Mean-reversion properties of implied volatilities

Abstract

In this paper, we present a new stylized fact for options whose underlying asset is a stock index. Extracting implied volatility time series from call and put options on the Deutscher Aktien index (DAX) and financial times stock exchange index (FTSE), we show that the persistence of these volatilities depends on the moneyness of the options used for its computation. Using a functional autoregressive model, we show that this effect is statistically significant. Surprisingly, we show that the diffusion-based stochastic volatility models are not consistent with this stylized fact. Finally, we argue that adding jumps to a diffusion-based volatility model help recovering this volatility pattern. This suggests that the persistence of implied volatilities can be related to the tails of the underlying volatility process: this corroborates the intuition that the liquidity of the options across moneynesses introduces an additional risk factor to the one usually considered.

Keywords:

• implied volatility
• stylized fact
• stocahstic volatility models
• volatility surface dynamics
• autoregressive models

JEL Classification :

• G13
• C01

Acknowledgements

We are grateful to Marie Brière, José Da Fonseca, Serge Darolles, Philippe Dumont, Antoine Jacquier and Nour Meddahi for very constructive comments. We thank the participants of the ‘Journée d'Econométrie’ seminar (Nanterre, Nov. 2007), the French Finance Association Annual meeting (Lille, France, 2008), the Forecasting Financial Markets 2008 Annual Meeting (Aix en Provence, France, 2008), the Augustin Cournot Doctoral Days (Strasbourg, France, 2008) and the PhD Seminar of the Sorbonne (France, 2008) for their questions and remarks. We also thank two anonymous referees for remarks and suggestions for improvements. The usual disclaimer nonetheless applies, and all errors remain ours. The paper previously circulated under the title ‘Smiled Dynamics of the Smile’.

Notes

The intrinsic value of an option is the value of the payoff function computed with the current value of the underlying asset. For example, with a payoff (S _T −K)⁺, the intrinsic value of an option price at time t is (S _t −K)⁺.

The implied variance is the square of the implied volatility. Thus and τ=T−t.

Note that this effect is not due to the fact that we use out-of-the-money options: the stylized fact presented here still holds in a less pronounced way when using in-the-money options instead. Tables and figures are available upon request. They are not included here, since they look fairly similar to what is presented in the out-of-the-money case.

The computation is tedious but straightforward, given the PDE obtained in the Heston case. Nonetheless, these calculations are presented in the two previous references and we chose not to report it in this paper.

Even if jumps are usually added in the Heston model to deal with the erratic behavior of implied volatilities for short maturities, it is also used to make the volatility process's tails heavier.