FRM Exam Part II · Volatility Smiles and Volatility Surfaces

Volatility Skew for Equity and Index Options Explained

Updated 11 October 2026 · Fact-checked

Equity volatility skew is the pattern where implied volatility falls as strike rises. Low-strike options carry the highest implied volatility: out-of-the-money puts and, via put-call parity, deep in-the-money calls at that strike. It implies a left-heavy, thin-right-tail distribution versus lognormal. Explanations are the leverage effect and crashophobia. To answer, compare strikes, read the slope, infer the tail.

Understand Volatility Skew for Equity and Index Options

Black-Scholes assumes one constant volatility for all strikes. If that were true, plotting implied volatility against strike would give a flat line. For equity and index options it does not. Implied volatility is highest at low strikes and declines as strike rises. This downward slope is the volatility skew, sometimes called a volatility smirk. The term "smile" usually describes a U-shape, as seen in foreign currency options. The equity pattern is a one-sided slope, not a U.

The skew tells you what the market believes about the distribution of the future stock price. A low-strike put pays off only if the price falls a lot. Pricing it with a higher volatility means the market gives more probability to large falls than a lognormal distribution would. A high-strike call pays off only on big rises. Its lower implied volatility means the market gives less probability to large rises. So the implied distribution has a heavier left tail and a thinner right tail than the lognormal distribution with the same mean and standard deviation.

Two explanations are tested. The leverage effect: when a firm's equity value falls, its debt-to-equity ratio rises, so equity becomes riskier and its volatility increases. When equity rises, leverage falls and volatility falls. So volatility is negatively related to the stock price, which produces the skew. The second is crashophobia: since the 1987 crash, traders have feared a sudden market-wide fall and bid up the price of out-of-the-money puts, especially on indices. Portfolio managers also buy puts to insure holdings, adding demand for low strikes.

The skew is also expressed by the relation between implied volatility and the strike relative to spot (K/S). Low K/S means high volatility. Keep the logic in one chain: low strike, higher implied volatility, more weight in the left tail, so put prices are higher than Black-Scholes with a flat volatility would give.

Key formulas to remember

Skew direction (equity): σ_implied falls as K rises; Downward-sloping. Low strikes have the highest implied volatility, high strikes the lowest.
Implied distribution vs lognormal: Left tail: heavier than lognormal; right tail: thinner than lognormal; Compared with a lognormal distribution with the same mean and standard deviation.
Flat Black-Scholes benchmark: σ_implied(K) = σ for all K; A constant volatility gives a flat line. Any slope or curve signals a departure from lognormality.
Leverage effect: Equity value ↓ ⇒ D/E ↑ ⇒ equity volatility ↑; Negative relationship between stock price and volatility.
Put-call parity consequence: c − p = S − K·e^(−rT) (no dividends); A European call and put with the same strike and maturity must have the same implied volatility, so the skew is the same from either side.

How to solve Volatility Skew for Equity and Index Options questions

Use this method for any question on equity skew, whether it gives numbers or asks for an explanation.

1Identify the underlying: equity or equity index options point to a downward skew. Foreign currency options point to a smile.
2Order the strikes from low to high and read the implied volatilities. Check that they fall as strike rises.
3Name the pattern: downward-sloping skew or smirk, not a symmetric smile.
4Translate to the implied distribution: higher volatility at low strikes means a heavier left tail. Lower volatility at high strikes means a thinner right tail than lognormal.
5Link to prices, strike by strike: a flat-volatility Black-Scholes model would underprice options at low strikes (out-of-the-money puts, and the in-the-money calls at the same strike) and overprice options at high strikes (out-of-the-money calls), relative to the market.
6Match the explanation asked: leverage effect (debt-to-equity rises as price falls) or crashophobia (demand for crash protection since 1987).
7Use put-call parity if the question mixes calls and puts: same strike and maturity means the same implied volatility.
8Check your answer's direction before choosing an option.

Quickest way: Skew in four checks

When to use it: Use when an MCQ gives a list of implied volatilities or asks which statement about equity options is correct.

Equity or index? Then expect high volatility at low strikes.
Low strike, high volatility, heavy left tail. High strike, low volatility, thin right tail.
If the option mentions a U-shape or both tails heavy, think currency options, not equities.
Reject any option saying the skew arises because the market expects big upward jumps.

Common mistakes in Volatility Skew for Equity and Index Options

Calling the equity pattern a symmetric smile.
Students merge the terms smile and skew and remember the currency U-shape.
Fix: Equity is a downward slope. Currency is a smile with both tails heavy. Link equity with left tail only.
Saying the right tail is heavier because high-strike calls look cheap.
Confusing low implied volatility with high probability.
Fix: Lower implied volatility at high strikes means less right-tail probability than lognormal, so a thinner right tail.
Stating the leverage effect backwards.
Students recall that leverage raises risk but lose the direction.
Fix: A falling stock price raises debt-to-equity, which raises equity volatility. Price down, volatility up.
Thinking calls and puts at the same strike have different implied volatilities.
Students look at put and call prices instead of implied volatility.
Fix: Put-call parity means European options with the same strike and maturity share one implied volatility.
Claiming the skew means the stock will fall.
Mixing up an implied distribution with a forecast.
Fix: The skew shows how the market prices tail risk and demand for protection, not a prediction of the direction.

Worked examples

Example 1

One-year options on an equity index (spot 4,000) show implied volatilities of 28% at strike 3,600, 22% at strike 4,000 and 18% at strike 4,400. Describe the skew and the implied distribution relative to lognormal.

Show the solution

Order strikes: 3,600, 4,000, 4,400. Implied volatilities: 28%, 22%, 18%.
Volatility falls as strike rises, so the skew is downward-sloping, typical of equity indices.
The low strike (3,600) has the highest volatility, so the market gives more probability to large falls than a lognormal distribution.
The high strike (4,400) has the lowest volatility, so the market gives less probability to large rises.
Conclusion: heavier left tail, thinner right tail.

Answer: Downward-sloping skew. The implied distribution has a heavier left tail and thinner right tail than the lognormal.

Example 2

A risk manager notes a firm's share price fell 40% and the firm's equity volatility rose sharply, while its debt was unchanged. Which explanation of the equity skew does this support, and why?

Show the solution

Use the stylised assumption that the firm's debt is constant and equity is the residual claim. In practice the market value of debt can also change.
Under this assumption, equity value fell 40% while debt stayed the same, so debt-to-equity rose.
A higher debt-to-equity ratio makes equity a riskier residual claim, so its volatility rises.
Price down and volatility up is a negative relation between stock price and volatility.
That negative relation implies low strikes (low prices) correspond to higher volatility, which gives a downward skew.
This is the leverage effect, not crashophobia, which concerns demand for crash protection.

Answer: The leverage effect: a falling price raises debt-to-equity and equity volatility, producing higher implied volatility at lower strikes (under the stylised assumption of constant debt).

Exam tips

Always tie the skew direction to the asset class: equity downward slope, currency smile.
Describe tails relative to the lognormal, not in absolute terms.
Know both explanations by name and the mechanism behind each, and that crashophobia is linked to the 1987 crash.
Expect statements mixing correct and reversed tail claims. Check left versus right carefully.
If asked about pricing, state it per strike: options at low strikes (out-of-the-money puts, and in-the-money calls at the same strike) are priced above flat-volatility Black-Scholes values, and options at high strikes (out-of-the-money calls) are priced below them.

Practice questions from Volatility Smiles and Volatility Surfaces

Volatility Skew for Equity and Index Options: frequently asked questions

Why is implied volatility higher for low-strike equity options?

The market assigns more probability to large falls than a lognormal distribution does. Leverage effects and demand for crash protection push up the prices of low-strike options, which shows up as higher implied volatility.

What is crashophobia?

It is the fear of a sudden large market fall, which grew after the 1987 crash. Investors pay more for out-of-the-money puts, raising implied volatility at low strikes.

What is the difference between a volatility smile and a volatility skew?

A smile is a U-shape where both low and high strikes have higher implied volatility, as in currency options. A skew is a one-sided slope, as in equities, where volatility falls as strike rises.

Does the skew mean the market expects a crash?

No. It reflects how the market prices tail risk and the demand for protection. It is a feature of the risk-neutral implied distribution, not a forecast.