Volatility Surface: Skew and Smile

A row of volatilities, not one

Black-Scholes takes one volatility as an input, and O9 used it that way: one number in, one price out. Run the formula backwards on every strike of a real chain and the number that comes back is different at almost every strike. Plot those implied volatilities against the strike and you have a curve; add the expiries and you have a **surface**. The surface is how the options market actually quotes risk, and the single “implied volatility” on a quote screen is one point on it — usually the at-the-money strike of one expiry. The curve has two common shapes. A **smile** is bowed up at both ends: options far from the money in either direction trade at higher implied volatility than options at the money, which is how a market says that large moves both ways are more likely than a normal distribution allows. A **skew** — traders also call it a smirk — slopes down from left to right: low strikes trade at high implied volatility and high strikes at low. Equity indexes show the skew. Many single stocks, currencies and commodities show something closer to a smile, sometimes tilted the other way. Two conventions make the curve comparable from day to day. Because a 4,500 strike means something different with the index at 5,000 than at 4,000, traders quote the curve by **moneyness** — strike as a percentage of the price — or by **delta**: the 25-delta put is the put whose delta is −0.25, roughly a one-in-four chance of finishing in the money. And they compress the slope into one number, the gap between the 25-delta put’s and the 25-delta call’s implied volatilities, called the **risk reversal** after the trade that buys one and sells the other. • Smile: both wings above the at-the-money level — large moves either way priced above a normal distribution. • Skew, or smirk: low strikes above high strikes — the downside priced above the upside; the equity-index shape. • Moneyness or delta, not the raw strike, is how the curve is compared across days. • The 25-delta risk reversal: the slope in one number. One index, one 30-day expiry, four strikes (the surface used in this lesson) — 4,500 put — 10% below the market: 24% implied volatility · 4,750 put — 5% below: 19% · 5,000 — at the money: 15% ← · 5,250 call — 5% above: 12%

Skew and the smile: the surface across strikes

Expiries are one direction of the surface. Strikes are the other, and the pattern there is the more surprising of the two: for most equity indexes, out-of-the-money **puts trade at a higher implied volatility than out-of-the-money calls**, at the same expiry. If implied volatility were a single forecast about the stock’s future movement, all the strikes would show the same number — a flat line across the chain — and the honest way to see what is happening is to ask what a flat line would mean: the market would be charging the same for a 5% fall and a 5% rise. It does not, and it has not for as long as the data exists. The pattern is called **skew** when it is asymmetric in this way, and a **smile** when both wings are bid above the at-the-money strike, which is what a single-name equity chain more often shows because a takeover or a biotech result can be just as violent upward as downward. The standard explanation is the **leverage effect**: when a stock falls, its debt-to-equity ratio rises and its equity becomes more volatile, so a decline is followed by more turbulence than a rally of the same size. Add to that the demand side — institutions and funds buy downside protection for reasons that have nothing to do with a forecast, and that steady bid is what makes the puts expensive. What follows for a learner is a discipline rather than a theory: **the skew is the price of insurance, not a prediction of a crash.** A chain showing 32% for the at-the-money strike and 44% for a 10%-out-of-the-money put is saying that a lot of people want that contract and few want to sell it, which is why a put can be expensive and still be the right purchase for a position that cannot afford the fall. The test is the same one O7 applied to the collar — what is the position paying for the floor, in dollars, and can it afford that every period it is renewed? Two practical consequences follow from the shape. The first is that the same protection can be bought at different prices along the wing: a put spread — long a nearer strike, short one further out — buys most of the protection for a fraction of the premium by giving away the extreme tail, which is precisely the part of the wing where implied volatility is highest. That is why put spreads and collars are the standard institutional hedge and an outright long put is not: the steepest part of the skew is the part the spread sells. The second is that skew has its own term structure, and it is usually steeper in the near months and flatter further out, because near-dated insurance is bought against specific events while long-dated insurance is bought against a general state of the world. A surface that stays steep all the way out is a market that expects trouble to persist; a surface that flattens out quickly is one where the fear is dated. Reading both directions together — expiry and strike — is what the practitioner calls reading the surface, and the useful habit is to look at it before choosing a structure rather than after. • Expiry direction: the front month carries the events inside it, so its implied volatility rises relative to the next expiry out. • Strike direction: equity puts usually quote above calls of the same expiry — skew — because of the leverage effect and the demand for protection. • Skew is the price of insurance, not a forecast; the test is the dollar cost of the floor, not whether it is “cheap”. • A put spread or a collar buys most of the floor for a fraction of the premium by selling the steepest part of the wing. • Skew has term structure too: steep near, flatter out means the fear is dated. The failure mode of the surface is treating either direction as a signal on its own. A high front-month implied volatility is not “expensive premium to sell” until the event inside it is identified, because the event is exactly what the seller is taking the other side of — and a steep put skew is not a warning that a crash is coming, it is a statement about who is buying and who is willing to sell.

October 1987: when the smile became a smirk

Before 1987, index options were priced much as the textbook says: implied volatility across strikes was close to flat. On Monday, October 19, 1987 the Dow Jones Industrial Average fell 22.6% in a single session — a move that a normal distribution, at the volatility of the time, places beyond any plausible probability. The options market never forgot it. From then on, out-of-the-money index puts traded at implied volatilities persistently above the at-the-money level, and the skew became a permanent feature of every equity-index market. Mark Rubinstein documented the change in 1994, and David Bates showed that index option prices after the crash went on embedding a fear of the next one. That history says what the skew is pricing. It is partly the **leverage effect** from the page before — volatility tends to rise when prices fall — which is real and visible in ordinary weeks. It is also the **jump**: the possibility that the market gaps down by an amount no continuous hedging can follow. A dealer who sells a 10%-out-of-the-money put can delta-hedge it against a slide (O10); nobody can delta-hedge it against a 20% opening gap. A risk that cannot be hedged away has to be paid for, and the payment appears as extra implied volatility on exactly the strikes a gap would hit. The third ingredient is who wants what. Pension funds, insurers and leveraged investors buy downside protection for reasons that have nothing to do with a forecast — mandates, capital rules, the cost a drawdown would impose on their own clients. Natural sellers are scarcer: index puts are mostly sold by dealers and specialised funds that want to be paid for carrying crash risk. Persistent demand meeting reluctant supply is a price, and in options the price is quoted in volatility. Studies of index options have found that sellers of puts have, on average, been more than compensated — the variance risk premium of O12, concentrated in the downside wing — with the bill presented all at once, in the crashes. • Leverage effect: volatility rises as prices fall, so the downside is priced as more volatile. • Jump risk: a gap cannot be hedged continuously, so the wing it would hit is charged for it. • Demand and supply: structural buyers of protection, few natural sellers. What each ingredient predicts about the curve — Leverage effect: A skew even in calm markets, steepening as prices fall · Jump risk: The far-out puts priced richest — the gap lives in the tail ← · Demand for protection: Steepening when hedging demand rises, whatever anyone forecasts

Measuring the slope, and what it does to a hedge

The working measure is the risk reversal: the 25-delta put’s implied volatility minus the 25-delta call’s, in volatility points. On a quiet day for the S&P 500 the gap is several points, and it widens as markets fall. Cboe also publishes the **SKEW Index**, launched in 2011, which reads the prices of far-out-of-the-money S&P 500 options and expresses them as a number that sits at 100 when no extra tail is priced and has mostly ranged between 100 and 150. The discipline of the page before applies to both: these measures say what insurance costs, not when the crash comes. A high SKEW reading has not been a reliable timer of declines — which is what you should expect of a price that rises whenever many people buy protection at once. The more useful consequence is what the slope does to the structures of O7 and O13. On this lesson’s surface the 4,750 put costs 21.99 index points at its own 19% — more than twice the 10.16 it would cost at the at-the-money 15%. That difference is the skew, and every structure that touches the downside either pays it or collects it. A **put spread** buys the 4,750 put and sells the 4,500, and because the 4,500 sits even further up the skew at 24% it sells for 7.95 points rather than the 0.38 a flat surface would give it: the spread costs 14.04, about 64% of the outright put, where on a flat surface it would cost 96%. A **collar** finances the same put by selling the 5,250 call — but the call sits on the cheap side of the curve at 12%, so it raises only 7.63, and the collar still costs 14.36 points where a flat surface would have paid a credit of 6.22. Read the pair carefully, because it reverses a common intuition. On a skewed index surface the put spread is the efficient way to buy protection, because it sells the most expensive wing; the collar is less efficient than a flat-volatility calculator suggests, because the call it sells is priced off the cheapest wing. A single stock with a symmetric smile changes the arithmetic again — which is why the surface is read before the structure is chosen, never after. • Risk reversal: 25-delta put volatility minus 25-delta call volatility — the slope in one number. • SKEW Index: 100 means no extra tail priced; mostly between 100 and 150. • Put spread: sells the richest wing, so the skew helps it. • Collar on an index: sells the cheapest wing, so the skew works against it. The same protection on two surfaces (index points, 30 days) — 4,750 put: skewed surface vs flat 15%: 21.99 vs 10.16 · 4,750/4,500 put spread: 14.04 vs 9.78 · Spread as a share of the outright put: 64% vs 96% ← · Collar: long 4,750 put, short 5,250 call: 14.36 debit vs 6.22 credit

Other markets, other smiles — and what the curve does when the price moves

The downward skew is an equity-index fact, not an options fact. Single stocks usually show a flatter, more symmetric curve, because a company can be taken over or win a drug approval as suddenly as it can disappoint: the upside wing carries its own jump. Stocks with heavy speculative call buying — the meme names of 2021 — have at times had the upside wing priced above the downside. Commodity options often tilt the other way from equities, because the violent move in a supply shock is a spike upward; crude oil’s fall to a negative futures price in April 2020 is the reminder that this is a tendency, not a law. In currency pairs the expensive wing belongs to whichever currency investors run to in a crisis, and the risk reversal changes sign when the crisis changes. The second practical question is what the curve does when the underlying moves, and there are two simple models of it. Under **sticky strike**, each strike keeps its implied volatility as the price moves, so after a fall the new at-the-money strike sits further up the old skew and at-the-money volatility rises. Under **sticky delta**, the whole curve slides along with the price, so at-the-money volatility stays where it was. Real markets sit between the two and change regime; in sharp index sell-offs at-the-money volatility usually rises faster than either model alone suggests, because the whole surface lifts while the price falls. A position that is long the wing and short the middle — or the reverse — makes or loses money according to which of these happens, which is why anyone holding a skew position watches the surface rather than just the price. The working rules are short. Read the skew before choosing a hedge. Price a protective structure in dollars on the real surface, never on one volatility. And treat a steepening skew as information about the price of insurance and the demand for it — the thing it measures — rather than a forecast of the event it insures. • Indexes: put skew. Single stocks: flatter or a smile, sometimes call-heavy. • Commodities: often call skew — supply shocks spike upward. • Currencies: the crisis currency’s wing is the expensive one. • Sticky strike vs sticky delta: two simple models of how the curve moves with the price. A skew trade is a position on the shape of the curve, and the shape can move against you while the price does exactly what you expected.

Sources

• The post-1987 shape of index option pricesRubinstein, “Implied Binomial Trees” (Journal of Finance, 1994)
• Crash fears priced into index options after 1987Bates, “Post-’87 Crash Fears in the S&P 500 Futures Option Market” (Journal of Econometrics, 2000)
• Why index puts are expensiveBondarenko, “Why Are Put Options So Expensive?” (Quarterly Journal of Finance, 2014)
• The SKEW Index and its methodologyCboe Global Markets, SKEW Index white paper
• The leverage effectBlack, “Studies of Stock Price Volatility Changes” (1976)