Abstract
Systemic risk, i.e., the risk that a local shock propagates throughout a given system due to contagion effects, is of great importance in many fields of our lives. In this summary article, we show how asymptotic methods for random graphs can be used to understand and quantify systemic risk in networks. We define a notion of resilient networks and present criteria that allow us to classify networks as resilient or non-resilient. We further examine the question how networks can be strengthened to ensure resilience. In particular, for financial systems we address the question of sufficient capital requirements. We present the results in random graph models of increasing complexity and relate them to classical results about the phase transition in the Erdös-Rényi model. We illustrate the results by a small simulation study.
References
- Alon, N. & Spencer, J. H. (2016), The Probabilistic Method, 4th edn, Wiley Publishing.Google Scholar
- Amini, H. & Fountoulakis, N. (2014), ‘Bootstrap percolation in power-law random graphs’, Journal of Statistical Physics155(1), 72–92. http://dx.doi.org/10.1007/s10955-014-0946-6Google Scholar
- Amini, H., Filipović, D. & Minca, A. (2015), ‘Systemic risk and central clearing counterparty design’, Swiss Finance Institute Research Paper pp. 13–34.Google Scholar
- Amini, H., Fountoulakis, N. & Panagiotou, K. (2014), ‘Bootstrap percolation in inhomogeneous random graphs’, arXiv:1402.2815Google Scholar
- Amini, H., Cont, R. & Minca, A. (2016), ‘Resilience to contagion in financial networks’, Mathematical Finance26(2), 329–365.MathSciNetCrossRefGoogle Scholar
- Armenti, Y., Crépey, S., Drapeau, S. & Papapantoleon, A. (2018), ‘Multivariate shortfall risk allocation and systemic risk’, SIAM Journal on Financial Mathematics9(1), 90–126.MathSciNetCrossRefGoogle Scholar
- Artzner, P., Delbaen, F., Eber, J. & Heath, D. (1999), ‘Coherent measures of risk’, Mathematical Finance9(3), 203–228.MathSciNetCrossRefGoogle Scholar
- Biagini, F., Fouque, J.-P., Fritelli, M. & Meyer-Brandis, T. (2019+), ‘A unified approach to systemic risk measures via acceptance sets’, Mathematical Finance.Google Scholar
- Bollobás, B. (2001), Random Graphs, Cambridge Studies in Advanced Mathematics, 2nd edn, Cambridge University Press, Cambridge.CrossRefGoogle Scholar
- Chalupa, J., Leath, P. L. & Reich, G. R. (1979), ‘Bootstrap percolation on a Bethe lattice’, Journal of Physics C: Solid State Physics12(1), L31–L35.CrossRefGoogle Scholar
- Chen, C., Iyengar, G. & Moallemi, C. (2013), ‘An axiomatic approach to systemic risk’, Management Science59(6), 1373–1388.CrossRefGoogle Scholar
- Chong, C. & Klüppelberg, C. (2018), ‘Contagion in financial systems: A Bayesian network approach’, SIAM Journal on Financial Mathematics9(1), 28–53.MathSciNetCrossRefGoogle Scholar
- Chung, F. & Lu, L. (2002), ‘Connected components in random graphs with given expected degree sequences’, Annals of Combinatorics6(2), 125–145.MathSciNetCrossRefGoogle Scholar
- Chung, F. & Lu, L. (2003), ‘The average distances in random graphs with given expected degrees’, Internet Mathematics1, 91–114.MathSciNetCrossRefGoogle Scholar
- Detering, N., Meyer-Brandis, T. & Panagiotou, K. (2015), ‘Bootstrap percolation in directed and inhomogeneous random graphs’, Electronic Journal of Combinatorics, 26(2), 2019, arXiv:1511.07993.
- Detering, N., Meyer-Brandis, T., Panagiotou, K. & Ritter, D. (2016), ‘Managing default contagion in inhomogeneous financial networks’, SIAM Journal on Financial Mathematics, 10(2), 2019, arXiv:1610.09542.
- Eisenberg, L. & Noe, T. H. (2001), ‘Systemic risk in financial systems’, Management Science47(2), 236–249.CrossRefGoogle Scholar
- Erdős, P. & Rényi, A. (1960), On the evolution of random graphs, in ‘Publication of the Mathematical Institute of the Hungarian Academy of Sciences’, pp. 17–61.Google Scholar
- Feinstein, Z., Rudloff, B. & Weber, S. (2017), ‘Measures of systemic risk’, SIAM Journal on Financial Mathematics8(1), 672–708.MathSciNetCrossRefGoogle Scholar
- Fouque, J.-P. & Langsam, J. A., eds (2013), Handbook on systemic risk, Cambridge University Press, Cambridge.zbMATHGoogle Scholar
- Gai, P. & Kapadia, S. (2010), ‘Contagion in Financial Networks’, Proceedings of the Royal Society A466, 2401–2423.MathSciNetCrossRefGoogle Scholar
- Gandy, A. & Veraart, L. A. M. (2017), ‘A Bayesian methodology for systemic risk assessment in financial networks’, Management Science63(12), 4428–4446.CrossRefGoogle Scholar
- Gilbert, E. N. (1959), ‘Random Graphs’, Ann. Math. Statist.30(4), 1141–1144. https://doi.org/10.1214/aoms/1177706098
- Hoffmann, H., Meyer-Brandis, T. & Svindland, G. (2016), ‘Risk-consistent conditional systemic risk measures’, Stochastic Processes and their Applications126(7), 2014–2037.MathSciNetCrossRefGoogle Scholar
- Hoffmann, H., Meyer-Brandis, T. & Svindland, G. (2018), ‘Strongly consistent multivariate conditional risk measures’, Mathematics and Financial Economics12(3), 413–444.MathSciNetCrossRefGoogle Scholar
- Hofstad, R. v. d. (2016), Random Graphs and Complex Networks, Vol. 1 of Cambridge Series in Statistical and Probabilistic Mathematics, Cambridge University Press.Google Scholar
- Hurd, T. R. (2016), Contagion! Systemic Risk in Financial Networks, Springer.Google Scholar
- Janson, S., Łuczak, T. & Rucinski, A. (2011), Random Graphs, Wiley.Google Scholar
- Janson, S., Łuczak, T., Turova, T. & Vallier, T. (2012), ‘Bootstrap percolation on the random graph \(G_{n,p}\)’, Ann. Appl. Probab.22(5), 1989–2047. http://dx.doi.org/10.1214/11-AAP822
- Kromer, E., Overbeck, L. & Zilch, K. (2016), ‘Systemic risk measures on general measurable spaces’, Mathematical Methods of Operations Research84(2), 323–357.MathSciNetCrossRefGoogle Scholar
- Kusnetsov, M. & Veraart, L. A. M. (2016), ‘Interbank Clearing in Financial Networks with Multiple Maturities’, Preprint.Google Scholar
- Weber, S. & Weske, S. (2017), ‘The joint impact of bankruptcy costs, cross-holdings and fire sales on systemic risk in financial networks’, Probability, Uncertainty and Quantitative Risk2(9), 1–38.MathSciNetGoogle Scholar
Copyright information
© Springer Nature Switzerland AG 2019