In [2] we
consider the problem of governing systemic risk in a banking system model. The
banking system model consists in an initial value problem for a system of
stochastic differential equations whose dependent variables are the log-monetary
reserves of the banks of the model as functions of time. The model introduced
in [2] generalizes previous models considered in [3], [1], [4] and describes an homogeneous population of banks. Two distinct mechanisms
are used to model the cooperation between banks and the cooperation between banks
and monetary authority. These mechanisms are regulated respectively by the
parameters α and γ. The governance of the banking system model
consists in choosing the values of the parameters α and γ and the
log-monetary reserves of the Òideal bankÓ as functions of time. In the model a
bank fails when its log-monetary reserves go below an assigned default level. We
call systemic risk or systemic event in a bounded time interval the fact that in
that time interval, at least a given fraction of the banks fails. We evaluate
the probability of systemic risk in a bounded time interval using statistical simulation.
The goal of the systemic risk governance is to keep the probability of systemic
risk in a time interval between two given thresholds. We propose a method to govern
systemic risk in a time interval based on the choice of the log-monetary
reserves of the Òideal bankÓ as a function of time and on the solution of an
optimal control problem for the mean field approximation of the banking system model.
The solution of the optimal control problem is used to determine the parameters
α and γ as functions of time. Some numerical examples are discussed.
In particular we simulate during a two year period the
governance of systemic risk in the next year. Governance decisions are taken at
the beginning of each quarter. The method to govern the systemic risk in the
next year proposed is tested in presence of positive and negative shocks acting
on the banking system. This website contains auxiliary material including
animations, an interactive application and an app that helps the understanding
of the paper [2]. A general reference to the work of the authors and of their coauthors in mathematical finance is the website: http://www.econ.univpm.it/recchioni/finance.
[1] R.
Carmona, J.-P. Fouque, L.-H.
Sun, Mean field games and systemic risk,
to appear in Communications in Mathematical Sciences, 2013.
[2] L.
Fatone, F. Mariani, M.C. Recchioni,
F. Zirilli, Systemic
risk governance in a dynamical model of a banking system, submitted 2014.
[3] J.-P. Fouque, L.-H. Sun, Systemic risk illustrated, in Handbook of Systemic Risk, J.-P. Fouque, J. Langsam Editors,
Cambridge University Press, Cambridge, U.K., 2013, pp. 444-452.
[4] J. Garnier, G. Papanicolaou, T.-W. Yang, Large
deviations for a mean field model of systemic risk, SIAM Journal on
Financial Mathematics, 4, pp. 151-184, 2013.
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