# Stylized Facts & Modelisation

Stylized Facts from data observations

Property: Mean Reversion.

Description: IV tends to mean revert around an average IV level.

Model: Orstein Uhlenbeck

Property:  Smile slope decreases as .

Description: Volatility slope behaves as the ATM IV.

Model: Model of Stochastic Volatility with 2 factors in order to control separately ATM and the skew.

Property: Smile curvature decreases as .

Description:  Volatility curve behaves as the ATM with a different speed of mean reversion.

Model: A two factor stochastic volatility enables this type of control.

Property: Put curvature is higher than call curvature.

Description: Dissymmetry between calls and puts.

Model: Jump model allows the generation of put prices more expensive than call prices.

Property: Smile dynamic.

Description: When the spot vibrates, volatility ATM approximately follows the smile (sticky strike).

Model: A mixture of model between local volatility and stochastic volatility allows this type of behavior.

The underlying philosophy for volatility modeling is to satisfy a series of constraints:

• Nature of the volatility

Volatility number  has to be compared with some historical measure

• Shape of the ATM volatility

ST volatility representation is mostly impacted by mean reversion effect.

LT volatility representation is impacted by stochastic IRs volatility.

• Shape of the skew and curvature

Downward slope is mainly due to volatility risk and in the case of stocks to credit risk.

It decreases as a function if the inverse of the square root of maturity.

Curvature decreases as the inverse of maturity.

• Extrapolation in time

LT smile is due to the cost of delta hedge and vega hedge when the volatility changes (vanna/volga).

• Dynamic of the volatility

The ATM volatility for equities around the ATM strike behaves as a sticky strike movement. In other words, the volatility associated with a given strike does not change when the spot moves.

All these facts and constraints must be taken into account in the volatility representation model. They include both the photography  and the dynamic type of constraints.