## ICIAM 2011: Contagion

**October 18, 2011**

Figure 1. Graph of the Brazilian interbank network, from [2].

**Richard Sowers and Kay Giesecke**

Contagion! The financial kind, not the medical kind. One can imagine that central bankers have as much affinity for the word "contagion" as the students of Hogwarts do for the name "Lord Voldemort." Defaults in one country can cause credit to tighten up, leading to defaults in another country, leading to a financial spiral. Because banks tend to share capital and funding, a default of one bank can lead to a default of another.

But what do we think of contagion? It sounds like something we applied mathematicians might have something to say about. After all, it sounds like what happens on an interconnected network. And that's the sort of thing we do.

The mathematical community is realizing that issues of contagion and financial networks are a rich source of challenging, appealing, and important problems. And they are indeed important. The modern financial world is interconnected, through pooling and cross-holdings of assets, trading of derivatives, multiple exchanges, and on and on. What happens in Singapore can affect what happens in Kansas City 15 minutes later (if not sooner).

A number of problems, models, and theorems are starting to emerge from the mathematical community. We can't be exhaustive, but we hope to give some idea of where headway is being made with the following list (which was heavily influenced by the program of ICIAM 2011).

- How does risk propagate through a financial network? In an empirical study of defaults in the Brazilian interbank system, Rama Cont and several co-authors (see [2]) have found that connectedness between banks follows a power law. They observe that network structure, not just size, determines which banks pose the most significant risk. See Figure 1.
- Another important question is how defaults are actually settled. A seminal paper by Eisenberg and Noe [6] posits that (under some regularity conditions) there is a unique natural payout structure when different creditors have to pay off their debts; some will be able to pay all, and others only part of their debts. Staum and Liu (see [10]) have used this as a starting point for a study of how systemic risk in financial networks should be quantified and allocated to individual institutions.
- How effective are central clearing counterparties, or CCPs? A number of markets require trading and posting of collateral through central counterparties. There is some regulatory support for CCPs as a means of mitigating the risk of contagion and reducing systemic risk. In recent work, however, Duffie and Zhu [5] suggest that unless they are carefully constructed, localization of risk in CCPs may in fact increase the systemic risk.
- How dangerous is contagion? The answer to this question involves feedback from a collection of assets back into itself. If enough defaults occur, business conditions worsen and more defaults are likely to occur. Defaults tend to be correlated and, indeed, self-exciting. In modeling and analyzing such systems, researchers develop tools for measuring and predicting systemic risk. See [1, 3, 4, 7–9].

It seems that the applied mathematics community is responding seriously to the challenges of financial contagion. We have many of the tools, and interest is increasing among the community. There are also glimmers of hope that such research could have a larger impact. The Office of Financial Research, established as part of the Dodd–Frank reform bill, has the goal of facilitating more sophisticated perspectives on financial networks. The mathematical community, with unique expertise on quantitative models of exactly the sort needed for progress on a number of the problems, has an important role to play.

**References**

[1] R. Carmona and S. Crépey, Particle methods for the estimation of Markovian credit portfolio loss distributions, Internat. J. Theoret. Appl. Finance, 13:4 (2010), 577–602.

[2] R. Cont, A. Moussa, and E.B. Santos, Network structure and systemic risk in banking systems, preprint.

[3] J. Cvitanic´, J. Ma, and J. Zhang, Law of large numbers for self-exciting correlated defaults, preprint, 2010.

[4] P. Dai Pra, W. Runggaldier, E. Sartori, and M. Tolotti, Large portfolio losses: A dynamic contagion model, Ann. Appl. Probab., 19 (2009), 347–394.

[5] D. Duffie and H. Zhu, Does a central clearing counterparty reduce counterparty risk?, Rev. Asset Pricing Studies, 2011, forthcoming.

[6] L. Eisenberg and T.H. Noe, Systemic risk in financial systems, Management Sci., 47:2 (2001), 236–249.

[7] I.O. Filiz, X. Guo, J. Morton, and B. Sturmfels, Graphical models for correlated defaults, Math. Finance, to appear.

[8] K. Giesecke, K. Spiliopoulos, and R.B. Sowers, Default clustering in large portfolios: Typical and atypical events, preprint.

[9] T. Ichiba and J.-P. Fouque, Stability in a model of inter-bank lending, preprint.

[10] M. Liu and J. Staum, Systemic risk components in a network model of contagion, preprint.

*Richard Sowers is a professor in the Department of Mathematics at the University of Illinois at Urbana–Champaign. Kay Giesecke is an assistant professor in the Department of Management Science and Engineering at Stanford University.*