The identification of the forces that drive stock returns and the dynamics of their associated volatilities is a major concern in empirical economics and finance. This analysis is particularly relevant for determining optimal hedging strategies based on whether shocks to the volatilities of returns of oil companies stock prices, relevant stock market indexes and oil spot and futures prices are high or low, and positively or negatively correlated. This paper investigates the correlations of volatilities in the stock price returns and their determinants for the most important integrated oil companies, namely Bp (BP), Chevron-Texaco (CVX), Eni (ENI), Exxon-Mobil (XOM), Royal Dutch (RD) and Total-Fina Elf (TFE). We measure the actual co-risk in stock returns and their determinants "within" and "between" the different oil companies, using multivariate cointegration techniques in modelling the conditional mean, as well as multivariate GARCH models for the conditional variances. We focus first on the determinants of the market value of each company using the cointegrated VAR/VECM methodology. Then we specifiy the conditional variances of VECM residuals with the Constant Conditional Correlation (CCC) multivariate GARCH model of Bollerslev (1990) and the Dynamic Conditional Correlation (DCC) multivariate GARCH model of Engle (2002). The "within" and "between" DCC indicate low to high/extreme interdependence between the volatilities of companies’ stock returns and the relevant stock market indexes or Brent oil prices.