Journal of Global Optimization

, Volume 75, Issue 3, pp 851–883

# Systemic risk governance in a dynamical model of a banking system

Article

## Abstract

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 as functions of time. The banking system model considered generalizes previous models studied in Fouque and Sun (in: Fouque and Langsam (eds) Handbook of systemic risk, Cambridge University Press, Cambridge, pp 444–452, 2013), Carmona et al. (Commun Math Sci 13(4):911–933, 2015), Garnier et al. (SIAM J Financ Math 4:151–184, 2013) and describes an homogeneous population of banks. Two distinct mechanisms are used to model the cooperation among banks and the cooperation between banks and monetary authority. These mechanisms are regulated respectively by the parameters $$\alpha$$ and $$\gamma$$. 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. The probability of systemic risk in a bounded time interval is evaluated using statistical simulation. A method to govern the probability of systemic risk in a bounded time interval is presented. The goal of the governance is to keep the probability of systemic risk in a bounded time interval between two given thresholds. The governance is based on the choice of the log-monetary reserves of a kind of “ideal bank” 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 determines the parameters $$\alpha$$ and $$\gamma$$ as functions of time, that is defines the rules of the borrowing and lending activity among banks and between banks and monetary authority. Some numerical examples are discussed. In particular during a two year period we consider the governance of systemic risk in the next year when governance decisions are taken quarterly. The systemic risk governance is tested in absence and in presence of positive and negative shocks acting on the banking system.

## Keywords

Systemic risk Mean field Stochastic optimal control Stochastic models

## References

1. 1.
Battiston, S., Gatti, D.D., Gallegati, M., Greenwald, B.C., Stiglitz, J.E.: Liaisons dangereuses: increasing connectivity, risk sharing, and systemic risk. J. Econ. Dyn. Control 36(8), 1121–1141 (2012)
2. 2.
Beale, N., Rand, D.G., Battey, H., Croxson, K., May, R.M., Nowak, M.A.: Individual versus systemic risk and the regulator’s dilemma. Proc. Natl. Acad. Sci. 108(31), 12647–12652 (2011)
3. 3.
Bisias, D., Flood, M., Lo, A., Valavanis, S.: A survey of systemic risk analytics. Annu. Rev. Financ. Econ. 4, 255–296 (2012)
4. 4.
Biagini, F., Fouque, J.P., Frittelli, M., Meyer-Brandis, T.: A unified approach to systemic risk measures via acceptance sets. Math. Finance 29(1), 329–367 (2019)
5. 5.
Carmona, R., Fouque, J.P., Sun, L.H.: Mean field games and systemic risk. Commun. Math. Sci. 13(4), 911–933 (2015)
6. 6.
Fatone, L., Mariani, F.: An assets-liabilities dynamical model of banking system and systemic risk governance, (2019). arxiv:1905.12431
7. 7.
Fouque, J.P., Sun, L.H.: Systemic risk illustrated. In: Fouque, J.P., Langsam, J. (eds.) Handbook of Systemic Risk, pp. 444–452. Cambridge University Press, Cambridge (2013)
8. 8.
Fouque, J.P., Langsam, J. (eds.) Handbook of Systemic Risk. Cambridge University Press, Cambridge (2013)Google Scholar
9. 9.
Gallavotti, G.: Statistical Mechanics: A Short Treatise. Springer, New York (1999)
10. 10.
Garnier, J., Papanicolaou, G., Yang, T.W.: Large deviations for a mean field model of systemic risk. SIAM J. Financ. Math. 4, 151–184 (2013)
11. 11.
Haldane, A.G., May, R.M.: Systemic risk in banking ecosystems. Nature 469, 351–355 (2011)
12. 12.
Hurd, T.R.: Contagion! Systemic Risk in Financial Networks, SpringerBriefs in Quantitative Finance. Springer, Berlin (2016)
13. 13.
Kolosov, G.E.: Optimal Design of Control Systems: Stochastic and Deterministic Problems. CRC Press, New York (1999)
14. 14.
Lintner, J.: The valuation of risk assets and the selection of risky investments in stock portfolios and capital budgets. Rev. Econ. Stat. 47, 13–37 (1965)
15. 15.
Lintner, J.: Security prices, risk and maximal gains from diversification. J. Finances 20, 587–615 (1965)Google Scholar
16. 16.
Markowitz, H.: Portfolio selection. J. Finance 7, 77–91 (1952)Google Scholar
17. 17.
May, R.M., Levin, S.A., Sugihara, G.: Complex systems: ecology for bankers. Nature 451, 893–895 (2008)
18. 18.
May, R.M., Arinaminpathy, N.: Systemic risk: the dynamics of model banking systems. J. R. Soc. Interface 7, 823–838 (2010)
19. 19.
Mossin, J.: Equilibrium in a capital asset market. Econometrica 35, 768–783 (1966)
20. 20.
Nier, E., Yang, J., Yorulmazer, T., Alentorn, A.: Network models and financial stability. J. Econ. Dyn. Control 31, 2033–2060 (2007)
21. 21.
Ross, S.A.: The arbitrage theory of capital asset pricing. J. Econ. Theory 13, 341–360 (1976)
22. 22.
Sharpe, W.F.: Capital asset prices: a theory of market equilibrium under conditions of risk. J. Finance 19, 425–442 (1964)Google Scholar