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

## Author

Listed:
• Lorella Fatone

() (Università di Camerino)

• Francesca Mariani

() (Università Politecnica delle Marche)

## 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 $$\alpha$$ α and $$\gamma$$ γ . 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 $$\alpha$$ α and $$\gamma$$ γ 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.

## Suggested Citation

• Lorella Fatone & Francesca Mariani, 2019. "Systemic risk governance in a dynamical model of a banking system," Journal of Global Optimization, Springer, vol. 75(3), pages 851-883, November.
• Handle: RePEc:spr:jglopt:v:75:y:2019:i:3:d:10.1007_s10898-019-00790-1
DOI: 10.1007/s10898-019-00790-1
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## References listed on IDEAS

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1. Lorella Fatone & Francesca Mariani, 2019. "An assets-liabilities dynamical model of banking system and systemic risk governance," Papers 1905.12431, arXiv.org.
2. John Lintner, 1965. "Security Prices, Risk, And Maximal Gains From Diversification," Journal of Finance, American Finance Association, vol. 20(4), pages 587-615, December.
3. Dimitrios Bisias & Mark Flood & Andrew W. Lo & Stavros Valavanis, 2012. "A Survey of Systemic Risk Analytics," Annual Review of Financial Economics, Annual Reviews, vol. 4(1), pages 255-296, October.
4. Francesca Biagini & Jean‐Pierre Fouque & Marco Frittelli & Thilo Meyer‐Brandis, 2019. "A unified approach to systemic risk measures via acceptance sets," Mathematical Finance, Wiley Blackwell, vol. 29(1), pages 329-367, January.
5. Ross, Stephen A., 1976. "The arbitrage theory of capital asset pricing," Journal of Economic Theory, Elsevier, vol. 13(3), pages 341-360, December.
6. William F. Sharpe, 1964. "Capital Asset Prices: A Theory Of Market Equilibrium Under Conditions Of Risk," Journal of Finance, American Finance Association, vol. 19(3), pages 425-442, September.
7. Nier, Erlend & Yang, Jing & Yorulmazer, Tanju & Alentorn, Amadeo, 2007. "Network models and financial stability," Journal of Economic Dynamics and Control, Elsevier, vol. 31(6), pages 2033-2060, June.
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Cited by:

1. Lorella Fatone & Francesca Mariani, 2020. "Systemic risk governance in a dynamical model of a banking system with stochastic assets and liabilities," Journal of Economic Interaction and Coordination, Springer;Society for Economic Science with Heterogeneous Interacting Agents, vol. 15(1), pages 183-219, January.

## More about this item

### Keywords

Systemic risk; Mean field; Stochastic optimal control; Stochastic models;

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