Implied Volatility Skew, Smirks, Smiles, Term Structure and Surface

In finance, the volatility smile is a long-observed pattern in which out of the money options tend to have higher implied volatilities than at the money options.

The pattern displays different characteristics for different markets and results from the probability of extreme moves. Equity options traded in American markets did not show a volatility smile before the Crash of 1987 but began showing one afterwards. Modelling the volatility smile is an active area of research in quantitative finance. Typically, a quantitative analyst will calculate the implied volatility from liquid vanilla options and use models of the smile to calculate the price of more exotic options. A closely related concept is that of term structure of volatility, which refers to how implied volatility differs for related options with different maturities. An implied volatility surface is a 3-D plot that combines volatility smile and term structure of volatility into a consolidated view of all options for an underlier.

In the Black-Scholes model, the theoretical value of a vanilla option is a monotonic increasing function of the Black-Scholes volatility. The price is a strictly increasing function of volatility. This means it is usually possible to compute a unique implied volatility from a given market price for an option. This implied volatility is best regarded as a rescaling of option prices which makes comparisons between different strikes, expirations, and underlyings easier and more intuitive.

When implied volatility is plotted against strike price, the resulting graph is typically downward sloping for equity markets, or valley-shaped for currency markets. For markets where the graph is downward sloping, such as for equity options, the term volatility skew is often used. For other markets, such as FX options or equity index options, where the typical graph turns up at either end, the more familiar term volatility smile is used. For example, the implied volatility for upside (i.e. high strike) equity options is typically lower than for at-the-money equity options. However, the implied volatilities of options on foreign exchange contracts tend to rise in both the downside and upside directions. In equity markets, a small tilted smile is often observed near the money as a kink in the general downward sloping implicit volatility graph. Sometimes the term smirk is used to describe a skewed smile.

Term structure of volatility

For options of different maturities, we also see characteristic differences in implied volatility. However, in this case, the dominant effect is related to the market's implied impact of upcoming events. For instance, it is well-observed that realized volatility for stock prices rises significantly on the day that a company reports its earnings.

Correspondingly, we see that implied volatility for options will rise during the period prior to the earnings announcement, and then fall again as soon as the stock price absorbs the new information. Options that mature earlier exhibit a larger swing in implied volatility than options with longer maturities. Other option markets show other behavior. For instance, options on commodity futures typically show increased implied volatility just prior to the announcement of harvest forecasts. Options on US Treasury Bill futures show increased implied volatility just prior to meetings of the Federal Reserve Board (when changes in short-term interest rates are announced).

Volatility term structures list the relationship between implied volatilities and time to expiration. The term structures provide another method for traders to gauge cheap or expensive options.

Implied volatility surface

It is often useful to plot implied volatility as a function of both strike price and time to maturity. The result is a 3-D surface whereby the current market implied volatility (Z-axis) for all options on the underlying is plotted against strike price and time to maturity (X & Y-axes). The implied volatility surface simultaneously shows both volatility smile and term structure of volatility. Option traders use an implied volatility plot to quickly determine the shape of the implied volatility surface, and to identify any areas where the slope of the plot (and therefore relative implied volatilities) seems out of line. The graph shows an implied volatility surface for a mixtrure of puts and calls on the CAC 40 index.

The code below shows how we can calculate the implied volatility for an option given a market price.

Athough it is possible to individually calculate the implied volatility for a series of strikes (skew, smile), expiry dates (Term Structure) or a combination of both (surface), the Quant Express libraries contain specific classes to simplify the task and fit the data to smooth curve. Examples are shown below: