# Dynamics of implied volatility surfaces

*Rama Cont* and
*José Da Fonseca* ()

*Quantitative Finance*, 2002, vol. 2, issue 1, 45-60

**Abstract:**
The prices of index options at a given date are usually represented via the corresponding implied volatility surface, presenting skew/smile features and term structure which several models have attempted to reproduce. However, the implied volatility surface also changes dynamically over time in a way that is not taken into account by current modelling approaches, giving rise to 'Vega' risk in option portfolios. Using time series of option prices on the SP500 and FTSE indices, we study the deformation of this surface and show that it may be represented as a randomly fluctuating surface driven by a small number of orthogonal random factors. We identify and interpret the shape of each of these factors, study their dynamics and their correlation with the underlying index. Our approach is based on a Karhunen-Loeve decomposition of the daily variations of implied volatilities obtained from market data. A simple factor model compatible with the empirical observations is proposed. We illustrate how this approach models and improves the well known 'sticky moneyness' rule used by option traders for updating implied volatilities. Our approach gives a justification for use of 'Vega's for measuring volatility risk and provides a decomposition of volatility risk as a sum of contributions from empirically identifiable factors.

**Date:** 2002

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