Financial fire sales as continuous-state complex contagion

Trading activities in financial systems create various channels through which systemic risk can propagate. An important contagion channel is financial fire sales, where a bank failure causes asset prices to fall due to asset liquidation, which in turn drives further bank defaults, triggering the next rounds of liquidation. This process can be considered as complex contagion, yet it cannot be modeled using the conventional binary-state contagion models because there is a continuum of states representing asset prices. Here, we develop a threshold model of continuous-state cascades in which the states of each node are represented by real values. We show that the solution of a multi-state contagion model, for which the continuous states are discretized, accurately replicates the simulated continuous state distribution as long as the number of states is moderately large. This discretization approach allows us to exploit the power of approximate master equations (AME) to trace the trajectory of the fraction of defaulted banks and obtain the distribution of asset prices that characterize the dynamics of fire sales on asset-bank bipartite networks. We examine the accuracy of the proposed method using real data on asset-holding relationships in exchange-traded funds (ETFs).