Dynamics of the implied volatility surface. Theory and empirical evidence

Abstract

I perform a regression analysis to test two of the most famous heuristic rules existing in the literature concerning the behavior of the implied volatility surface. These rules are the sticky delta rule and the sticky strike rule. I present a new specification to test the sticky strike rule that allows for dynamics in the implied volatility surface. In the empirical application, I use monthly implied volatility surfaces corresponding to the IBEX 35 index. The estimation results show that the extended specification for the sticky strike rule presented in this article better represents the behavior of the implied volatility under this rule. Furthermore, there is not one rule that is the most appropriate at all times to explain the evolution of the implied volatility surface. Depending on the market situation, one rule may be more appropriate than another. In particular, when the underlying asset displays trend, the sticky delta rule tends to prevail against the sticky strike rule. Conversely, when the underlying asset moves in range, then the sticky strike rule tends to predominate.

Keywords:

• Sticky delta rule
• Sticky strike rule
• Implied volatility surface
• Volatility skew

JEL Classification:

• C2
• C23
• G1
• G10

Acknowledgements

I thank Markit Group Limited for allowing me to use the implied volatility data in the empirical application. The content of this paper represents the author's personal opinion and does not reflect the views of BBVA.

Notes

†These authors also consider the square root of time rule. This rule is related to the extrapolation of implied volatilities for maturities and strike prices for which there is no market. Since the main objective of this article is to study the evolution of the implied volatility surface, I will focus on the sticky strike and the sticky delta rules.

†See Balland (2002) for a detailed study of the arbitrage-free models that are consistent with each of the heuristic rules corresponding to the behavior of the implied volatility surface.

‡I assume that the implied volatility is constant, so that there is no vega risk.

†See Carr (2002) for a similar result when the risk-free rate and the dividend yield are different from zero. For a detailed study of the robustness of the Black–Scholes (1973) formula in a stochastic volatility framework, see El Karoui et al. (1998).

†Since I include a constant term in the regression equation, the parameters δ _i , for i = 2,…, T _N , account for the differential aggregate effect corresponding to observation i with respect to the first observation.