Systemic Risk

George PapanicolaouStanford University

2012 SIAM Annual MeetingSIAM Conference on Financial Mathematics & Engineering

July 9, 2012

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Outline:

1. Why systemic risk? What is systemic risk? Key word:Interconnectivity.

2. Mean field models of systemic risk (very simple form ofinterconnectivity).

3. Large deviations for mean field models (dynamic phase transitions).

4. Algorithmic trading (high-frequency trading), statistical arbitrage:How should they be assessed from the viewpoint of (i) traders, (ii)investors, (iii) systemic risk (regulators)?

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What is Systemic Risk?

• Overall failure of large scale, interconnected systems that is triggeredby seemingly unimportant events. A type of instability specific tohighly interconnected systems. A phase transition.

• Cascading failure of interconnected components. Spreading out (inspace and/or time) of a ”contagion”.

• Examples: Banking systems, power grids, large engineering systems,large companies, ...

• A research area that is aligned with the emergence of UncertaintyQuantification as a new direction in large-scale scientific computing.The merging of stochastic modeling and analysis (little computing)and large scale scientific computing (with little stochasticmethodology). An emerging part of mainstream FinancialMathematics.

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Systemic risk as a dynamic phase transition

Consider evolving systems with a large number of inter-connectedcomponents, each of which can be in a normal state or in a failed state.We want to study the probability of overall failure of the system, that is,its systemic risk.There are three effects that we want to model and that contribute to thebehavior of systemic risk:

• The intrinsic stability of each component

• The strength of external random perturbations to the system

• The degree of inter-connectedness or cooperation betweencomponents

Application to banks: They cooperate, and by spreading the risk of(credit) shocks between them they can operate with less restrictiveindividual risk policies (capital reserves). However, this increases the riskthat they may all fail, that is, the systemic risk.

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A quote from the world of Marcroeconomics

Risk and Global Economic Architecture: Why Full FinancialIntegration May Be Undesirable. (Joseph E. Stiglitz, February2010)”Integration of global financial markets was supposed to lead to greaterfinancial stability, as risks were spread around the world. The financialcrisis has thrown doubt on this conclusion.A failure in one part of the global economic system caused a globalmeltdown. The recent crisis has shown that in the absence of appropriategovernment intervention, privately profitable transactions may lead tosystemic risk.This paper provides a general analytic framework within which we cananalyze the optimal degree (and form) of financial integration. Withinthis general framework, full integration is not in general optimal. Indeed,faced with a choice between two polar regimes, full integration orautarky, in the simplified model autarky may be superior.”

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From: Haldane (09), ”Rethinking the Financial Network”

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A basic, bistable mean-field model (GPY-12)

A simple model with which the local-global risk tradeoff can be analyzed:

• The variables xj (t), j = 1, . . . ,N (”risk” variables), satisfy the SDEs

dxj (t) = −h∂

∂yV(xj (t)

)dt+ θ

(x (t) − xj (t)

)dt+σdwj (t)

• Here V (y) is a potential with two stable states. Without noise, the componentsxj (t) stay in these states, one of which denotes say the normal state and theother the failed state.

• A typical but not unique choice of V (y) is V (y) = 14y

4 − 12y

2. The parameterh controls the probility with which xj jumps from one state to the other.

• {wj}Nj=1are independent Brownian motions and σ is the strength of the

random perturbations.

• x = 1N

∑Ni=1 xi is the mean-field, which we take (define) as the systemic

variable, and θ(x− xj

)is the interaction force, with θ > 0 the cooperative

interaction parameter.

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Why this model?

• The three parameters, h, σ and θ, control the three effects we wantto study: (i) intrinsic stability, (ii) random perturbations, and (iii)the degree of cooperation, respectively.

• Why mean field interaction? Because it is perhaps the simplestinteraction that models cooperative behavior. And it can begeneralized to include diversity, as explained later, as well as othermore complex interactions such as hierarchical ones.

• This model was studied extensively by D. Dawson and J. Gartner inthe eighties. There is a considerable amount of work on mean fieldmodels in recent years because a lot can be done with themanalytically. They are among the best studied interacting particlemodels. Early mathematical work goes back to H. McKean in thesixties. Introduced much earlier to model the physics of phasetransitions (dynamic Curie-Weiss model).

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Schematic for the risk of one component

-1 0 +1

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Status

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The large system limit (N→∞)

• The empirical risk density XN (t) := 1N

∑Nj=1 δxj(t)(·) converges weakly, in

probability, as N→∞ to u (t, ·), the solution of the nonlinear Fokker-Planckequation (Dawson 1983) :

∂

∂tu = h

∂

∂y[U (y)u] +

1

2σ2 ∂

2

∂y2u− θ

∂

∂y

{[∫yu (t,dy) − y

]u

}U (y) =

d

dyV (y) .

• Existence of bi-stable equilibrium states in the limit: Given θ and h, there existsa critical value σc such that u has one stable equilibrium for σ > σc, and hastwo stable equilibria for σ < σc.

• Simplification: If h is small, then u can have the bi-stable states if and only if3σ2 < 2θ.

• Explanation: The condition 2θ > 3σ2 means that the system interactiondominates the noise, and therefore component cooperation dominates. Incontrast, with strong noise forces, all xj’s act more as independent componentsand roughly one half are in one state and the rest are in the other state.

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Local stability analysis

We consider the probability of failure using the standard fluctuationtheory of a single agent. We assume that xj(0) = −1 for all j and xj(t)’sare in the vicinity of −1 so that we can linearize

xj(t) = −1 + δxj(t), x(t) = −1 + δx(t), δx(t) =1

N

N∑j=1

δxj(t).

For V(y) = 14y

4 − 12y

2, δxj(t) and δx(t) satisfy the linear SDEs:

dδxj = −(θ+2h)δxjdt+θδxdt+σdwj, dδx = −2hδxdt+σ

N

N∑j=1

dwj,

with δxj(0) = δx(0) = 0. δxj(t) and δx(t) are Gaussian processes andthe mean and variance functions are easily calculated. We are especiallyinterested in the case of large N. We have that for all t > 0,Eδxj(t) = Eδx(t) = 0 and Varδx(t) = σ2

N(1 − e−4ht). In addition,

Varδxj(t)→ σ2

2(θ+2h) (1 − e−2(θ+2h)t) as N→∞, uniformly in t > 0.

We know that σ2/N and σ2/2(θ+ 2h) should be sufficiently small sothat the linearizations are legitimate.

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What does linearization tell us?

Two things:

First: Increasing cooperation (θ) makes each agent believe that they candeal with larger external shocks (σ)

Second: The systemic effect of each agent’s action is invisible orunknown to them.

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Transition to failure and its probability

• For σ < σc (or 3σ2 < 2θ for small h), the value of the systemic riskremains around x ≈ ±ξb.

• Because of the randomness, the transition in (0, T) (or the systemcollapse in the risk sense):

x (0) ≈ −ξb, x (T) ≈ ξb

happens with nonzero probability.

• Systemic Risk Question: What is the probability of this happening?

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Large deviation principle

• The asymptotic probabilities, for large N, can be computed through a large deviationprinciple.

• [Dawson-Gartner, 1987] M1 (R) is the space of probability measures on R, and A is a set ofM1 (R)-valued continuous process on [0,T ]. Then

P (XN ∈A) ≈ exp

(−N inf

φ∈AIh (φ)

)where

Ih (φ) =1

2σ2

∫T0

sup

f:⟨φ,(∂∂yf)2⟩6=0

⟨∂∂tφ−L∗φφ−hM∗φ, f

⟩2⟨φ,(∂∂yf)2⟩ dt

L∗ψφ =1

2σ2 ∂

2

∂y2φ− θ

∂

∂y

{[∫yψ (t,dy) − y

]φ

}

M∗φ =∂

∂y

[(y3 − y

)φ]

.

• To compute the transition probability, A is the set of all continuous transition paths:

A ={φ : [0,T ]→M1 (R) ,Eφ(0)X = −ξb,Eφ(T)X = ξb

}.

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Small h (intrinsic stability) analysis of the rate function

• Why consider small h?

• The problem is nonlinear and infinite-dimensional, and is generallyintractable.

• If h is small then the problem can be reduced to a finite-dimensionalproblem. For V (y) = − 1

4y4 + 1

2y2, it is a four-dimensional problem.

• Numerical simulations show that the probability of transitions isalmost zero even for moderate h.

• When h = 0, the systemic risk is effectively a Brownian motion:

x (t) =σ√Nw (t) .

We might expect, therefore, that for small h, a transition path ofempirical densities is Gaussian with a small perturbation:

A=

φ= p+hq : p(t,y) =1√

2πb2 (t)exp

[(y−a(t))2

2b2 (t)

],a(0) = −ξb ,a(T) = ξb

.

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Small h asymptotic analysis of the rate function

• (Theorem:) For h small, the large deviation problem is solvableapproximately by a ”Chapman-Enskog” expansion, and thetransition probability is

P (XN ∈A)

≈ exp

(−N

σ2T

[2

(1 − 3

σ2

2θ

)+

24h

σ2

(σ2

2θ

)2 (1 − 2

σ2

2θ

)+O(h2)

]).

• Here are some comments of this result:

• A large system is more stable than a small system.• In the long run (T large), a transition will happen.• Increase of the intrinsic stabilization parameter h reduces systemic

risk.• Role of increase of cooperation (θ larger but σ2/θ fixed) on systemic

transition: Increases it.

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Statistical arbitrage, high-frequency trading, liquidity andsystemic risk

• With the automation of exchanges (electronic trading, last 10-15years) and the increasing use of high-speed, computer-driven,algorithmic trading, new strategies have been developed thatexecute trades based on complex and sophisticated trading signals.

• Example: Market-neutral investing using ”noise residuals”, which isa generalization of ”pairs-trading” (Bouchaud-Potters-Laloux(2005), Avellaneda-Lee (2009)). Buy low-sell high trading (high orlow frequency) using fluctuation signals, and betting on their meanreversion.

• Simpler, long-short, neutral portfolio investing, exploiting high vslow frequency market structure (Fernholz-Maguire (07)). Liquidityproviders of different types, optimal unwinding of large positions, etc

• How have markets evolved with advancing technology (hardware,software, strategic, ...)?

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From: Menkveld (11) (Jones (02)), ”Electronic Tradingand Market Structure”: Historic Bid-Ask/Trans-cost

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From: Menkveld (11): Bid-Ask since 2003

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Algorithmic trading increases market liquiditiy

• There is considerable empirical evidence (Menkveld (11),Hendreshott-Jones-Menkveld (11)) that algorithmic trading hasincreased marked liquidity dramatically in the last ten years.

• Is increased market liquidity an unquestionable figure or merit? Ithas reduced transaction costs dramatically, has it not?

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Diversity (”significant” subspace of empirical covariancematrix or daily returns of SP500) and volatility (VIX, fromoptions)

1992 1994 1996 1998 2000 2002 2004 2006 20080

10

20

30

40

50

60

70

80

90

Year

Number of Significant Eigenvectors vs VIX

VIXNumber of Eigenvectors

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Liquidity, volatility, diversity, did the risk go?

• Liquidity and volatility go inevitably together. Diversity is a deeperproperty of markets, reflecting correlations, a more global property.

• How have investors (of all kinds) adapted to changing markets?There is empirical evidence that they are suspicious of reducedtransaction costs and increased liquidity.

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From: Menkveld (11): May 6, 2010 Flash-Crash

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Concluding remarks

Modern financial markets are highly interconnected systems with manysources of instability that are very poorly understood and difficult totrack.

• A simple mean field model and large deviations give some idea ofhow reducing local risk by sharing (diversification) increases overall,systemic risk. Can something like this be done for liquidity andalgorithmic trading?

• Can dynamic control mechanisms reduce systemic risk? What if thecontrollers must rely on imperfect information? Who may be thecontrollers (regulators)?

• Can transaction fees (Tobin tax) stabilize markets in the systemicrisk sense?

• Who will do the challenging basic research here (dynamic randommatrix theory, flows on random networks, ...)?

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