Bibliography

(eng): We propose a comprehensive treatment of the leverage effect, i.e. the relationship between returns and volatility of a specific asset, focusing on energy commodities futures, namely Brent and WTI crude oils, natural gas and heating oil. After estimating the volatility process without assuming any specific form of its behavior, we find the volatility to be long-term dependent with the Hurst exponent on a verge of stationarity and non-stationarity. To overcome such complication, we utilize the detrended cross-correlation and the detrending moving-average cross-correlation coefficients and we find the standard leverage effect for both crude oils and heating oil. For natural gas, we find the inverse leverage effect. Additionally, we report that the strength of the leverage effects is scale-dependent. Finally, we also show that none of the effects between returns and volatility is detected as the long-term cross-correlated one.