On the effect of heterogeneity on flocking behavior and systemic risk

Fei Fang; Yiwei Sun; Konstantinos Spiliopoulos

doi:10.1515/strm-2016-0013

Article

On the effect of heterogeneity on flocking behavior and systemic risk

Fei Fang , Yiwei Sun and Konstantinos Spiliopoulos

Published/Copyright: July 7, 2017

Published by

Become an author with De Gruyter Brill

Submit Manuscript Author Information

From the journal Statistics & Risk Modeling Volume 34 Issue 3-4

Abstract

The goal of this paper is to study organized flocking behavior and systemic risk in heterogeneous mean-field interacting diffusions. We illustrate in a number of case studies the effect of heterogeneity in the behavior of systemic risk in the system, i.e., the risk that several agents default simultaneously as a result of interconnections. We also investigate the effect of heterogeneity on the “flocking behavior” of different agents, i.e., when agents with different dynamics end up following very similar paths and follow closely the mean behavior of the system. Using Laplace asymptotics, we derive an asymptotic formula for the tail of the loss distribution as the number of agents grows to infinity. This characterizes the tail of the loss distribution and the effect of the heterogeneity of the network on the tail loss probability.

Keywords: Systemic risk; flocking behavior; heterogeneity; interacting diffusions

MSC 2010: 60F15; 60K35

Funding source: National Science Foundation

Award Identifier / Grant number: DMS 1550918

Funding statement: This work was partially supported by the National Science Foundation (NSF) CAREER award DMS 1550918.

Appendix

In the Appendix we prove Lemma 3.4. In order to obtain the expansion of VT2 with respect to δ, we first obtain an expansion of ϱT⁢eM⁢t. Based on the form of M, we rewrite it as M=-α¯⁢M¯-δ⁢α¯⁢N. Here M¯=I-μ⁢ϱT, where μ is the K-dimensional column vector with each component 1 and N=ci⁢(δi⁢j-ρj), i,j=1,…,K. Therefore we have

eM⁢t=∑n=0∞(-α¯⁢t)nn!⁢(M¯+δ⁢N)n.

We focus on Taylor expansion with respect to δ≪1. Note the five facts ∑i=1Kρi=1, M¯n=M¯, M¯T⁢ϱ=0⋅μ, M¯T⁢NT⁢ϱ=NT⁢ϱ and M¯T⁢(NT)2⁢ϱ=(NT)2⁢ϱ. Below we first present the coefficients of the Taylor expansion of VT2 with respect to δ up to second order.

For the zeroth order, i.e., for O⁢(δ0)=O⁢(1), the coefficient is

ϱT⁢∑n=0∞(-α¯⁢t)nn!⁢M¯n=∑n=0∞(-α¯⁢t)nn!⁢ϱT⁢M¯=ϱT.

For the first order, i.e., for O⁢(δ1), the coefficient is

ϱT⁢∑n=1∞(-α¯⁢t)nn!⁢δ⁢N⁢M¯n-1=∑n=1∞(-α¯⁢t)nn!⁢δ⁢ϱT⁢N⁢M¯n-1=δ⁢[-α¯⁢t1!⁢ϱT⁢N+(e-α¯⁢t-1+α¯⁢t1!)⁢ϱT⁢N⁢M¯]=δ⁢(e-α¯⁢t-1)⁢ϱT⁢N.

For the second order, i.e., for O⁢(δ2), the coefficient is

ϱT⁢{(-α¯⁢t)22!⁢(δ⁢N)2+∑n=3∞(-α¯⁢t)nn!⁢[δ2⁢N⁢M¯n-2⁢N+(δ⁢N)2⁢M¯n-2+(n-3)⁢δ2⁢(N⁢M¯)2]}

=ϱT⁢{(-α¯⁢t)22!⁢(δ⁢N)2+∑n=3∞(-α¯⁢t)nn!⁢[δ2⁢N2+(δ⁢N)2+(n-3)⁢δ2⁢N2]}

=ϱT⁢δ2⁢[∑n=2∞(-α¯⁢t)nn!⁢(n-1)⁢N2]

=ϱT⁢δ2⁢[(-α¯⁢t)⁢∑n=2∞(-α¯⁢t)n-1(n-1)!⁢N2-∑n=2∞(-α¯⁢t)nn!⁢N2]

=ϱT⁢δ2⁢[(-α¯⁢t)⁢(e-α¯⁢t-1)-(e-α¯⁢t-1+α¯⁢t)]⁢N2

=δ2⁢ϱT⁢(1-e-α¯⁢t-α¯⁢t⁢e-α¯⁢t)⁢N2.

Finally, we get the result

ϱT⁢eM⁢t=ϱT+δ⁢(e-α¯⁢t-1)⁢ϱT⁢N+δ2⁢(1-e-α¯⁢t-α¯⁢t⁢e-α¯⁢t)⁢ϱT⁢N2+O⁢(δ3).

Therefore, we have^[1]

ϱT⁢eM⁢t⁢R-1⁢(ϱT⁢eM⁢t)T=ϱT⁢R-1⁢ϱ+δ⁢(e-α¯⁢t-1)⁢(ϱT⁢R-1⁢NT⁢ϱ+ϱT⁢N⁢R-1⁢ϱ)+δ2⁢(1-e-α¯⁢t-α¯⁢t⁢e-α¯⁢t)⁢(ϱT⁢R-1⁢(NT)2⁢ϱ+ϱT⁢N2⁢R-1⁢ϱ)+δ2⁢(e-α¯⁢t-1)2⁢ϱT⁢N⁢R-1⁢NT⁢ϱ+O⁢(δ3).

Then we can simplify as follows:

ϱT⁢eM⁢t⁢R-1⁢(ϱT⁢eM⁢t)T=ϱT⁢R-1⁢ϱ+2⁢δ⁢(e-α¯⁢t-1)⁢ϱT⁢N⁢R-1⁢ϱ+2⁢δ2⁢(1-e-α¯⁢t-α¯⁢t⁢e-α¯⁢t)⁢ϱT⁢N2⁢R-1⁢ϱ+δ2⁢(e-α¯⁢t-1)2⁢ϱT⁢N⁢R-1⁢NT⁢ϱ+O⁢(δ3).

Noting now the two important facts that, for j=1,…,K,

(ϱT⁢N)1⁢j=ρj⁢cj-ρj⁢∑i=1Kρi⁢ci

and

(N⁢R-1⁢ϱ)j⁢1=cj⁢σj2-cj⁢∑i=1Kρi⁢σi2,

we finally obtain

ϱT⁢R-1⁢ϱ=∑i=1Kρi⁢σi2,ϱT⁢N⁢R-1⁢ϱ=∑i=1Kρi⁢ci⁢σi2-∑i=1Kρi⁢ci⁢∑i=1Kρi⁢σi2,ϱT⁢N2⁢R-1⁢ϱ=∑i=1Kρi⁢ci2⁢σi2-∑i=1Kρi⁢σi2⁢∑i=1Kρi⁢ci2-∑i=1Kρi⁢ci⁢∑i=1Kρi⁢ci⁢σi2+∑i=1Kρi⁢σi2⁢(∑i=1Kρi⁢ci)2,ϱT⁢N⁢R-1⁢NT⁢ϱ=∑i=1Kρi⁢ci2⁢σi2-2⁢∑i=1Kρi⁢ci⁢∑i=1Kρi⁢ci⁢σi2+∑i=1Kρi⁢σi2⁢(∑i=1Kρi⁢ci)2.

Now, we set

A=∑i=1Kρi⁢σi2,B=2⁢δ⁢(∑i=1Kρi⁢ci⁢σi2-∑i=1Kρi⁢ci⁢∑i=1Kρi⁢σi2),C=2⁢δ2⁢[∑i=1Kρi⁢ci2⁢σi2-∑i=1Kρi⁢σi2⁢∑i=1Kρi⁢ci2-∑i=1Kρi⁢ci⁢∑i=1Kρi⁢ci⁢σi2+∑i=1Kρi⁢σi2⁢(∑i=1Kρi⁢ci)2],D=δ2⁢[∑i=1Kρi⁢ci2⁢σi2-2⁢∑i=1Kρi⁢ci⁢∑i=1Kρi⁢ci⁢σi2+∑i=1Kρi⁢σi2⁢(∑i=1Kρi⁢ci)2].

Consequently, we get

∫0TϱT⁢eM⁢s⁢R-1⁢(ϱT⁢eM⁢s)T⁢𝑑s=∫0T[A+B⁢(e-α¯⁢s-1)+C⁢(1-e-α¯⁢s-α¯⁢s⁢e-α¯⁢s)+D⁢(e-2⁢α¯⁢s+1-2⁢e-α¯⁢s)]⁢𝑑s.

Based on this, we get the approximation

VT2=∫0TϱT⁢eM⁢s⁢R-1⁢(eM⁢s)T⁢ϱ⁢𝑑s≈[12⁢α¯⁢(2⁢B-4⁢C-3⁢D)+(A-B+C+D)⁢T+(2⁢Cα¯+C⁢T-Bα¯+2⁢Dα¯)⁢e-α¯⁢T-D2⁢α¯⁢e-2⁢α¯⁢T]=V^T2⁢(δ).

Then, with αk=α¯⁢(1+δ⁢ck), we find that

α¯=∑i=1Kρi⁢αi=∑i=1Kρi⁢α¯⁢(1+δ⁢ci)=α¯+δ⁢α¯⁢∑i=1Kρi⁢ci,

where α¯>0,δ>0. Equivalently, we have obtained

∑i=1Kρi⁢ci=0.

Plugging the latter expression into A,B,C,D and using ∑i=1Kρi⁢ci=0, we conclude the proof of the lemma.

Acknowledgements

The authors would like to thank the reviewer for a critical review that significantly improved the paper.

References

[1] J.-P. Fouque and T. Ichiba, Stability in a model of interbank lending, SIAM J. Financial Math. 4 (2013), no. 1, 784–803. 10.1137/110841096Search in Google Scholar

[2] J.-P. Fouque and J. A. Langsam, Handbook on Systemic Risk, Cambridge University Press, Cambridge, 2013. 10.1017/CBO9781139151184Search in Google Scholar

[3] J.-P. Fouque and L.-H. Sun, Systemic risk illustrated, Handbook on Systemic Risk, Cambridge University Press, Cambridge (2013), 444–452. 10.1017/CBO9781139151184.023Search in Google Scholar

[4] J. Garnier, G. Papanicolaou and T.-W. Yang, Diversification of fiancial networks may increase systemic risk, Handbook on Systemic Risk, Cambridge University Press, Cambridge (2013), 432–443. 10.1017/CBO9781139151184.022Search in Google Scholar

[5] J. Garnier, G. Papanicolaou and T.-W. Yang, Large deviations for a mean field model of systemic risk, SIAM J. Financial Math. 4 (2013), no. 1, 151–184. 10.1137/12087387XSearch in Google Scholar

[6] K. Giesecke, K. Spiliopoulos and R. B. Sowers, Default clustering in large portfolios: Typical events, Ann. Appl. Probab. 23 (2013), no. 1, 348–385. 10.1214/12-AAP845Search in Google Scholar

[7] I. Karatzas and S. E. Shreve, Brownian Motion and Stochastic Calculus, Grad. Texts in Math. 113, Springer, New York, 1988. 10.1007/978-1-4684-0302-2Search in Google Scholar

[8] S. Motsch and E. Tadmor, A new model for self-organized dynamics and its flocking behavior, J. Stat. Phys. 144 (2011), no. 5, 923–947. 10.1007/s10955-011-0285-9Search in Google Scholar

[9] K. Spiliopoulos, Systemic risk and default clustering for large financial systems, Large Deviations and Asymptotic Methods in Finance, Springer Proc. Math. Stat. 110, Springer, Cham (2015), 529–557. 10.1007/978-3-319-11605-1_19Search in Google Scholar

[10] K. Spiliopoulos and R. B. Sowers, Default clustering in large pools: Large deviations, SIAM J. Financial Math. 6 (2015), no. 1, 86–116. 10.1137/130944060Search in Google Scholar

Received: 2016-8-12

Revised: 2017-6-4

Accepted: 2017-6-6

Published Online: 2017-7-7

Published in Print: 2017-9-1

You are currently not able to access this content.

Articles in the same Issue

https://doi.org/10.1515/strm-2016-0013

Keywords for this article

Systemic risk; flocking behavior; heterogeneity; interacting diffusions