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In this work we propose a new and general approach to build dependence in multivariate Lévy processes. We fully characterize a multivariate Lévy process whose margins are able to approximate any Lévy type. Dependence is generated by one or more common sources of jump intensity separately in jumps of any sign and size and a parsimonious method to determine the intensities of these common factors is proposed. Such a new approach allows the calibration of any smooth transition between independence and a large amount of linear dependence and provides greater flexibility in calibrating nonlinear dependence than in other comparable Lévy models in the literature. The model is analytically tractable and a straightforward multivariate simulation procedure is available. An empirical analysis shows an accurate multivariate fit of stock returns in terms of linear and nonlinear dependence. A numerical illustration of multi-asset option pricing emphasizes the importance of the proposed new approach for modeling dependence. [ABSTRACT FROM PUBLISHER]