Explain what is meant by a smile and by a skew in implied volatility.
The idea
Implied volatility is the volatility that, put into the Black-Scholes-Merton formula, reproduces an option's market price. If the model's assumptions held, with lognormal prices and constant volatility, every option on one underlying and maturity would give the same implied volatility whatever its strike. In practice they do not, and a plot of implied volatility against strike for one maturity is the volatility smile. A smile is U-shaped, with higher implied volatility at both low and high strikes than near the money. A skew, also called a smirk, is a mostly downward-sloping pattern, with the highest implied volatility at the lowest strikes. Both describe how the market prices options relative to the model benchmark. A flat line would mean the market agrees with a lognormal distribution. Do not confuse either shape with the term structure, which varies maturity rather than strike.
A practice question
What does a volatility smile plot for a given underlying asset?
- Implied volatility against time to maturity, for options that share the same strike price.Why not: That is the volatility term structure; a smile fixes maturity and varies the strike.
- Historical volatility against strike price, for options that share the same maturity.Why not: Uses realised volatility; the smile is built from implied volatilities backed out of option prices.
- Implied volatility against strike price, for options that share the same maturity. - correct
- Option premium against strike price, for options that share the same maturity.Why not: Plots prices rather than implied volatilities; a premium curve always slopes down for calls and up for puts.
A volatility smile is the relationship between the Black-Scholes-Merton implied volatility and the strike price at one maturity. Changing the maturity instead gives the term structure.
A harder one
Three-month implied volatilities (in percent) are observed at strikes of 80, 90, 100, 110 and 120 for two underlyings. Underlying X: 27.0, 23.0, 20.0, 18.0, 17.0. Underlying Y: 12.4, 11.0, 10.2, 10.9, 12.2. Which description fits best?
- X shows a U-shaped smile and Y shows a skew, because X has the wider range of volatilities.Why not: Classifies by the size of the range, not by its shape; X is monotonic and Y turns back up.
- X shows a skew that falls as the strike rises, and Y shows a U-shaped smile centred near the 100 strike. - correct
- Both show skews, because the volatility at strike 80 is higher than at strike 120 in each case.Why not: Compares only the two end strikes; Y falls and then rises, so its middle is the minimum.
- Both show smiles, because both are non-flat and the model only allows a smile.Why not: Any non-flat pattern is not a smile; the shape (monotonic or U) decides the label.
X falls monotonically from 27 to 17, a skew. Y has its minimum (10.2) at the money and rises on both sides, a smile. The label depends on the shape, not on the end points alone.
The hardest tier
One week ago the 3-month implied volatilities of an index were 26 (90% strike), 22 (100%) and 19 (110%). Today they are 30 (90%), 25 (100%) and 21 (110%). Measure the skew as the 90% strike volatility minus the 110% strike volatility. Which conclusion is correct?
- The skew was unchanged at 7 points because the at-the-money volatility rose by 3 points.Why not: Treats the at-the-money move as a uniform shift; the 90% strike rose by 4 and the 110% strike by 2.
- The skew steepened from 7 to 9 volatility points, even though the whole curve moved up. - correct
- The skew flattened because every implied volatility rose, so strikes are priced more alike.Why not: A level rise says nothing about slope; the gap between the wings widened.
- The skew steepened by 4 points, the increase in the 90% strike volatility.Why not: Counts only the low-strike move and ignores that the 110% volatility also rose by 2 points.
Skew before = 26 - 19 = 7; after = 30 - 21 = 9, so it steepened by 2 points. The curve shifted up (ATM +3) and also tilted: protection against falls became relatively dearer.
In the app: 6 practice questions on this outcome, spaced review that brings it back before you forget it.
Start free →More in Volatility Smiles and Volatility Surfaces
- Show what put-call parity implies for the implied volatilities of calls and puts.
- Link the form taken by a smile or skew to the price distribution it implies for the underlying, and to the resulting option prices.
- Characterise how currency returns are distributed, and trace what that implies for implied volatility and the pricing of FX options.
- Describe how the smile looks for options on stocks and for options on currencies, and put forward plausible explanations of why.
- Set out other ways in which the volatility smile can be described.
- Characterise volatility term structures and volatility surfaces, and show how option pricing can make use of them.
- Show how an option's Greek letter risk measures are affected by the volatility smile.
- Show how one jump in an asset's price affects a volatility smile.