Volatility Smile - Volatility Smiles and Implied Volatility

Volatility Smiles and Implied Volatility

In the Black–Scholes model, 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 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.

Market practitioners use the term implied-volatility to indicate the volatility parameter for 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, and short 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)