Rama Cont, Jose da Fonseca
Although the Black Scholes formula is widely used by market practitioners, when applied to (``vanilla'') call and put options it is very often reduced to a means of quoting option prices in terms of another parameter, the implied volatility. The implied volatility of call options at a given date is a function of the strike price and exercice date: this function is called the implied volatility surface.
Two features of this surface have captured the attention of researchers in financial modeling. First, the non-flat instantaneous profile of the surface, whether it be a ``smile'', ``skew'' or the existence of a term structure, point out to the insufficiency of the Black Scholes model for matching a set of option prices at a given time instant. Second, the level of implied volatilities changes with time, deforming the shape of the implied volatility surface. The evolution in time of this surface captures the evolution of prices in the options market.
We present here an empirical study of this evolution : how does the implied volatility surface actually behave? How can this behavior be described in a parsimonious way?
Our study is based on time series of implied volatilities of FTSE, SP500 and DAX options. Starting from market prices for European call and put options, we describe a non-parametric smoothing procedure for constructing a time series of smooth implied volatility surfaces. We then study some of the statistical properties of these time series. By applying a Karhunen-Loéve decomposition to the daily variations of the volatility surface, we extract the principal deformations of the surface, study the time series of the corresponding factor loadings and examine their correlation with the underlying asset.
Based on these findings, we show that the implied volatility surface can be modeled as a mean reverting random field with a covariance structure matching the empirical observations. We propose a two factor model for the evolution of the surface which adequately captures the observed properties, indicate simple methods for estimating model coefficients. This model extends and improves the well known "constant smile" model used by practitioners. The consequences of these findings for measurement and management of volatility risk are then outlined.