Volatility Smile

Volatility smile
In finance, the volatility smile is a long-observed pattern in which at-the-money options tend to have lower implied volatilities than in- or out-of-the-money options. This results from the underlying assumption that asset returns have normal distributions. 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.[1]

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 complex exoticoptions.

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 underlying.
Volatility smiles and implied volatility
In the Black–Scholesmodel, the theoretical value of a vanilla option is a monotonic increasing function of the volatility of the underlying. Furthermore, except in the case of American options with dividends whose early exercise could be optimal, 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 impliedvolatilities 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.

Market practitioners use the term implied-volatility to indicate the volatility parameterfor ATM (at-the-money) option. Adjustments to this value is undertaken by incorporating the values of Risk Reversal and Flys (Skews) to determine the actual volatility measure that may be used for options with a delta which is not 50.

Callx = ATM + 0.5 RRx + Flyx

Putx = ATM - 0.5 RRx + Flyx

Risk reversals are generally quoted X% delta risk reversal and essentially is Long X% delta call, andshort X% delta put.

Butterfly, on the other hand, is Y% delta fly which mean Long Y% delta call, Long Y% delta put, short one ATM call and short one ATM put. (small hat shape).

Implied volatility and historical volatility
It is helpful to note that implied volatility is related to historical volatility, but the two are distinct. Historical volatility is a direct measure of the movement ofthe underlying’s price (realized volatility) over recent history (e.g. a trailing 21-day period). Implied volatility, in contrast, is determined by the market price of the derivative contract itself, and not the underlying. Therefore, different derivative contracts on the same underlying have different implied volatilities as a function of their own supply and demand dynamics. For instance, the...