Volatility Smile

A volatility smile is the pattern seen when the implied volatility of options on one underlying asset, all with the same expiry, is plotted against strike price. Implied volatility is lowest for strikes near the current price and rises for strikes further away in either direction, giving a curve shaped like a smile. The term is also used loosely for any variation of implied volatility across strikes, including the one-sided slope more precisely called the volatility skew.

The smile is evidence that markets do not price options as the Black-Scholes-Merton model assumes. That model uses a single volatility for an asset, so it predicts a flat line. The symmetric smile is most typical of options on major currency pairs. Equity index options show a skew, and the smile on short-dated options is more pronounced than on long-dated ones.

An option far from the money pays off only after a large move. If such options trade at higher implied volatilities than the model's single figure, the market is pricing large moves as more likely than a lognormal distribution allows. In statistical terms the distribution of future prices implied by option prices has fat tails, or excess kurtosis. A symmetric smile indicates heavy tails on both sides, and a skew indicates that one tail is heavier.

The link can be made exact. Douglas Breeden and Robert Litzenberger showed in 1978 that the full set of option prices across strikes determines the market-implied probability distribution of the asset price at expiry. The smile is therefore a compact description of that distribution.

In the foreign exchange market, the smile for each expiry is summarized by three numbers: the at-the-money volatility, the risk reversal and the butterfly. The 25-delta butterfly measures the curvature of the smile:

• $\sigma_{25\Delta \, \text{call}}$ and $\sigma_{25\Delta \, \text{put}}$ are the implied volatilities of the out-of-the-money call and put with deltas of 0.25 in absolute value.
• $\sigma_{\text{ATM}}$ is the implied volatility of the at-the-money option.

Suppose the 25-delta call trades at 11.0 percent, the 25-delta put at 12.0 percent and the at-the-money option at 10.0 percent. The butterfly is $(11.0 + 12.0) / 2 - 10.0 = 1.5$ volatility points, showing that the wings are on average 1.5 points above the centre. The risk reversal, the call volatility minus the put volatility, is $11.0 - 12.0 = -1.0$, showing that the smile tilts toward puts. From these three quotes a dealer can reconstruct the whole curve.

• Volatility is not constant. Actual volatility changes over time. When volatility is itself uncertain, large moves become more likely than under a fixed level, which raises the value of options away from the money.
• Prices jump. Markets sometimes move discontinuously on news. Jumps add weight to the tails, especially at short horizons, which is why short-dated smiles are steep.
• Supply and demand. Hedgers and speculators concentrate their buying in particular strikes, and dealers charge for the risk of supplying them.
• Hedging costs. Options far from the money are harder to hedge, because their sensitivity to volatility itself changes as volatility moves, an effect measured by vomma.

Several families of model extend the basic framework so that it fits observed prices. Jump-diffusion models, introduced by Robert Merton in 1976, add random jumps to the price. Stochastic volatility models, of which Steven Heston's 1993 model is the best known, let volatility follow its own stochastic process. Local volatility models, developed by Bruno Dupire and by Emanuel Derman and Iraj Kani in 1994, make volatility a function of price and time chosen to match every quoted option exactly. The SABR model of 2002 is widely used to interpolate smiles in interest rate markets.

Each fits today's smile but implies different behaviour for the smile in future, and so gives different hedges and different prices for exotic options.

Traders treat the smile as a pricing tool. To value an option at a strike that is not quoted, they read its implied volatility from the fitted curve and insert it into the standard formula. A butterfly spread is a direct position on the curvature of the smile, and risk reversals are positions on its tilt. Risk managers recalculate portfolios under shifts in the level, tilt and curvature of the smile, not only its level.

The smile must also satisfy consistency conditions. If the curve is too sharply bent at some strike, a butterfly could be bought for a negative price, which would be an arbitrage. Curve-fitting methods are designed to rule this out.

A smile is built from quoted prices, and quotes for options far from the money are sparse and have wide bid-ask spreads, so the wings of the curve are uncertain. The smile describes prices, which include risk premiums, and should not be read as a pure forecast of probabilities. Its shape shifts with market conditions, and a curve fitted at one moment can be out of date within minutes in a fast market. Together with the pattern across expiries, the smiles for all maturities make up the volatility surface.