Volatility Smile

The volatility smile is an empirical pattern where implied volatility varies across different strike prices for options with the same expiration date on the same underlying. In many markets, the IV is lowest for at-the-money options and rises for in-the-money and out-of-the-money options, creating a U-shaped curve that resembles a smile. Related patterns—volatility skew and volatility term structure—describe IV varying across moneyness and expiration.

The smile pattern

In a perfect Black-Scholes model world with constant volatility, all options on the same underlying and expiration should have the same implied volatility, regardless of strike.

In practice, this does not happen. The plot of implied volatility vs. strike price looks like a smile (U-shaped) or a skew (lopsided). For example:

• ATM $100 call/put: IV = 18%
• $95 OTM put: IV = 22%
• $105 OTM call: IV = 21%

The OTM options have higher implied volatility than the ATM options. The market is pricing in a higher probability of large moves at the extremes.

Volatility skew vs. smile

Smile: Symmetric; IV rises on both sides of the strike equally. More common in indices and currencies.

Skew: Asymmetric; IV rises more on one side. Most common in single stocks, where OTM puts have much higher IV than OTM calls (fear of crashes, not rallies).

Why the smile exists

Several factors contribute:

1. Jump risk: Stock prices can gap overnight (earnings, news). Jumps are non-log-normal. Options priced under jump models have higher IV for OTM options, which are sensitive to jumps.

2. Leverage effect: As stock prices fall, equity volatility rises (companies become riskier). This creates skew: OTM puts are more valuable because falls are more volatile.

3. Demand imbalances: After a crash (e.g., 2008), demand for OTM puts surges, pushing their IV higher. Supply-demand creates skew.

4. Model limitations: Black-Scholes assumes log-normal prices and constant volatility. Real markets have fatter tails (more extreme moves) and stochastic volatility, both of which create smile.

Trading the smile

Volatility smile traders:

1. Identify mispricings: If a model predicts IV should be flat but the smile is steep, they identify where the market is over- or under-pricing moves.

2. Sell overpriced volatility: Buy ATM options, sell OTM options (higher IV), betting the smile flattens.

3. Buy underpriced volatility: The reverse; buy OTM options, sell ATM, betting the smile steepens or the OTM options become more expensive.

4. Skew trades: Buy OTM puts, sell OTM calls (or vice versa) to express a view on skew changes.

Volatility term structure

Related to the smile is the volatility term structure—how IV varies across expiration dates. Near-term options might have IV of 20%; 6-month options might have 25%. This reflects different volatility expectations for the near vs. far future.

The term structure shifts with regime changes. Before earnings, the near-term IV spikes; after earnings (once uncertainty resolves), it may fall faster than longer-term IV.

Stochastic volatility models

To model the smile properly, quants use stochastic volatility models (e.g., Heston, SABR, local volatility) that allow volatility to change over time and vary by price level. These models fit the smile better than Black-Scholes and improve hedging.

See also