces skew in a model; without it, random volatility only fattens both tails equally. (The models are compared in pricing models beyond Black-Scholes.)
Jumps and crash fear. Robert Merton’s 1976 jump model shows that the possibility of sudden large moves raises the value of deep out-of-the-money, short-dated options. Index crashes are jumps that cannot be diversified away, so the market charges for them.
Demand pressure. Nicolas Bollen and Robert Whaley (“Does Net Buying Pressure Affect the Shape of Implied Volatility Functions?”, Journal of Finance, 2004) found that changes in implied volatility “are directly related to net buying pressure from public order flow,” and that S&P 500 option volatilities “are most strongly affected by buying pressure for index puts,” while single-stock option volatilities are “dominated by call option demand.” Dealers who sell protection need compensation for holding risk they cannot fully hedge.
A risk premium. Investors who fear crashes will pay more than expected losses for insurance, the subject of the variance risk premium.

Skew also differs by underlying. Single-stock smiles are typically flatter than index skews: Jens Jackwerth’s 1999 literature review of option-implied distributions cites Toft and Prucyk (1997) finding “more gently downward sloping smiles” in individual stock options, which is consistent with Bollen and Whaley’s finding that call demand dominates single-stock options.

Worked example: a hypothetical index surface

To make the numbers concrete, take a hypothetical index at 5,000, three-month options (T = 0.25), a 4% rate and a 1.3% dividend yield, so the forward is about 5,034. We give it a skew of the typical equity shape: 18% at the money, rising for lower strikes, falling for higher ones (implied volatility = 18% - 0.30 x ln(K/F) + 0.40 x ln(K/F)^2).

Strike	Type	Implied vol	Price with skew (index points)	Price at flat 18%	Skew / flat	Delta
4,000	put	27.0%	10.89	0.68	16.1x	-0.04
4,250	put	24.2%	20.42	4.78	4.3x	-0.07
4,500	put	21.9%	40.84	21.63	1.9x	-0.14
4,750	put	19.9%	83.22	68.23	1.2x	-0.26
5,000	put	18.2%	164.03	162.02	1.0x	-0.45
5,250	call	16.8%	84.55	95.29	0.89x	0.32
5,500	call	15.7%	26.28	40.24	0.65x	0.14
5,750	call	14.7%	5.49	14.76	0.37x	0.04

The skew is barely visible in at-the-money prices and dominates the wings. The 20% out-of-the-money put costs sixteen times what a flat-volatility model would charge, while the 15% out-of-the-money call costs about a third.

The summary measures for this expiry:

Measure	Strike	Implied vol
25-delta put	4,730	20.0%
ATM (forward)	5,034	18.0%
25-delta call	5,337	16.4%
25-delta risk reversal		-3.6 points
25-delta butterfly		+0.2 points
10-delta put / call	4,374 / 5,571	23.0% / 15.4%
10-delta risk reversal		-7.6 points

What the skew says about probabilities

Option prices imply probabilities. The derivative of the put price with respect to strike, scaled by e^(rT), is the risk-neutral probability of finishing below that strike; the second derivative is the density, which Dupire’s 1994 Risk paper uses as the starting point for local volatility. Computing both distributions numerically:

Risk-neutral probability index finishes below	With skew	Flat 18%
4,250 (-15%)	5.5%	3.3%
4,500 (-10%)	11.7%	11.5%
4,750 (-5%)	23.7%	27.4%
5,250 (+5%)	66.0%	69.6%
5,500 (+10%)	85.6%	84.8%

The skewed surface puts two-thirds more probability on a 15% fall than the lognormal does, but less on a moderate 5% decline, and less on modest gains. This is the left-skewed, fat-tailed shape Jackwerth and Rubinstein recovered from post-crash prices. These are risk-neutral probabilities: they include the price of insurance, so they overstate the real-world likelihood of a crash by whatever risk premium investors pay.

Term structure

The expiry dimension has its own logic. Implied variance (volatility squared, times time) adds up across periods, so two at-the-money volatilities imply a forward volatility between them. Bruno Dupire’s 1994 paper “Pricing with a Smile” opens with exactly this example on the Nikkei 225: six-month options at 20% and one-year options at 18%. The one-year total variance is 0.18^2 x 1 = 0.0324; the first six months use 0.20^2 x 0.5 = 0.0200; so the second half-year carries 0.0124 of variance, a forward volatility of sqrt(0.0124 / 0.5) = 15.7%.

Three shapes are common:

Upward sloping (short-dated below long-dated), the usual state in calm markets, since volatility tends to revert toward a long-run level.
Inverted (short-dated above long-dated), during stress, when the market prices an immediate shock that it expects to fade.
Humped around events, when a known date such as an earnings release or a central-bank meeting sits inside one expiry and not the next. These are covered in volatility term structure and events.

Skew also has a term structure. Short-dated equity skews are steeper per unit of moneyness than long-dated ones; Jackwerth’s review notes that smile patterns “become more pronounced as the time-to-expiration shortens.” A steep one-month skew and a gentle two-year skew are not a contradiction: over two years many small moves dilute the effect of a single crash.

Sticky strike, sticky delta, and the implied tree

The surface’s shape today does not tell you how it will move tomorrow, and that movement determines hedge ratios. Derman’s “Regimes of Volatility” lays out three rules for a negative skew:

Rule	What stays fixed when the index moves	ATM vol as index rises	Delta vs Black-Scholes
Sticky strike	Each strike keeps its implied vol	Falls (slides down the skew)	Equal
Sticky delta (sticky moneyness)	Each moneyness or delta keeps its vol	Unchanged	Higher
Sticky implied tree (local volatility)	One local volatility surface for the index	Falls twice as fast	Lower

Derman called sticky strike “a poor man’s attempt to preserve the Black-Scholes model” and noted that only the implied-tree model gives a consistent, arbitrage-free view, while “the other models are closer to heuristic rules than theories.” He also matched each to a market regime: sticky strike for a range-bound market, sticky delta for a market trending without a change in realized volatility, and the implied tree when jumps, especially downward ones, are likely.

Worked example: three deltas for one option

Take the three-month at-the-money call on our hypothetical index (strike 5,034, implied volatility 18.0%). Its Black-Scholes vega is about 993 index points per 1.00 of volatility (9.93 per vol point). With the skew slope above, the delta under each rule:

Rule	How implied vol changes with the index	Delta
Sticky strike	Not at all	0.516
Sticky delta	Rises about 0.006 vol points per index point as the index rises	0.576
Local volatility	Falls about 0.006 vol points per index point as the index rises	0.457

The ordering (local volatility below sticky strike below sticky delta) is exactly the one Derman’s framework predicts for a negative skew. A desk short 100 of these calls would hedge with index exposure equal to 45.7, 51.6 or 57.6 contracts’ worth of the underlying depending on the rule: a 26% difference in hedge size for the same position.

Now move the index down 5% to 4,750 and ask what the 5,034 call is worth under each rule (ignoring the passage of time):

Rule	New implied vol for the 5,034 strike	New ATM vol	Call value (points)	Per contract ($100 multiplier)
Sticky strike	18.0%	19.6%	77.33	$7,733
Sticky delta	16.6%	18.0%	65.70	$6,570
Local volatility	19.5%	about 21.1%	90.15	$9,015

The call started at 178.81 points. The marks after the move differ by 24.45 points, about $2,445 per contract, between the sticky-delta and local-volatility views. That gap is model risk, and it is why a book that is “delta neutral” under one rule can show a large loss under another.

What actually happened in Derman’s sample is instructive. Before the October 27, 1997 drop, volatilities by strike were roughly constant (sticky strike). After the drop, “all option volatilities moved in the opposite direction to the index,” consistent with the implied tree. In the calm rally from about 950 to 1,100 in early 1998, strike volatilities stayed fixed and at-the-money volatility slid from about 25% to below 19%, which Derman argued “seems to have been irrational, or at best an oversight”: sticky delta would have kept it stable. Then, in one week as the index neared 1,100, every strike’s volatility jumped about five points, back to where sticky delta would have left it.

Risk reversals as a trade

A risk reversal buys an out-of-the-money call and sells an out-of-the-money put (or the reverse). With delta-neutral sizing it is a near-pure position on skew. In our example, buy the 25-delta call (5,337 strike, 16.4%) and sell the 25-delta put (4,730 strike, 20.0%):

Call costs 58.63 points; put brings in 78.66 points
Net: a credit of 20.03 points ($2,003 per contract)
Delta: about +0.50, which a desk would hedge by selling index exposure
Net vega: about zero, because the two legs have nearly equal vega

With a flat 18% surface the same structure would cost a debit of 8.80 points. The skew is worth 28.83 points on this structure. Because net vega is near zero, the delta-hedged risk reversal does not care much about the overall level of volatility; it cares about the tilt. If the skew steepens by two points (put vol up one, call vol down one), the position loses about $1,586 per contract, roughly the 7.93-point vega of each leg times one point, twice. Selling the risk reversal for a credit is therefore short skew, and its worst day is a sell-off in which put volatility jumps while call volatility falls.

What can go wrong

Treating the surface as static. A position hedged at sticky-strike deltas will be under-hedged or over-hedged if the market switches regime, as it did in October 1997 and August 1998.
Wing illiquidity. Far out-of-the-money strikes trade rarely, so their marks are often extrapolations. Derman noted that the large swings in his 750-strike volatilities were “most likely a consequence of the illiquidity of the option.”
Arbitrage in fitted surfaces. A carelessly interpolated surface can imply negative probabilities (a butterfly with negative value) or calendar arbitrage (total variance falling with maturity). Our worked surface was checked for a positive density at every strike.
Skew and level move together. In a crash, put volatility rises more than at-the-money volatility, which rises more than call volatility. Short-skew and short-volatility positions lose together, at the same moment margin requirements rise.
Reading risk-neutral probabilities as forecasts. The 5.5% risk-neutral chance of a 15% fall in our example includes a premium for insurance. It is a price, not a prediction.

The short version

The volatility surface is the market’s grid of implied volatilities across strikes and expiries. Equity index skew, with puts priced at higher volatilities than calls, appeared after the 1987 crash and has persisted, as Rubinstein and Jackwerth documented and Derman measured through the 1997 and 1998 sell-offs. It reflects the negative correlation between prices and volatility, the risk of jumps, demand for index puts, and a risk premium. The term structure adds forward volatilities and event bumps. How the surface moves (sticky strike, sticky delta, or local volatility) changes hedge ratios materially, and risk reversals isolate the skew as a tradable exposure with losses concentrated in sell-offs.