  We study the impact of regulatory capital constraints on fire sales and
financial stability in a large banking system using a mean field game model. In
our model banks adjust their holdings of a risky asset via trading strategies
with finite trading rate in order to maximize expected profits. Moreover, a
bank is liquidated if it violates a stylized regulatory capital constraint. We
assume that the drift of the asset value is affected by the average change in
the position of the banks in the system. This creates strategic interaction
between the trading behavior of banks and thus leads to a game. The equilibria
of this game are characterized by a system of coupled PDEs. We solve this
system explicitly for a test case without regulatory constraints and
numerically for the regulated case. We find that capital constraints can lead
to a systemic crisis where a substantial proportion of the banking system
defaults simultaneously. Moreover, we discuss proposals from the literature on
macroprudential regulation. In particular, we show that in our setup a systemic
crisis does not arise if the banking system is sufficiently well capitalized or
if improved mechanisms for the resolution of banks violating the risk capital
constraints are in place.
