Impact of Contingent Payments on Systemic Risk in Financial Networks
aa r X i v : . [ q -f i n . M F ] D ec Impact of Contingent Payments on Systemic Risk in FinancialNetworks
Tathagata Banerjee ∗ Zachary Feinstein † December 14, 2018
Abstract
In this paper we study the implications of contingent payments on the clearing wealth in anetwork model of financial contagion. We consider an extension of the Eisenberg-Noe financialcontagion model in which the nominal interbank obligations depend on the wealth of the firmsin the network. We first consider the problem in a static framework and develop conditions forexistence and uniqueness of solutions as long as no firm is speculating on the failure of otherfirms. In order to achieve existence and uniqueness under more general conditions, we introducea dynamic framework. We demonstrate how this dynamic framework can be applied to problemsthat were ill-defined in the static framework.
The global financial crisis of 2007-2009 proved the need to study and understand how failuresand losses spread through the financial system. This effect, in which the distress of one bankputs the financial health of other banks in jeopardy, is called financial contagion. In an eraof globalization and tight interconnections among the various financial entities, this type ofcontagion can spread rapidly causing a systemic crisis. Hence a thorough analysis of the differentfactors and mechanisms are of paramount importance.However, the 2007-2009 financial crisis proved that not just banks, but insurance companiesare also part of the financial system and hence linkages formed between banks and insurancecompanies can act as potential channels of financial contagion. These linkages are formed andresolved in a way that is different from normal bank loans. A typical example of such a linkageis a credit default swap [CDS]. A credit default swap is a contract in which a buyer pays apremium to a seller in order to protect itself against a potential loss due to the occurrence ofa credit event that affects the value of the contract’s underlying reference obligation, e.g., acorporate or sovereign bond. The contract specifies the credit events that will trigger paymentfrom the seller to the buyer. Whereas such instruments can be used to hedge risks, they mayalso be used for speculative purposes to put a short position on the credit markets.The important role that such contingent linkages play is demonstrated by the financial crisisof 2007-2009. As that crisis unfolded, AIG faced bankruptcy after the failure of Lehman Brothersdue to the large payouts it was required to make on its CDS contracts referencing Lehman andmortgage backed securities. When the crisis hit, the sudden calls to pay out the CDS contractsput great pressure on AIG, which traditionally had a thin capital base. Consequently AIGhad to be rescued by the U.S. Department of Treasury so as to avoid jeopardizing the financial ∗ Washington University in St. Louis, Department of Electrical and Systems Engineering, St. Louis, MO 63130,USA. † Washington University in St. Louis, Department of Electrical and Systems Engineering, St. Louis, MO 63130,USA. [email protected] ealth of firms which bought CDSs from AIG. However, despite the importance of these linkages,current models are unable to account for the conditional payments that an insurance or creditdefault swap contract would require. We refer to [8] for a preliminary study of the insuranceand reinsurance market.Our modeling follows the setting of the seminal paper of [17]. That work proposes a weightedgraph to model the spread of defaults in the financial system. In this model, banks’ liabilities aremodeled through the edges. The banks use their liquid assets to pay off these liabilities; unpaidliabilities may cause other banks to default as well. Under simple conditions, they provideexistence and uniqueness of the clearing payments and develop an algorithm for computing thesame. The base model of [17] has been extended in many directions to capture complexitiesin the financial system: Bankruptcy costs have been explored in [19, 39, 18, 27, 5, 10], cross-holdings in [19, 18, 5], extension to illiquid assets in [12, 36, 26, 3, 11, 5, 4, 21, 23, 22]. Empiricalstudies on the spread of contagion has been done in [20, 42, 16, 27].As far as the authors are aware, theoretical work on contingent payments and CDS inrelation to systemic risk has not been explored much. [40, 41] show that the clearing vectorin the presence of generalized CDS contracts is not well-defined and need not exist. Theyfurther propose a static setting to model CDS payments and give sufficient conditions on thenetwork topology for existence of a clearing solution. [31] considers such a model in a staticframework and proposes a method to rewrite some classes of network topologies as an Eisenberg-Noe system. [38, 15] modeled CDS payments, but most of the reference entities are requiredto be external to the financial system. [29] modeled reinsurance networks and studied theimplications of network topologies on existence and uniqueness of the liabilities and clearingpayments. A different approach has been taken in [28] in which a stochastic setting is used toanalyze contagion caused by credit default swaps. The role of credit default swaps in causingfinancial contagion has been captured in several empirical studies, see e.g. [37, 33].The current work aims to provide a generalized theoretical framework in which to studycredit default swaps and other contingent payments in the Eisenberg-Noe setting. We focus onexistence and uniqueness of the clearing payments under contingent payments without presup-posing the nature of those payments or strong assumptions on the network topology. This is incontrast to the aforementioned literature on CDS network models in which there is no guaranteethat the realized networks would obey the required conditions. Hence it is paramount to de-velop results for a general network, irrespective of the topology. We do this by first consideringthe problem in a static framework where all claims are settled simultaneously. In such a set-ting we find that uniqueness of the clearing solution follows so long as no firm is “speculating”on another firm’s failure. However, with speculation the problem in a static setting no longersatisfies the sufficient mathematical properties for uniqueness. In order to overcome this issue,we introduce a dynamic framework. This setting ensures both existence and uniqueness of aclearing solution under the usual conditions from [17]. This dynamic framework is similar tothe discrete time systems considered in [6, 9].This work is organized in the following way: First, in Section 2, we will introduce the math-ematical and financial setting. In Section 3, we develop the static framework for incorporatingcontingent payments such as insurance and CDS, provide results on existence and develop condi-tions for uniqueness that are intimately related to considerations of insurance versus speculation.Further we demonstrate some shortcomings inherent to the static framework with contingentpayments. In Section 4, we introduce a discrete time dynamic framework and discuss existenceand uniqueness results. Additionally we demonstrate how this framework can be applied toproblems that were ill-defined in the static framework through numerical examples. We begin with some simple notation that will be consistent for the entirety of this paper. Let x, y ∈ R n for some positive integer n , then x ∧ y = (min( x , y ) , min( x , y ) , . . . , min( x n , y n )) ⊤ , − = − ( x ∧ x + = ( − x ) − .Throughout this paper we will consider a network of n financial institutions. We will denotethe set of all banks in the network by N := { , , . . . , n } . Often we will consider an additionalnode 0, which encompasses the entirety of the financial system outside of the n banks; this node0 will also be referred to as society or the societal node. The full set of institutions, includingthe societal node, is denoted by N := N ∪ { } . We refer to [25, 27] for further discussion ofthe meaning and concepts behind the societal node.We will be extending the model from [17] in this paper. In that work, any bank i ∈ N mayhave obligations L ij ≥ j ∈ N . We will assume that no firm hasany obligation to itself, i.e., L ii = 0 for all firms i ∈ N , and the society node has no liabilitiesat all, i.e., L j = 0 for all firms j ∈ N . Thus the total liabilities for bank i ∈ N is given by¯ p i := P j ∈N L ij ≥ π ij := L ij ¯ p i if ¯ p i > p i = 0, we will let π ij = n for all j ∈ N \{ i } and π ii = 0. On theother side of the balance sheet, all firms are assumed to begin with some amount of assets x i ≥ i ∈ N . The resultant clearing payments , under a pro-rata payments environment,satisfy the fixed point problem p = ¯ p ∧ (cid:0) x + Π ⊤ p (cid:1) . (1)That is, each bank pays the minimum of what it owes: ¯ p i and what it has: x i + P j ∈N π ji p j .The resultant vector of wealths for all firms is given by V = x + Π ⊤ p − ¯ p. (2)Noting that these payments can be written as a simple function of the wealths ( p = [¯ p − V − ] + ),we provide the following proposition. We refer also to [43, 7, 6] for similar notions of utilizingclearing wealth instead of clearing payments. Proposition 2.1 (Proposition 2.1 of [6]) . p ∈ [0 , ¯ p ] is a clearing payment in the Eisenberg-Noesetting if and only if p = [¯ p − V − ] + for some V ∈ R n +1 satisfying the following fixed pointproblem V = x + Π ⊤ [¯ p − V − ] + − ¯ p. (3) Vice versa, V ∈ R n +1 is a clearing wealths if and only if V is defined as in (2) for some clearingpayments p ∈ [0 , ¯ p ] as defined in the fixed point problem (1) . Due to the equivalence of the clearing payments and clearing wealths provided in Proposi-tion 2.1, we are able to consider the Eisenberg-Noe system as a fixed point of equity and lossesrather than payments.In [17] results for the existence and uniqueness of the clearing payments (and thus for theclearing wealths as well) are provided. In fact, it can be shown that there exists a unique clearingsolution in the Eisenberg-Noe framework so long as L i > i ∈ N . We will takeadvantage of this result later in this paper. This is a reasonable assumption (as discussed in,e.g., [27, 25]) as obligations to society include, e.g., deposits to the banks. Remark 2.2.
The analysis presented in this paper can be extended to include illiquid assets asdiscussed in, e.g., [12, 4, 21, 23], to include financial derivatives on illiquid assets. This wouldallow for obligations to depend on the price of the illiquid assets, e.g., for hedging using putoptions.
Let us now consider the case when the nominal liabilities between financial institutions dependexplicitly on the wealths of the firms. This is, for instance, the case with insurance, creditdefault swaps, reserve requirements with a central bank, or the default waterfall enacted bycentral counterparties; see Examples 3.2-3.5 for more details of those cases. s a general setting, this corresponds to the situation in which the nominal liabilities L ij : R n +1 → R + from bank i ∈ N to j ∈ N is a mapping from the vector of bank wealths into theobligations; as mentioned above, we will assume that L i ≡ L ii ≡ i ∈ N .That is, dependent on the actualized wealths V ∈ R n +1 of all institutions in the system, thenominal liabilities will adjust to be L ( V ) ∈ R ( n +1) × ( n +1)+ a nonnegative matrix with 0 diagonal.In the case that the societal node is not desired, then this can be incorporated by setting L i ≡ i ∈ N . Thus we consider a static setting for these contingent payments, i.e., we assumeall claims are resolved simultaneously and the nominal liabilities L account for all layers ofcontingent claims. Assumption 3.1.
The nominal liabilities L ij : R n +1 → R + are bounded with upper bound ¯ L ij ≥ for all institutions i, j ∈ N . Example 3.2.
Consider a static network model of external assets x ∈ R n +1+ and liabilities L ∈ R ( n +1) × ( n +1)+ (with corresponding total liabilities ¯ p and relative liabilities Π ). Firm j ∈ N purchased an insurance contract from firm i ∈ N on the event that firm k ∈ N does notpay its obligations in full to firm j ; this is encoded in the nominal liabilities function L ij ( V ) = L ij + X k ∈N η kij ( V ) L kj ( V ) P l ∈N L kl ( V ) V − k . In the above equation we set the parameter η kij : R n +1 → [0 ,
1] to denote the level of insuranceoffered by the contract. Logically we impose the condition that η iij ≡ i ∈ N and j ∈ N so as a firm is not insuring against itself. We further impose a tree structure on theinsurance, that is insurers will not directly insure nonpayments from other insurers in a cyclicalmanner. This is codified in the condition that η k ij η k k j . . . η ik m j = 0 for all i, j, k , · · · , k m ∈N . This tree structure immediately implies the uniqueness of the nominal liabilities matrix L : R n +1 → R ( n +1) × ( n +1)+ . These conditions are related to the “green core” system in [40]. Inthe case that η kij ( V ) >
1, this is the situation of over-insurance which no longer is considered“insurance” in the strict legal sense; see Example 3.3 for this more general setting. Moregenerally, over-insurance is implied by the condition P i ∈N η kij ( V ) >
1, i.e., the total amountof insurance on any payment should be bounded by 1. Though explained as a single insurancecontract, multiple such contracts may be layered so that one financial institution may haveinsurance against the failures of multiple counterparties. The simplest insurance contracts aresuch that η kij ≡ ˆ η kij ∈ [0 , τ kij , e.g., η kij ( V ) = ˆ η kij h L kj ( V ) − L kj ( V ) P l ∈N L kl ( V ) V − k + τ kij i + h L kj ( V ) − L kj ( V ) P l ∈N L kl ( V ) V − k i + . Also within this framework we allow for reinsurance contracts; that is, insurance contracts thatpay out once payments from an insurer reach a certain threshold so as to contain the losses forthe insurer itself.
Example 3.3.
As in Example 3.2, consider an initial static network model with asset andliability parameters ( x, L ). Though similar to an insurance policy, a firm may purchase creditdefault swaps. Firm j ∈ N purchased a credit default swap [CDS] from firm i ∈ N on thefailure of firm k ∈ N is encoded in the formula L ij ( V ) = L ij + X k ∈N η kij ( V ) V − k . In this example we define η kij : R n +1 → R + without restriction on the number of swaps purchasedor the existence of an insurable interest. In such a way we allow for so-called “naked” CDSswhere the payments to firm j are not based on any insurable interest in firm k . xample 3.4. As in Example 3.2, consider an initial static network model with asset andliability parameters ( x, L ). We will now consider a system in which all firms must pay towardsa centralized stability fund. That is, prior to the start some amount y ∈ [0 , x ] of the externalassets are provided from each firm used in the stability fund. In the case of failures this fundwould support the defaulting firms. Consider this centralized fund to be denoted as node B andlet N B = N ∪ { B } . This system can be described in which the bailout fund is capitalized priorto clearing or as part of clearing. If the bailout is collected prior to clearing than this system isdescribed by external assets of x i − y i ≥ i ∈ N and P i ∈N y i ≥ B and liabilities of L ij ( V ) = L ij ∀ i, j ∈ N , L Bi ( V ) = V − i ∀ i ∈ N , L iB ( V ) = 0 ∀ i ∈ N . The payments to this stability fund can also be made as a part of clearing. In this case theexternal assets are x and liabilities are L ij ( V ) = L ij ∀ i, j ∈ N , L Bi ( V ) = V − i ∀ i ∈ N , L iB ( V ) = y i ∀ i ∈ N . This can be extended further by setting the payments to the stability fund y to itself be afunction of the wealth of each institution. This allows for concepts such as pooled reserverequirements to be encoded into our general framework. Example 3.5.
The final general conceptual example we wish to present is the situation ofintroducing a central counterparty [CCP] . In this setting, the network topology follows a starshape, i.e., firms only have liabilities to and from some centralized CCP node. The true CCPrules, however, also include what is called a default waterfall. The default waterfall kicks inwhen the CCP is unable to pay out in full through the initial collected liabilities and marginpayments. In such a case the remaining solvent firms are forced to provide more liquidity to theCCP node. In a broad sense, this fits within the general framework considered herein as theobligations to the CCP are directly dependent on the wealths of all firms in the system. CCPsare described in more detail in [1, 35, 13, 15].As in the construction of the Eisenberg-Noe setting [17], the total and relative liabilities willimplicitly be functions of the system wealths as well, i.e.,¯ p i ( V ) = X j ∈N L ij ( V ) (4) π ij ( V ) = L ij ( V )¯ p i ( V ) if ¯ p i ( V ) > n if ¯ p i ( V ) = 0 , i = j p i ( V ) = 0 , i = j (5)for firms i, j ∈ N and system equities V ∈ R n +1 .With this contingent setting we can define the extension of the Eisenberg-Noe framework asthe fixed point problem V = x + Π( V ) ⊤ [¯ p ( V ) − V − ] + − ¯ p ( V ) . (6)That is, the wealths are the sum of external assets and payments from other banks minus thepayments owed. This could equivalently be defined directly as the payments as is done in [17],we choose to consider the wealths directly in this work as it is easier to consider the examples,e.g., insurance payments. The realized payments can be defined (as discussed previously withoutcontingent payments) by p = [¯ p ( V ) − V − ] + . Proposition 3.6.
Under Assumption 3.1, any fixed point wealth V ∈ R n +1 of (6) lies withinthe compact set Q ni =1 [ x i − P j ∈N ¯ L ij , x i + P j ∈N ¯ L ji ] .Proof. The result is immediate by the boundedness properties of Assumption 3.1.
Corollary 3.7.
Under Assumption 3.1, there exists an equilibrium wealth of (6) if L ij : R n +1 → R + is continuous as a function of wealths for all firms i, j ∈ N . roof. This follows from the compactness argument of Proposition 3.6 and the Brouwer fixedpoint theorem.Though in Corollary 3.7 we have proven the existence of an equilibrium solution to (6), thisneed not be a unique solution. The following example illustrates a simple network with multipleequilibria. Further, Corollary 3.7 and (6) implicitly assume that there are no bankruptcy costs.With such costs (as introduced in [39]), Corollary 3.7 will no longer apply. See also Remark 3.12for a discussion on sufficient conditions to guarantee existence of a clearing wealths vector underbankruptcy costs.
Example 3.8.
Consider the network with n = 3 banks, and without the societal node. Thisnetwork is depicted in Figure 1. Banks 1 and 3 have x = x = 0 external assets and bank 2begins with x = 3 /
16 external assets. We consider the case in which L = L ≡ L ( V ) = V − . No otherexposures exist within this system. The system of wealths must therefore satisfy V = V − V − (1 + V − − V − ) + V = 316 + (1 − V − ) + − (1 + V − ) V = 11 + V − (1 + V − − V − ) + − . It can be shown that the following are both equilibrium wealths of the contingent network: • V = (0 , / , ⊤ , i.e., payments are given by p = ¯ p ( V ) − V − = (0 , , ⊤ , and • V = (3 / , − / , − / ⊤ , i.e., payments are given by p = ¯ p ( V ) − V − = (0 , / , / ⊤ . x = 0 x = x = 0 L ( V ) = V − L ≡ L ≡ We will now impose additional properties upon the financial system to have stronger existenceresults, culminating in uniqueness of the clearing solutions. These results provide monotonicityof the wealth of the banks in the financial system. The first of such properties, defined as anonspeculative property, is provided below in Definition 3.9.
Definition 3.9.
Firm i ∈ N is called nonspeculative if x i + X j ∈N π ji ( V )[¯ p j ( V ) − V − j ] + − ¯ p i ( V ) is nondecreasing in V ∈ R n +1 . The network N is called nonspeculative if all firms i ∈ N arenonspeculative. e call the property in Definition 3.9 “nonspeculative” as it provides conditions so that firm i ∈ N does not benefit from (i.e., speculate on) the failure of another firm. We do, however,allow for firm i to hedge its exposure to other firms. This exposure can be either direct orindirect. Remark 3.10.
The nonspeculative framework considered herein is similar to, and can be con-sidered as an extension of the properties considered in [40]. In that work, the monotonicityproperty, in the definition of nonspeculative systems that we consider, is specified for creditdefault swaps. Properties on solutions, which we will derive from this nonspeculative property,are considered as a function of the network topology in [40]; in fact, the topological featuresrequired in [40] guarantee that the “green core” system is inherently nonspeculative.
Lemma 3.11.
Under Assumption 3.1, any nonspeculative system has a greatest and least equi-librium wealth V ↑ ≥ V ↓ satisfying (6) existing within the compact space Q i ∈N [ x i − P j ∈N ¯ L ij , x i + P j ∈N ¯ L ji ] . Additionally, under all clearing vectors the value of the equity of each node of thefinancial system is the same, that is, if V and ˆ V are any two clearing wealths then V + = ˆ V + .Proof. By the nonspeculative property we can apply the Tarski fixed point theorem to get theexistence of a maximal and minimal fixed point V ↑ ≥ V ↓ . By Proposition 3.6 we have that suchsolutions must exist within the provided compact space.Now we will show the uniqueness of the positive equities by proving that ( V ↑ ) + = ( V ↓ ) + .By definition we know that ( V ↑ ) + ≥ ( V ↓ ) + , so as in [17, Theorem 1] we will prove that thetotal positive equity in the system remains constant. Let V ∈ R n +1 be some equilibrium wealthsolution, then since ¯ p ≡ X i ∈N V + i = X i ∈N x i + X j ∈N π ji ( V )[¯ p j ( V ) − V − j ] + − ¯ p i ( V ) + = X i ∈N x i + X j ∈N π ji ( V )[¯ p j ( V ) − V − j ] + − [¯ p i ( V ) − V − i ] + = X i ∈N x i + X j ∈N [¯ p j ( V ) − V − j ] + X i ∈N π ji ( V ) − X i ∈N [¯ p i ( V ) − V − i ] + = X i ∈N x i + X j ∈N [¯ p j ( V ) − V − j ] + − X i ∈N [¯ p i ( V ) − V − i ] + = X i ∈N x i . Therefore P i ∈N ( V ↑ i ) + = P i ∈N ( V ↓ i ) + and thus ( V ↑ ) + = ( V ↓ ) + . Remark 3.12.
The inclusion of bankruptcy costs to this setting, in much the same way asaccomplished in [39, 40], would guarantee the existence of a maximal and minimal clearingwealths vector under the assumptions of Lemma 3.11. Much as in [40], without the nonspec-ulative assumption, existence of a solution may not exist since Corollary 3.7 will no longerapply.We will now give additional properties for the societal node 0 to satisfy.
Assumption 3.13.
All firms i ∈ N have strictly positive obligations to society, i.e., L i : R n +1 → R ++ . Additionally, the obligations to society depend only on the negative wealths of allfirms, i.e., L i ( V ) = L i ( − V − ) for all i ∈ N . Definition 3.14.
The societal node is called strictly nonspeculative if X j ∈N π j ( V )[¯ p j ( V ) − V − j ] + s strictly increasing in V ∈ R n +1 − . The network N is called strictly nonspeculative if the systemis nonspeculative and the societal node is strictly nonspeculative. We call this property strictly nonspeculative since it provides the condition that society doesstrictly worse as any firm defaults by any additional amount. This requires that society cannever perfectly hedge its risk, and as a consequence is strictly not speculating on any firm’sfailure. This is a reasonable property as society should always be exposed to banking failuresto some degree through, e.g., deposits and the payments necessary for deposit insurance.
Corollary 3.15.
Under Assumptions 3.1 and 3.13, any strictly nonspeculative system has aunique equilibrium wealth of (6) existing within the compact space Q i ∈N [ x i − P j ∈N ¯ L ij , x i + P j ∈N ¯ L ji ] .Proof. Using Lemma 3.11 we have the existence of greatest and least fixed points V ↑ ≥ V ↓ .Let us assume there exists some firm i ∈ N such that 0 > V ↑ i > V ↓ i (otherwise uniqueness isguaranteed by nonexistence of such a firm as well as the uniqueness of the positive equities).By the definition of the equilibrium wealth of the society node V ↑ = X j ∈N π j ( V ↑ )[¯ p j ( V ↑ ) − ( V ↑ j ) − ] + > X j ∈N π j ( V ↓ )[¯ p j ( V ↓ ) − ( V ↓ j ) − ] + = V ↓ . However, immediately we know that the societal node has positive equity, therefore by Lemma3.11 it must follow that V ↑ = V ↓ , which is a contradiction so uniqueness must follow.We will now provide a version of the fictitious default algorithm (as discussed in, e.g., [17,39, 4, 21, 5]) for the contingent payments described in (6). Algorithm 3.16 provides the maximalfixed point V ↑ under the conditions of Lemma 3.11, that is, for a network of nonspeculativebanks. It can easily be modified to provide the minimal fixed point V ↓ instead. Under theassumptions of Corollary 3.15 this algorithm results in the unique equilibrium wealths. Algorithm 3.16.
Consider the setting of Lemma 3.11 such that L ( V ) = L ( − V − ) for every V ∈ R n +1 . The greatest clearing wealths V ↑ can be found in at most n iterations of the followingalgorithm. Initialize k = 0, D = ∅ , and V = x + Π(0) ⊤ ¯ p (0) − ¯ p (0).1. Increment k = k + 1;2. Denote the set of insolvent banks by D k = { i ∈ N | V k − i < } ;3. If D k = D k − then terminate;4. Define the matrix Λ ∈ { , } ( n +1) × ( n +1) so thatΛ ij = ( i = j ∈ D k V k = ˆ V is the maximal solution of the following fixed point problem:ˆ V = x + Π(Λ ˆ V ) ⊤ [¯ p (Λ ˆ V ) + Λ ˆ V ] + − ¯ p (Λ ˆ V )in the domain Q i ∈N [ x i − P j ∈N ¯ L ij , x i + P j ∈N ¯ L ji ];6. Go back to step 1.In Algorithm 3.16 at most n iterations are needed, as opposed to n + 1 as would generally bestated for the fictitious default algorithm of [17]. This is due to the fact that, by definition, thesocietal node 0 has no obligations and therefore cannot default. The additional condition that L ( V ) = L ( − V − ) for Algorithm 3.16 corresponds to the case in which the nominal liabilities only epend on the set of solvent institutions and the shortfall of each insolvent institution. This issatisfied for, e.g., CDSs as described in Example 3.3.We now consider a simple sensitivity analysis of the clearing wealths V under uncertaintyin the initial endowments x ∈ R n +1+ . In so doing we are able to consider similar comparativestatics results as in Lemma 5 of [17]. Proposition 3.17.
Consider the setting of Corollary 3.15 such that the nominal liabilities L ij : R n +1 → R + are continuous for every pair of firms i, j ∈ N . The unique clearing wealths V : R n +1+ → R n +1 are continuous and nondecreasing as a function of the initial endowments x ∈ R n +1+ .Proof. The proof is presented in the appendix.
We wish to conclude our discussion of the system with contingent obligations under simultaneousclaims by considering shortfalls to this approach. In the subsequent section we will consider adynamic approach to overcome these shortcomings, as well the issues on existence and uniquenessconsidered above and by [40, 41].Consider a setting in which a firm takes out an insurance contract on its own failure. Thissetting is of particular interest as it is inherently the type of contingent payment owed to a centralcounterparty as part of the default waterfall in the CCP framework considered in Example 3.5or the stability fund discussed in Example 3.4. To further simplify this setting, and again tomake it relevant with regards to the CCP framework, we will consider the case in which thefirm(s) offering insurance have enough assets to make all payments in full. Allow the firm takingout the insurance contracts to be firm 1. If the sum total of all contingent payments are exactlyenough to make firm 1 whole again, i.e., all contingent payments to firm 1 sum up to V − , thenin principle firm 1 will never default. However, in equilibrium, this is not what the insurancepayments will be; in fact, if the insurance payments in a fixed point were to make firm 1 wholethen no insurance payments would be made and the initial shortfall would be realized oncemore. Therefore, in equilibrium, it must be the case that firm 1 will default even if they arepaid the insurance. As a simple demonstrative example, if firm 1 only has obligations to thesocietal node (allowing for us to ignore all feedback effects from firm 1 paying more and havinga higher recovery rate through the network) then the insurance will add up to exactly half offirm 1’s initial shortfall in the fixed point as the new shortfall for firm 1 will be equal to thecontingent payments that are being made.However, while this conceptual problem with a firm taking out an insurance contract on itsown losses is important, it can conceivably be overcome by providing a sufficiently complicatedstructure to the contingent payments. A more subtle, but pernicious, flaw is that this contingentpayment system is speculative by construction. Namely, if the wealth of firm 1 is lowered, noother firm does better (firm 1 will pay out less and the insurance companies will have higherclaims to pay), but firm 1 itself improves its wealth. This is due to the nonspeculative propertybeing constructed in which firm 1 does not directly get hit by its own lower wealth, but wouldonly occur in network effects that would be on the second order, not in evidence in the singleiteration of the definition. Thus, even though the network is constructed from the notion thatno firm benefits in the case of defaults (and this would be evidenced in any equilibrium), themonotonicity of the nonspeculative property is a stronger construct that cannot be satisfied soeasily from a conceptual standpoint.The above described problems could, in specific circumstances (e.g., a single insurer andonly one contingent payment contract or a “green core” system from [40]), be overcome byreformulating the payments appropriately. However, in the general case with each contractincorporating no speculation from a financial perspective, this system would have the aforemen-tioned shortcomings. These challenges, along with the inability to deal with speculative systemsin general, stem from structural issues in such a static framework. Specifically, insurance, andcontingent payments more generally, are paid on specific claims, not simultaneous to the claim eing made. This necessitates a dynamic approach to this problem, which we will discuss in thesubsequent section. As detailed above, the static, simultaneous claims, model presented has both mathematical andeconomic issues that the authors are not aware of any way to overcome in a general setting.These problems are associated with the presence of, potentially, infinite cycles. Much as with[30], these cycles could alternate between two states, particularly for speculative systems. Thatis, for instance, insurance is paid out because a bank is insolvent, but because of this insurancepayment the firm is no longer insolvent and no payment would be necessary. As in [30], we willconsider an algorithmic approach to this issue. We thus propose a simple dynamic framework.Additionally, we consider this setting to be more realistic than the static setting consideredabove and by [40] as the financial system does not include the payment of, e.g., a CDS on theobligation inherent in that contract.
We adapt the framework introduced in [6] for the purposes of constructing a simple dynamicframework for contingent payments. Consider a discrete set of clearing times T , e.g., T = { , , . . . , T } for some (finite) terminal time T < ∞ or T = N . For processes we will use thenotation from [14] such that the process Z : T → R n has value of Z ( t ) at time t ∈ T and history Z t := ( Z ( s )) ts =0 . As an explicit extension to [6], we consider the external (incoming) cash flow x : T × R ( n +1) ×| T | → R n +1+ and nominal liabilities L : T × R ( n +1) ×| T | → R ( n +1) × ( n +1)+ to befunctions of the clearing time and prior wealths. For simplicity, we will consider x ( t, · ) := x ( t )to be independent of the prior wealths, though it may still depend on time. The distinguishingfeature of this model compared to the static Eisenberg-Noe model (or the static contingentpayment model above and in [40]) is that the system parameters may depend on prior times.For example, if firm i has positive equity at time t − V i ( t − >
0) then these surplusassets are available to firm i at time t in order to satisfy its obligations. In the contingentsetting, the wealths of all banks at time t − t as well.We note that in [6] all unpaid obligations from a prior time are assumed to roll forwardautomatically. That is, if firm i has negative wealth at time t − T . Herein,with the explicit consideration of the contingent payments, we may “zero out” a firm beforethe terminal date if it is deemed to default in much the same as in, e.g., [9]. While we canincorporate the notion of loans from [9] as well, we will restrict our analysis to debts rollingforward in time so as to simplify the discussion.As noted above, in addition to the structure from [6], the nominal liabilities will explicitlydepend on the clearing wealths of the prior time(s), i.e., L : T × R ( n +1) ×| T | → R ( n +1) × ( n +1)+ .Often, to make this difference explicit especially in examples, we consider the full nominalliabilities L to be a combination of two components: a non-contingent component L : T → R ( n +1) × ( n +1)+ which is only a function of clearing times and a contingent component L c : T × R ( n +1) ×| T | → R ( n +1) × ( n +1)+ which is a function of both the clearing times and the past history(but only encodes the contingent payments based on the past history). That is, L = L + L c .As a descriptive consideration of the contingent obligations L c , consider the insurance-based(Example 3.2) or credit default swap (Example 3.3) scenarios of the previous section. Forinstance, a bank j may purchase a credit default swap from bank i on the failure of firm k asdescribed in Example 3.3. As opposed to the simultaneous claims setting in Section 3, in thisdynamic setting we consider an order of operations. That is, first firm k must fail at time t − t . This delay in payments is areflection of the real financial system in which there is a time between a claim being made by ank j to i and the payment on that claim. The payment due to this credit default swap wouldbe incorporated in L cij but not L ij .Even with this important distinction, we can use the same methodology as in [6] to proveexistence and uniqueness of the clearing wealths in this setting. The following assumption, withthe concept taken from [6] guarantees that all firms are solvent at the start of the system andthat the system is a regular network as described by [17]. Assumption 4.1.
Before the time of interest, all firms are solvent and liquid. That is, V i ( − ≥ for all firms i ∈ N . Additionally, all firms have positive external cash flow or obligations tosociety at all times t ∈ T , i.e., x i ( t ) + L i ( t ) > for all firms i ∈ N and all times t ∈ T . To incorporate the possibility of firms defaulting before the terminal time, let N t ( V t − ) ⊆ N denote the firms that are paying obligations at time t ∈ T based on the history of wealths upto time t −
1. In particular, we will assume that N ( V − ) := N and N t +10 ( V t ) ⊆ N t ( V t − )for any time t and any wealths process V . That is, all firms are deemed solvent at time 0 asin Assumption 4.1 and no firm recovers from default. This notion allows for a consideration inmuch the same manner as [9]. Mathematically this does not require further consideration thanin [6] as N t only depends on the history up to time t −
1. With this notation we can define L ij ( t, V t − ) = 0 for all firms j ∈ N and i
6∈ N t ( V t − ). With the notion of an auction from [9] itwill also follow that L ji ( t, V t − ) = 0 for all firms j ∈ N and i
6∈ N t ( V t − ). We define the totalliabilities and relative liabilities at time t ∈ T as¯ p i ( t, V t − ) := X j ∈N L ij ( t, V t − ) + V i ( t − − π ij ( t, V t − ) := L ij ( t,V t − )+ π ij ( t − ,V t − ) V i ( t − − ¯ p i ( t,V t − ) if ¯ p i ( t, V t − ) > n if ¯ p i ( t, V t − ) = 0 , j = i p i ( t, V t − ) = 0 , j = i ∀ i, j ∈ N . Then the clearing wealths must satisfy the following fixed point problem in time t wealths: V ( t ) = V ( t − + + x ( t )+Π( t, V t − ) ⊤ diag( { i ∈N t ( V t − ) } ) (cid:2) ¯ p ( t, V t − ) − V ( t ) − (cid:3) + − ¯ p ( t, V t − ) . (7)We proceed to reformulate the problem as in [6]. We consider a process of cash flows c andfunctional relative exposures A . These we define by c ( t, V t − ) := x ( t ) + L ( t, V t − ) ⊤ ~ − L ( t, V t − ) ~ a ij ( t, V t ) := ( π ij ( t, V t − ) if ¯ p i ( t, V t − ) ≥ V i ( t ) − , i ∈ N t ( V t − ) L ij ( t,V t − )+ a ij ( t − ,V t − ) V i ( t − − V i ( t ) − else ∀ i, j ∈ N . (8)That is, we consider c ( t, V t − ) ∈ R n +1 to be the vector of book capital levels at time t ,i.e., the new wealth of each firm assuming all other firms pay in full. We can also consider c i ( t, V t − ) to be the net cash flow for firm i at time t . We define the functional matrix A : T × R ( n +1) ×| T | → [0 , ( n +1) × ( n +1) to be the relative exposure matrix. That is, a ij ( t, V t ) V i ( t ) − provides the (negative) impact that firm i ’s losses have on firm j ’s wealth at time t ∈ T . Thisis in contrast to Π, the relative liabilities, in that it endogenously imposes the limited expo-sures concept. This equivalent formulation provides mathematical simplicity to the analysis. Abroader discussion of this reformulation is provided in [6].Thus the fixed point equation reduces to V ( t ) = V ( t −
1) + c ( t, V t − ) − A ( t, V t ) ⊤ V ( t ) − + A ( t − , V t − ) ⊤ V ( t − − . (9)With this setup we now wish to extend the existence and uniqueness results of [17] to discretetime. orollary 4.2. Let ( c, L ) : T × R ( n +1) ×| T | → R n +1 × R ( n +1) × ( n +1)+ define a dynamic financialnetwork such that every bank has cash flow at least at the level dictated by nominal interbankliabilities, i.e., c i ( t, V t − ) ≥ P j ∈N L ji ( t, V t − ) − P j ∈N L ij ( t, V t − ) for all times t ∈ T and allwealth processes V , and so that every bank owes to the societal node at all times t ∈ T , i.e., L i ( t, V t − ) > for all banks i ∈ N , times t ∈ T , and wealths V . Under Assumption 4.1, thereexists a unique solution of clearing wealths V : T → R n +1 to (9) .Proof. This follows directly from the proof of Theorem 3.2 of [6].With the construction of the existence and uniqueness of the solution, we now want toemphasize the fictitious default algorithm from [17] to construct this clearing wealths vectorover time. This algorithm is nearly identical to that presented in [6]. We note that at eachtime t this algorithm takes at most n iterations as is the case for the fictitious default algorithmoriginally presented in [17]. Thus with a terminal time T , this algorithm will construct the fullclearing solution over T in nT iterations. Algorithm 4.3.
Under the assumptions of Theorem 4.2, the clearing wealths process V : T → R n +1 can be found by the following algorithm. Initialize t = − V ( − ≥ t = max T :1. Increment t = t + 1.2. Initialize k = 0, V = V ( t −
1) + c ( t, V t − ), and D = ∅ . Repeat until convergence:(a) Increment k = k + 1;(b) Denote the set of illiquid banks by D k := (cid:8) i ∈ N t ( V t − ) | V k − i < (cid:9) .(c) If D k = D k − then terminate and set V ( t ) = V k − .(d) Define the matrix Λ k ∈ { , } n × n so that Λ kij = ( i = j ∈ D k V k = ( I − Π( t, V t − ) ⊤ Λ k ) − (cid:0) V ( t −
1) + c ( t, V t − ) + A ( t − , V t − ) ⊤ V ( t − − (cid:1) .As in [6], in step (2e) of the fictitious default algorithm we are able to replace A ( t, V t ) withΠ( t, V t − ). This is beneficial as it allows us to directly compute V k without requiring a fixedpoint problem. We additionally note that the inclusion of defaulted banks only required thechange that the fictitious set of illiquid banks is a subset of N t ( V t − ) at each time t . Remark 4.4.
The dynamic framework provides a flexible way to deal with contingent payments.In particular, we can have as many time steps as the number of contingent payment layers inthe network. For example, to consider insurance we need to have two time points to incorporatethe nominal claims and the insurance claims triggered by the clearing of these nominal claims.For reinsurance markets, we need three time steps, the third one to incorporate the reinsuranceclaims triggered by the clearing of the insurance claims. We feel this hierarchical resolution ofthe claims is widely observed in reality.
Remark 4.5.
One of the advantages of the dynamic framework is that it provides a naturalway to include bankruptcy costs. This is a deviation from the static framework where we mightnot have existence of solutions for bankruptcy costs. However in the dynamic framework wecan always determine the time point when the equity of a bank reaches zero and include thebankruptcy costs for the successive time periods. Hence the solution will exist and be unique.
Remark 4.6.
We can provide much stronger sensitivity results in this case, as compared tothe static case. Since in this approach at every time step we get an Eisenberg-Noe system, thesensitivity results are a sequential application of Section 4 of [17]. Directional derivatives of thestatic Eisenberg-Noe approach have been considered in [32, 24].We wish to finish this section by remarking on when the dynamic framework presented hereinwill provide a clearing solution from the simultaneous claim setting in the prior section. emark 4.7. In general, the clearing solutions of the simultaneous claims framework will notcoincide with the terminal clearing wealths of the dynamic framework. These notions will,however, coincide if the relative liabilities are kept constant as a function of wealths and time.For a more detailed discussion see Section 5 of [6]. Other settings, as evidenced by the examplesprovided in the next section, may provide sufficient conditions for the dynamic framework toprovide a clearing solution from the simultaneous clearing setting. In particular, this will occurif the contingent payments do not strongly feedback into the network itself, e.g. if insuranceis owed to an already solvent firm. However, we want to emphasize that the conditions underwhich the static and dynamic solutions coincide are very restrictive and in general this will notbe the case. This is appropriate given the shortcomings of the static setting as expressed inSection 3.3.
We now wish to provide three illustrative examples to demonstrate the value of the discretetime setting as a model over the static setting presented in Section 3. These three examplescorrespond to simple networks in which the static setting has no clearing wealths, has multipleclearing wealths, and has a poor interpretation of the clearing wealths respectively. We willshow that in all three situations the discrete time model presented above provides a uniqueclearing wealth for which the interpretation of the results is as anticipated.
Example 4.8.
We wish to consider a small network example in which the financial systemdoes not admit a clearing solution in the static setting, but a unique and financially meaningfulsolution in the dynamic setting. In this case we will consider a digital CDS. That is, in the casethe CDS is triggered, the payment is a fixed strictly positive value (herein set to be 1), otherwiseit pays out nothing. Immediately we can see that this is not a continuous payout and thereforedoes not automatically provide a clearing solution in the static setting (see Corollary 3.7),however we will still need to prove that there does not exist any solution.Consider the network with n = 3 banks, and without the societal node, depicted in Figure2. That is, bank 1 begins with x = 1, bank 2 with x = 0, and bank 3 with x = 2 in externalassets. We consider the case in which L ≡ L ≡ . L ( V ) = { V < } . No other exposures exist withinthis system. The system of wealths must therefore satisfy V = 1 + ( { V < } − V − ) + − V = (2 − V − ) + − . V = 2 + (1 . − V − ) + − { V < } . To show that no clearing solution exists to this system, we will consider the two possible settings:bank 2 is solvent or bank 2 has negative wealth.1. Assume bank 2 is solvent, i.e. { V < } = 0. We can compute a unique solution to theclearing wealths V = ( − , − . , ⊤ . However, since this violates our assumption that V ≥
0, this cannot be a clearing solution to the full problem.2. Assume bank 2 is insolvent, i.e. { V < } = 1. We can compute a unique solution tothe clearing wealths V = (0 , . , . ⊤ . However, since this violates our assumption that V <
0, this cannot be a clearing solution to the full problem.As no other possible clearing solutions can exist, it must be the case that there does not exista clearing solution to this static financial system.Now we wish to consider the same example but in the discrete time framework with T = { , } . Here we will consider all possible divisions of the external assets over the two time points.Formally, define x ǫ (0) = ( ǫ, , ⊤ and x ǫ (1) = (1 − ǫ, , ⊤ for any ǫ ∈ [0 , ǫ ∈ [0 , = 1 x = 0 x = 2 L ≡ L ≡ . L ( V ) = { V < } Figure 2: Example 4.8: A graphical representation of the network model with 3 banks which hasno clearing in a static setting. as required by Assumption 4.1. In any scenario, define L (0) = 2 and L (0) = 1 . L (1 , V ) = { V (0) < } with no other new obligations at time 1. Further,all scenarios will be assumed to start from zero wealths (thus satisfying Assumption 4.1). Wecan easily compute the unique clearing wealths under x ǫ (assuming no firms are removed fromthe system) as V ǫ (0) = ( ǫ − , ǫ − . , ǫ + 1) ⊤ and (noting that V ǫ (0) < ǫ ∈ [0 , V ǫ (1) = (0 , . , . ⊤ . We note that this clearing solution is identical to the proposed static wealths under the assumption that bank 2 is insolvent. Additionally, the final wealths areindependent of the choice of ǫ . Example 4.9.
Consider again Example 3.8 with three banks. In the static solution this wasencoded by the parameters: external assets of x = (0 , / , ⊤ and sparse liabilities providedby L = L ≡ L ( V ) = V − . Two clearing solutions existed, V ∗ = (0 , / , ⊤ and V ∗ = (3 / , − / , − / ⊤ .Now we wish to consider the same example but in the discrete time framework with T = { , } . Here we will consider all possible divisions of the external assets over the two time points.Formally, define x ǫ (0) = (0 , ǫ, ⊤ and x ǫ (1) = (0 , / − ǫ, ⊤ for any ǫ ∈ (0 , /
16] to guaranteethe uniqueness of the clearing solutions as a regular network from [17] (and as required fromAssumption 4.1 and since no societal node is included in this example). In any scenario, define L (0) = L (0) ≡ L (1 , V ) = V (0) − with no other newobligations at time 1. Further, all scenarios will be assumed to start from zero wealths (thussatisfying Assumption 4.1). We can easily compute the unique clearing wealths under x ǫ as V ǫ (0) = (0 , ǫ, ⊤ and V ǫ (1) = (0 , / , ⊤ . We note that this clearing solution is identical tothe first clearing wealths solution of the static system and is independent of the choice of ǫ . Example 4.10.
Finally, we want to consider a simple financial system to demonstrate the issuesdiscussed in Section 3.3 surrounding the static framework. We will then use this same networkin the discrete time framework to find a unique, financially meaningful, clearing solution. Todo so, consider a bank who takes out an insurance payment on its own losses. As discussed inSection 3.3, while the insured bank may, rightly, assume that their total losses will be madewhole, in a static setting this will not happen. However, in the dynamic framework this doesoccur appropriately.Consider the network with n = 3 banks, and without the societal node, depicted in Figure3. That is, bank 1 begins with x = 1, bank 2 with x = 0, and bank 3 with x = 2 inexternal assets. We consider the case in which L ≡ L ≡ . L ( V ) = V − . No other exposures exist within this system. The system of wealths musttherefore satisfy V = 1 + ( V − − V − ) + − V = (2 − V − ) + − . V = 2 + (1 . − V − ) + − V − . ithout the insurance payment, the first bank will default with wealths of V = ( − , − . , ⊤ .However, if the insurance is paid out in full then the first bank is made whole and the resultantwealths are V = (0 , . , . V = ( − . , , ⊤ . That is, bank 1 will have a shortfall midwaybetween its wealth with and without the insurance being paid. This, though, is not the notionthat a firm purchasing insurance would expect as it cannot make them whole. x = 1 x = 0 x = 2 L ≡ L ≡ . L ( V ) = V − Figure 3: Example 4.10: A graphical representation of the network model with 3 banks which haspoor interpretation in a static setting.
Now we wish to consider the same example but in the discrete time framework with T = { , } . Here we will consider all possible divisions of the external assets over the two time points.Formally, define x ǫ (0) = ( ǫ, , ⊤ and x ǫ (1) = (1 − ǫ, , ⊤ for any ǫ ∈ [0 , ǫ ∈ [0 , L (0) = 2 and L (0) = 1 . L (1 , V ) = V (0) − with no other new obligations at time 1. Further,all scenarios will be assumed to start from zero wealths (thus satisfying Assumption 4.1). Wecan easily compute the unique clearing wealths under x ǫ (assuming no firms are removed fromthe system) as V ǫ (0) = ( ǫ − , ǫ − . , ǫ + 1) ⊤ and (noting that V ǫ (0) < ǫ ∈ [0 , V ǫ (1) = (1 − ǫ, . , . ǫ ) ⊤ . We note that, in the case that ǫ = 1, this clearing solution isidentical to the proposed static wealths when the insurance is paid in full. As opposed to theprior examples, here the final wealths are a function of ǫ . In this paper we consider an extension of the network model of [17] to include contingentpayments viz. insurance and CDSs with endogenous reference entities. We first study thesecontingent payments in a static, simultaneous claims, framework and develop conditions toprovide existence and uniqueness of the clearing wealths. Further, sensitivity analysis andfinancial implications are considered in this setting. We find that the static framework is suitableonly for a certain class of networks and we cannot guarantee the existence of a clearing solutionbeyond these systems. Indeed the problem often becomes ill-defined from a financial standpoint.Hence we introduce the dynamic framework and show that we can get existence and uniquenessunder very mild assumptions. Further we show that the problems which could not be solved inthe simultaneous claims framework can be studied with this dynamic approach.A clear extension of this model would be to include illiquid assets as discussed in, e.g.,[4, 21, 23] along with financial derivatives on these illiquid assets, i.e., options. These derivativesfall under the general class of contingent payments and can be used a tool for either hedging(insurance) or speculation. Due to the possibility of speculation, in such a setting a firm mayhave incentives to attempt to precipitate a fire sale and collect profit from the derivatives. Proof of Proposition 3.17
Proof.
Firstly, as in (6), the clearing wealths as a function of initial endowments are defined by V ( x ) = x + Π( V ( x )) ⊤ [¯ p ( V ( x )) − V ( x ) − ] + − ¯ p ( V ( x )) . We will prove continuity by utilizing the closed graph theorem (see, e.g., [2, Theorem 2.58])noting that Proposition 3.6 provides us with the condition that the clearing wealths map into acompact set. Theorem 4 of [34] immediately provides the monotonicity of the clearing wealths.Fix x ∈ R n +1+ and let X = x + [ − , n +1 be a closed compact neighborhood of x in the fullEuclidean space R n +1 . Then we can define V x : X → R n +1 as the restriction (and possibleexpansion to negative terms) of the domain of V to X . The graph of V x is given by:graph V x = (ˆ x, ˆ V ) ∈ X × Y i ∈N [ x i − − X j ∈N ¯ L ij , x i + 1 + X j ∈N ¯ L ji ] | ˆ V = ˆ x + Π( ˆ V ) ⊤ [¯ p ( ˆ V ) − ˆ V − ] + − ¯ p ( ˆ V ) . To see that graph V x is closed let (ˆ x k , ˆ V k ) k ∈ N ⊆ graph V x → (ˆ x, ˆ V ), then immediatelyˆ V = lim k →∞ ˆ V k = lim k →∞ h ˆ x k + Π( ˆ V k ) ⊤ [¯ p ( ˆ V k ) − ( ˆ V k ) − ] + − ¯ p ( ˆ V k ) i = ˆ x +Π( ˆ V ) ⊤ [¯ p ( ˆ V ) − ˆ V − ] + − ¯ p ( ˆ V )by continuity of the nominal liabilities matrix L . Therefore by the closed graph theorem weimmediately recover that V x is continuous for any x ∈ R n +1+ , which implies that V is continuousat any x as well and thus V : R n +1+ → R n +1 is a continuous mapping. References [1] Viral Acharya and Alberto Bisin. Counterparty risk externality:centralized versus over-the-counter markets.
Journal of Economic Theory , 149:153–182, 2014.[2] Charalambos D. Aliprantis and Kim C. Border.
Infinite Dimensional Analysis: A Hitch-hiker’s Guide . Springer, 2007.[3] Hamed Amini, Damir Filipovi´c, and Andreea Minca. Systemic risk with central counter-party clearing. Swiss Finance Institute Research Paper No. 13-34, Swiss Finance Institute,2015.[4] Hamed Amini, Damir Filipovi´c, and Andreea Minca. Uniqueness of equilibrium in a pay-ment system with liquidation costs.
Operations Research Letters , 44(1):1–5, 2016.[5] Kerstin Awiszus and Stefan Weber. The joint impact of bankruptcy costs, cross-holdingsand fire sales on systemic risk in financial networks.
Probability, Uncertainty and Quanti-tative Risk , 2(9):1–38, 2017.[6] Tathagata Banerjee, Alex Bernstein, and Zachary Feinstein. Dynamic clearing and conta-gion in financial networks. 2018. Working paper.[7] Paolo Barucca, Marco Bardoscia, Fabio Caccioli, Marco D’Errico, Gabriele Visentin, Ste-fano Battiston, and Guido Caldarelli. Network valuation in financial systems. 2016. Work-ing paper.[8] Jose Blanchet and Yixi Shi. Stochastic risk networks: Modeling, analysis and efficientmonte carlo. 2012. Working paper.[9] Agostino Capponi and Peng-Chu Chen. Systemic risk mitigation in financial networks.
Journal of Economic Dynamics and Control , 58:152–166, 2015.[10] Agostino Capponi, Peng-Chu Chen, and David D. Yao. Liability concentration and systemiclosses in financial networks.
Operations Research , 64(5):1121–1134, 2016.
11] Nan Chen, Xin Liu, and David D. Yao. An optimization view of financial systemic risk mod-eling: The network effect and the market liquidity effect.
Operations Research , 64(5):1089–1108, 2016.[12] Rodrigo Cifuentes, Hyun Song Shin, and Gianluigi Ferrucci. Liquidity risk and contagion.
Journal of the European Economic Association , 3(2-3):556–566, 2005.[13] Rama Cont. The end of the waterfall: Default resources of central counterparties.
Journalof Risk Management in Financial Institutions , 8(4):365–389, 2015.[14] Rama Cont and David-Antoine Fourni´e. Functional Itˆo calculus and stochastic integralrepresentation of martingales.
The Annals of Probability , 41(1):109–133, 2013.[15] Rama Cont and Andreea Minca. Credit default swaps and systemic risk.
Annals of Oper-ations Research , 247:523–547, 2016.[16] Rama Cont, Amal Moussa, and Edson Bastos e Santos. Network structure and systemic riskin banking systems. In
Handbook on Systemic Risk , pages 327–368. Cambridge UniversityPress, 2013.[17] Larry Eisenberg and Thomas H. Noe. Systemic risk in financial systems.
ManagementScience , 47(2):236–249, 2001.[18] Matthew Elliott, Benjamin Golub, and Matthew O. Jackson. Financial networks andcontagion.
American Economic Review , 104(10):3115–3153, 2014.[19] Helmut Elsinger. Financial networks, cross holdings, and limited liability. ¨OsterreichischeNationalbank (Austrian Central Bank) , 156, 2009.[20] Helmut Elsinger, Alfred Lehar, and Martin Summer. Risk assessment for banking systems.
Management Science , 52(9):1301–1314, 2006.[21] Zachary Feinstein. Financial contagion and asset liquidation strategies.
Operations ResearchLetters , 45(2):109–114, 2017.[22] Zachary Feinstein. Obligations with physical delivery in a multi-layered financial network.2018. Working paper.[23] Zachary Feinstein and Fatena El-Masri. The effects of leverage requirements and fire saleson financial contagion via asset liquidation strategies in financial networks.
Statistics andRisk Modeling , 2017.[24] Zachary Feinstein, Weijie Pang, Birgit Rudloff, Eric Schaanning, Stephan Sturm, andMackenzie Wildman. Sensitivity of the Eisenberg–Noe clearing vector to individual in-terbank liabilities.
SIAM Journal on Financial Mathematics , 2018. To appear.[25] Zachary Feinstein, Birgit Rudloff, and Stefan Weber. Measures of systemic risk.
SIAMJournal on Financial Mathematics , 8(1):672–708, 2017.[26] Prasanna Gai and Sujit Kapadia. Contagion in financial networks. Bank of EnglandWorking Papers 383, Bank of England, 2010.[27] Paul Glasserman and H. Peyton Young. How likely is contagion in financial networks?
Journal of Banking and Finance , 50:383–399, 2015.[28] Sebastian Heise and Reimer K¨uhn. Derivatives and credit contagion in interconnectednetworks.
The European Physical Journal B , 85(4):115, 2012.[29] Ariah Klages-Mundt and Andreea Minca. Cascading losses in reinsurance networks. 2018.Working paper.[30] Michael Kusnetsov and Luitgard A. M. Veraart. Interbank clearing in financial networkswith multiple maturities. 2018. Working paper.[31] Matt Leduc, Sebastian Poledna, and Stefan Thurner. Systemic risk management in financialnetworks with credit default swaps.
Journal of Network Theory in Finance , 3(3):19–39,2017.
32] Ming Liu and Jeremy Staum. Sensitivity analysis of the Eisenberg-Noe model of contagion.
Operations Research Letters , 35(5):489–491, 2010.[33] Sheri M Markose, Simone Giansante, Mateusz Gatkowski, and Ali Rais Shaghaghi. Toointerconnected to fail: Financial contagion and systemic risk in network model of CDS andother credit enhancement obligations of us banks. Technical Report DP 683, EconomicsDepartment, University of Essex, 2010.[34] Paul Milgrom and John Roberts. Comparing equilibria.
American Economic Review ,84(3):441–459, 1994.[35] David Murphy. The systemic risk of otc derivatives central clearing.
Journal of RiskManagement in Financial Institutions , 5(3):319–334, 2012.[36] Erland Nier, Jing Yang, Tanju Yorulmazer, and Amadeo Alentorn. Network models andfinancial stability.
Journal of Economic Dynamics and Control , 31(6):2033–2060, 2007.[37] Mark E. Paddrik, Sriram Rajan, and Peyton Young. Contagion in the CDS market. OFRWorking Paper 16-12, Office of Financial Research, 2016.[38] Michelangelo Puliga, Guido Caldarelli, and Stefano Battiston. Credit default swaps net-works and systemic risk.
Scientific Reports , 4, 2014.[39] L. C. G. Rogers and L. A. M. Veraart. Failure and rescue in an interbank network.
Man-agement Science , 59(4):882–898, 2013.[40] Steffen Schuldenzucker, Sven Seuken, and Stefano Battiston. Default ambiguity: Creditdefault swaps create new systemic risks in financial networks. 2017. Working Paper.[41] Steffen Schuldenzucker, Sven Seuken, and Stefano Battiston. Finding Clearing Payments inFinancial Networks with Credit Default Swaps is PPAD-complete. In Christos H. Papadim-itriou, editor, ,volume 67 of
Leibniz International Proceedings in Informatics (LIPIcs) , pages 32:1–32:20,Dagstuhl, Germany, 2017. Schloss Dagstuhl–Leibniz-Zentrum fuer Informatik.[42] Christian Upper. Simulation methods to assess the danger of contagion in interbank mar-kets.
Journal of Financial Stability , 7(3):111–125, 2011.[43] Luitgard A.M. Veraart. Distress and default contagion in financial networks. 2018. Workingpaper., 7(3):111–125, 2011.[43] Luitgard A.M. Veraart. Distress and default contagion in financial networks. 2018. Workingpaper.