Volatility Skew

Volatility skew is the uneven distribution of implied volatility across options that share an underlying asset and an expiry date but differ in strike price. If the standard option pricing model held exactly, all such options would have the same implied volatility. In practice they do not. In equity index markets, options with low strikes carry markedly higher implied volatilities than options with high strikes, so a plot of implied volatility against strike slopes downward.

The term is used both for this sloping shape and for any measure of its steepness. A roughly symmetric, U-shaped pattern is called a volatility smile, and the lopsided equity pattern is sometimes called a smirk. The skew across strikes, combined with the pattern across expiries, forms the volatility surface.

Several conventions are used. One compares options at fixed distances from the current price, such as the implied volatility at 90 percent of the price minus that at 110 percent. Another compares options with the same delta, which adjusts for the volatility and maturity of the asset. The most common is the 25-delta skew:

• $\sigma_{25\Delta \, \text{put}}$ is the implied volatility of the out-of-the-money put with a delta of -0.25.
• $\sigma_{25\Delta \, \text{call}}$ is the implied volatility of the out-of-the-money call with a delta of 0.25.

If the 25-delta put trades at 24 percent implied volatility and the 25-delta call at 18 percent, the skew is 6 volatility points. Dividing by the at-the-money volatility, say 20 percent, gives a normalized skew of 0.30, which allows comparison between calm and turbulent periods. A skew rank places the current reading within its range over the past year. In currency markets the same quantity, with the sign convention reversed, is quoted as a risk reversal.

Before the stock market crash of October 1987, implied volatilities of index options in the United States were fairly flat across strikes. Afterwards a pronounced skew appeared and has persisted ever since. The crash showed that very large one-day falls were possible, and the market began to price them.

Several explanations are offered, and they are not mutually exclusive.

• Demand for protection. Investors who own shares buy an out-of-the-money put option as insurance, and many finance it by selling calls. Persistent buying of puts and selling of calls raises the price of one and lowers the other.
• Volatility rises when prices fall. Share prices and volatility are negatively related. If the index drops toward a low strike, volatility is likely to be higher by then, and options at that strike are priced accordingly.
• Fatter left tail. The Black-Scholes-Merton model assumes a lognormal distribution of future prices. The skew implies that the market assigns a higher probability to large falls than that distribution allows, that is, negative skewness and fat tails.
• Limited supply. Sellers of downside protection face large potential losses and demand a premium for providing it.

The direction of skew depends on which move the market fears. In individual shares the pattern resembles that of indices but is usually flatter. In many commodities the skew runs the other way: supply disruptions cause sudden price rises, so high-strike calls carry higher implied volatility. Shares subject to speculative buying can show the same call skew. In currency options, the smile is closer to symmetric, and the sign of the risk reversal shows which currency the market expects to fall in a crisis.

The skew determines the relative cost of common positions. A collar, which buys a put and sells a call against a shareholding, is more expensive when the skew is steep, because the put bought is dear and the call sold is cheap. A bear put spread, which buys one put and sells a lower-strike put, benefits from selling the option with the higher implied volatility. Traders who believe the skew is too steep or too flat take positions through risk reversals and ratio spreads, hedged against moves in the underlying.

Skew also affects risk management. The delta of an option depends on how implied volatility is assumed to change as the price moves, and the sensitivity that captures this interaction is vanna. Dealers debate whether volatility stays fixed at each strike or moves with the price, and the answer changes the hedge.

A steep skew is often read as a sign of fear, and exchanges publish indices that summarize it. The reading needs care. Skew reflects the price of protection, which combines the perceived probability of a fall with the premium investors will pay to avoid its consequences. It has not proved a reliable predictor of crashes.

Quoted skew also depends on the measure chosen, the expiry and the quality of prices for options far from the money, which trade less and have wide bid-ask spreads. Comparisons should use the same definition throughout.