  We propose a simple model of inter-bank borrowing and lending where the
evolution of the log-monetary reserves of $N$ banks is described by a system of
diffusion processes coupled through their drifts in such a way that stability
of the system depends on the rate of inter-bank borrowing and lending. Systemic
risk is characterized by a large number of banks reaching a default threshold
by a given time horizon.
  Our model incorporates a game feature where each bank controls its rate of
borrowing/lending to a central bank. The optimization reflects the desire of
each bank to borrow from the central bank when its monetary reserve falls below
a critical level or lend if it rises above this critical level which is chosen
here as the average monetary reserve. Borrowing from or lending to the central
bank is also subject to a quadratic cost at a rate which can be fixed by the
regulator. We solve explicitly for Nash equilibria with finitely many players,
and we show that in this model the central bank acts as a clearing house,
adding liquidity to the system without affecting its systemic risk. We also
study the corresponding Mean Field Game in the limit of large number of banks
in the presence of a common noise.
