Editing Volatility smile (section)

==Implied volatility==
In the [[Black–Scholes]] model, the theoretical value of a [[vanilla option]] is a [[Monotonic function#Monotonicity in calculus and analysis|monotonic increasing function]] of the volatility of the underlying asset. This means it is usually possible to [[root finding algorithm|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 are 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.

===Formula===

: <math>\operatorname{Call}x = \mathrm{ATM} + 0.5\operatorname{RR}x + \operatorname{Fly}x</math>

: <math>\operatorname{Put}x = \mathrm{ATM} - 0.5 \operatorname{RR}x + \operatorname{Fly}x</math>

where:

* <math>\operatorname{Call}x</math> is the implied volatility at which the ''x''%-delta call is trading in the market
* <math>\operatorname{Put}x</math> is the implied volatility of the ''x''%-delta put
*ATM is the At-The-Money Forward volatility at which ATM Calls and Puts are trading in the market
* <math>\operatorname{RR}x = \operatorname{Call}x - \operatorname{Put}x</math>
* <math>\operatorname{Fly}x = 0.5(\operatorname{Call}x + \operatorname{Put}x) - \mathrm{ATM}</math>

[[Risk reversal]]s are generally quoted as ''x''% delta risk reversal and essentially is Long ''x''% delta call, and short ''x''% delta put.

[[Butterfly (options)|Butterfly]], on the other hand, is a strategy consisting of:
−''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).