Introducing time dependent dynamics in modeling the implied volatility surface

The purpose of this study was to investigate whether introducing time dependent surface dynamics would improve the modeling of implied volatility surfaces. The aim of the research is therefore to see whether models which incorporate these dynamics perform better than a benchmark model that uses an implied volatility surface with a constant shape, of which a single point is modeled over time. Three different models are used; principal component analysis, the Heston model and a regression model based on terms which are functions of moneyness and maturity. All three models and the benchmark model are calibrated on the implied volatility surface at the end of each month in the period January 2005 till June 2012, for both the Standards \& Poor 500 (SPX index) and the Euro Stoxx 50 (SX5E index). Afterwards the calibrated parameters that follow from these models are modeled over time using regression models. These regression models are mostly AR(1) models with additional independent variables like the return of the SPX index, realized volatility, VIX index or the consumer price index. A final comparison of the performance of these models is made in terms of the mean absolute error over time and surface. Two of the three models are an improvement over the benchmark model in modeling the implied volatility surfaces of both the SPX index and the SX5E index. The Heston model is only a slight improvement over the benchmark model, while the regression model shows better improvements. The regression models contains five terms; a constant, moneyness divided by the square root of maturity, moneyness squared divided by maturity, maturity and moneyness times the square root of maturity. The principal component analysis shows an almost constant mean absolute error, which is higher than the mean absolute error of the other models (and also the benchmark model). Since two of the three models that allow the shape of the volatility surface to change over time are an improvement over the benchmark model, it can be concluded that introducing time dependent surface dynamics does improve the modeling of implied volatility surfaces.