The importance of adequately modeling credit risk has once again been highlighted in the recent financial crisis. Defaults tend to cluster around times of economic stress due to poor macro-economic conditions, but also by directly triggering each other through contagion. Although credit default swaps have radically altered the dynamics of contagion for more than a decade, models quantifying their impact on systemic risk are still missing. Here, we examine contagion through credit default swaps in a stylized economic network of corporates and financial institutions. We analyse such a system using a stochastic setting, which allows us to exploit limit theorems to exactly solve the contagion dynamics for the entire system. Our analysis shows that CDS, when used to expand banks' loan books (arguing that CDS would offload the additional risks from banks' balance sheets), can actually lead to greater instability of the entire network in times of economic stress, by creating additional contagion channels. This can lead to considerably enhanced probabilities for the occurrence of very large losses and very high default rates in the system. Our approach adds a new dimension to research on credit contagion, and could feed into a rational underpinning of an improved regulatory framework for credit derivatives.