JOINT MODELLING OF S&P500 AND VIX INDICES WITH ROUGH FRACTIONAL ORNSTEIN-UHLENBECK VOLATILITY MODEL

In this paper, we study the joint modelling problem of S&P500 and VIX indices, under rough volatility dynamics by a stochastic model with continuous paths. Our aim is to improve the future values' forecast of S&P500 index using the VIX index estimates. The present study is built on the estimation with the rough volatility models of the noise component which is included in financial models. The main stylized facts of the volatility can be captured well by fractional Brownian motions with a Hurst index, lower than 0.5. The H parameter governs the realized volatility roughness of time series. In the rough volatility approach, the Hurst exponent H is estimated by using the scaling properties of the volatility series. We describe the log-volatility of S&P500 index using a rough fractional Ornstein-Uhlenbeck model. The VIX index is a measure of the market's expected volatility on the S&P 500 Index. When the rBergomi model is empirically calibrated to daily data of the proxy, realized volatility and the VIX index, it is found that the VIX index is rough with H < 0.3 and consistent with daily implied volatility. The findings suggest that the VIX index is consistent with daily implied volatility of S&P500 and also rescaled version of the VIX index can be used to model the volatility process of S&P500 index. Finally, price estimates of S&P500 can be properly approached by using a Rough Fractional Ornstein-Uhlenbeck model of VIX index which is an implied volatility process.