Univariate and Multivariate Value-at-Risk: Application and Implication in Energy Markets

This study investigated the cross-markets price changes, volatility, and shock transmission mechanism among gasoline, crude oil, and diesel spot markets. An asymmetric time-varying volatility model is used to reveal the hidden dynamic shock transmission mechanism among the markets. An iterative optimization Newton–Raphson algorithm is used in the nonlinear estimation procedures by updating the outer product of the gradient vector. The estimated results are used in quantifying the cross-market risk, optimal portfolio holding, and hedging among the energy markets.