On the Joint Dynamics of the Spot and the Implied Volatility Surface

In this paper, we revisit the "Smile Dynamics" problem. In a previous work, Bergomi built a class of linear stochastic volatility models in which he specified the joint dynamics between the underlying and its instantaneous forward variances. The author introduced a quantity, which he called the Skew Stickiness ratio, in order to relate two quantities of interest: the first quantity is the correlation between the increments of the at-the-money implied volatility of maturity T and the log-returns of the underlying, while the second quantity is the implied skew of the same maturity T. In our work, we continue the study of the Skew stickiness ratio both from theoretical and empirical point of view. First, we provide a method to estimate the SSR (skew stickiness ratio) from option prices, this measure is called the implied SSR as it is conducted under the risk-neutral pricing measure Q. Next to that, we recall how to measure the realized SSR under the real-world probability measure P and we point out empirically that there is a discrepancy between the implied SSR and the realized SSR. The empirical study shows also that the implied SSR, in the limit of short maturities, can take a value superior to 2 which is in discordance with the results obtained in linear stochastic volatility models. For this reason, we show that the positive quantity (SSR -2) is coherent with the presence of jumps in a stochastic volatility model