Natural-gas and electric power systems and their corresponding markets have evolved over time independently. However, both systems are increasingly interdependent since combined cycle gas turbines that use natural gas to produce electricity increasingly couple them together. Therefore, suitable analysis techniques are most needed to comprehend the consequences on market outcomes of an increasing level of integration of both systems. There is a vast literature on integrated natural-gas and electric power markets assuming that the two markets are operated centrally by a single operator. This assumption is often untrue in the real world, which necessitates developing models for these interdependent yet independent markets. In this vein, this paper addresses the gap in the literature and provides analytical Nash-Cournot equilibrium models to represent the joint operation of natural-gas and electric power markets with the assumption that the market participants in each market make their own decisions independently seeking the maximum profits, as often is the case in the real world. We develop an analytical equilibrium model and apply the Karush-Kuhn-Tucker (KKT) approach to obtain Nash-Cournot equilibria for the interdependent natural-gas and electric power markets. We use a double-duopoly case to study the interaction of both markets and to derive insightful analytical results. Moreover, we derive closed-form analytical expressions for spot-market equilibria in both natural-gas and electric power markets, which are relevant and of practical significance for decision makers. We complement the double-duopoly study with a detailed sensitivity analysis.