Abstract: In this work we apply univariate and multivariate linear regressions to model the relation between at-the-money local volatility and realized volatility of stocks on the example of Microsoft shares.

Local volatility is extracted from the set of Vanilla option prices on Microsoft stocks by assuming that Microsoft stock price follows Dupire local volatility process. At-the-money local volatility at different maturities is then used in linear regression predictor while realized volatility is a resulting variable.

To handle the ill-posed character of Dupire calibration problem we use genetic algorithm of optimization. To obtain two local volatility datasets (regression inputs) two runs of the calibration are executed as we want to reflect the random nature of the genetic algorithm that can give slightly different values of local volatility for different runs.

The model validation is performed by predicting out-of-sample realized volatility using local volatility and comparing it to real world values of the realized volatility. The statistical significance of local volatility is measured as a predictor of realized volatility at different maturities in the article.

It is concluded that in all models the local volatility at longer maturities proves to be significant predictor of realized volatility (whether we predict realized volatility in a short time interval or in a longer one). Therefore it makes sense to predict the volatility on the market by calibrating local volatility from the options with longer maturities.