Dynamic Default Contagion in Heterogeneous Interbank Systems
DDynamic Default Contagion in Interbank Systems
Zachary Feinstein ∗ Andreas Søjmark † October 30, 2020
Abstract
In this work we provide a simple setting that connects the structural modelling approachof Gai–Kapadia interbank networks with the mean-field approach to default contagion. Toaccomplish this we make two key contributions. First, we propose a dynamic default contagionmodel with endogenous early defaults for a finite set of banks, generalising the Gai–Kapadiaframework. Second, we reformulate this system as a stochastic particle system leading to alimiting mean-field problem. We study the existence of these clearing systems and, for themean-field problem, the continuity of the system response.
Keywords: systemic risk; financial networks; default contagion; mean-field model.
More than a decade after the collapse of Lehman Brothers and the threat of contagious defaultsthroughout the global financial system in 2008, systemic risk is still of vital importance tostudy. Systemic risk is the risk of financial contagion, i.e., when the distress of one institutionspreads due to interlinkages in balance sheets. In this work we focus on the modelling of defaultcontagion, which occurs if the failure of one bank or institution to repay its debts in full causesother defaults, triggering a chain reaction of failing banks. This may occur through a networkof interbank obligations as studied in the seminal works of [10, 15, 24] in a static, network-basedsetting. More specifically, in those works, the default of a bank causes direct impacts to thebalance sheets of other banks in the financial system. This loss of capital can cause other banksto default, thus spreading the original shock further throughout the system.Most work on network models for default contagion have focused on a static setting. However,banking balance sheets are highly dynamic and subject to fluctuations due to, e.g., marketmovements. Indeed the conclusion of [10] gives a discussion of how to include multiple clearingdates and time dynamics, which is studied in [6, 12]. Additionally, [20] considers a similarapproach to a financial model with multiple maturities. Most prior works on dynamic networkmodels consider a discrete time setting [6, 12, 20]. As far as we are aware, the only two extensionsof the Eisenberg–Noe framework [10] to continuous time are [2, 25]. In this work we are interestedin such a continuous time representation with early defaults which neither [2, 25] allow.Moreover, we are interested in making a precise connection between this dynamic balancesheet framework and the recent probability literature on mean-field approaches to contagionmodelling [18, 19, 21, 22, 23]. This, of course, also links to the financial mathematics literatureon mean-field models for systemic risk, as exemplified by [5, 7, 13], although these works havefocused on ‘flocking effects’ modelled by mean-reversion rather than default contagion. Whenformulating their models, both these strands of literature abstract away any considerations ofthe precise financial assumptions and underlying balance sheet mechanisms. This leaves animportant gap that we aim to address with this work. ∗ Stevens Institute of Technology, School of Business, Hoboken, NJ 07030, USA. [email protected] † Imperial College London, Department of Mathematics, London, SW7 2AZ, UK. [email protected] a r X i v : . [ q -f i n . M F ] O c t ur main contribution is two-fold. First, as detailed in Section 2, we propose a dynamicdefault contagion model with early defaults that generalises the Gai-Kapadia framework [15]under recovery of face value with historical price accounting (explained in detail below). Wethen prove existence of a greatest clearing solution in this financial setting and provide a discus-sion of how defaults spread through the system of banks (Prop. 2.2). By itself, the introductionof early defaults in this dynamic structural (balance sheet based) network model is novel in theliterature. Second, as detailed in Section 3, starting from our finite bank setting, we proposea tractable framework for reformulating the model as a stochastic particle system , giving thegreatest clearing capital solution when the contagion is governed by a suitable ‘cascade con-dition’ (Prop. 3.1). This then leads us to a limiting mean-field problem , yielding a succinctrepresentation of the system, which is shown to evolve continuously in time under a constrainton the interactions (Thm. 3.2). If this constraint is violated, jumps may survive the passage tothe limit, and we end the paper by discussing a possible characterisation of such jumps in termsof a mean-field analogue of the finite cascade condition; see (3.15). As such, this work providesan intriguing starting point for reconciling the structural (finite bank) balance sheet literaturewith the recent dynamic continuous-time mean-field approaches for default contagion. In order to study defaults prior to maturity, we need to consider the valuation of any obligationsdue after the default event. Within the corporate debt literature, three primary notions of suchvaluation are considered (see, e.g., [17]). Under recovery of face value [RFV] , in case of adefault event, holders of bonds (of the same issuer) recover a fraction of the face value of theheld bonds regardless of maturity. In the (static time) systemic risk literature, this formulationcorresponds with the default contagion proposed in, e.g., [15]. The other notions are recoveryof treasury [RT] (recovery of a fraction of the present value of a risk-free bond with the samematurity; this is equivalent to RFV under risk-free rate of r = 0 ) and recovery of market value[RMV] (recovery of a fraction of the value of the bond from just prior to default; this formulationcorresponds with the default contagion process of [10, 24] assuming a notion of network valuationadjustments as in, e.g., [3, 4]). In this work we focus on RFV as it is often considered moreaccurate than RMV in approximating the realised recovery rates for corporate bonds [16, 17].As hinted with the notions of valuation in default (especially RMV), we also need somenotion of valuing bonds that have not yet matured. This has been studied with a networkvaluation adjustment by, e.g., [3, 4], in a single time step setting with perfect information ofthe entire financial system. Herein we assume only limited information is known by each bankabout its counterparties. Therefore in order to determine the probability of a future defaultevent only historical information about those obligations is known; as such historical priceaccounting will be utilised. That is, the default probability of any institution is assumed basedon its historical rate (e.g., based on the credit rating of a bond). Within this work, to simplifymatters, we will assume the historical default probability is 0 and, thus, this probability isutilised in the banking book until the default realises. Throughout this section we will consider a financialsystem with n ∈ N financial institutions. This system does not include the central bank orother financial entities not included within this system; we will consider such an entity, calledthe “societal node” and denote it by node . Notationally, N = { , , ..., n } is the set of banksand N = N ∪ { } includes the societal node.In order to construct a continuous-time model with maturity T < ∞ we will begin byconsidering the stylised balance sheet for a generic bank i in our system. This balance sheetshould be viewed as a dynamic and stochastic version over times [0 , T ] of the static setting of[15] with filtered probability space (Ω , F , ( F t ) t ∈ [0 ,T ] , P ) with risk-neutral measure P . Throughouttime, all assets are of only two types: interbank assets and external assets; all liabilities are eitherinterbank (and thus assets for another bank j in the system) or external and owed to the societal ssets Liabilities External (Mark-to-Market) E [ x i ( T ) | F t ] Interbank (Solvent) (cid:80) j ∈A t L ji ( T ) Interbank (Insolvent) (cid:80) j ∈N\A t (cid:16) (1 − R ) L ji ( τ j )+ RL ji ( T ) (cid:17) Total (cid:80) j ∈N L ij ( T ) Capital K i ( t ) (a) Stylised balance sheet for bank i at time t . Time t C ap i t a l K Bank 1Bank 2Bank 3 (b)
Realisation of a 3 bank system driven by correlatedGBMs with a contagious default at t ≈ . . Figure 1:
Stylised balance sheet at time t and a realisation of this system over time. node . For simplicity, we will set the risk free rate to 0, i.e., r = 0 . This stylised balance sheetis depicted in Figure 1a.Specifically, we will consider the book values for bank i ∈ N . Let x i ( T ) ∈ L ( F T ) bethe value of the external assets for this bank at maturity. The value of this external asset attime t ∈ [0 , T ] is the discounted (risk-neutral) expected value of the value at maturity, i.e., E [ x i ( T ) | F t ] . Throughout this work we will take the external asset processes x i to follow anon-negative (Itô) process; if the drift component of x i is constructed from the risk-free rate r = 0 then x i ( t ) = E [ x i ( T ) | F t ] , otherwise these values may differ.In contrast, the liabilities (and thus also interbank assets) have fixed and known (i.e., deter-ministic) coupon payments. Notationally, let L ij ( t ) be the sum total of all coupon paymentsowed to bank j ∈ N up through time t (i.e., on [0 , t ] ). In this way, L ij ( T ) denotes the totalobligations through the maturity time T owed to bank j . For simplicity, we will additionallyassume the interbank liabilities to be continuous in time, i.e., t ∈ [0 , T ] (cid:55)→ L ij ( t ) is continuous.When valuing interbank assets, there are two settings to consider: old obligations and future coupon payments. Consider a fixed time t ∈ [0 , T ] as the “present,” i.e., all obligations L ji ( t ) from bank j to bank i are considered old whereas the obligations L ji ( T ) − L ji ( t ) due after t willbe considered future obligations. We follow the simple assumption that every solvent firm makesall its current (and past) coupon payments on schedule. To simplify notation, let A t denote theset of banks that are solvent at time t (to be discussed further below). If bank j ∈ A t is solventat time t , then the value of the interbank assets L ji ( t ) for bank i is exactly L ji ( t ) as thesepayments have been actualised. (In [2] it is possible for solvent firms to have a cash shortfall, weassume a functioning and well-capitalised external repurchase agreement market to overcomeany need to consider the specific cash account.) Consider now bank j ∈ N \A t insolvent whichdefaults on its obligations at some time τ j ≤ t . As stated already all payments L ji ( τ j ) due beforedefault are paid (and therefore marked) in full. Following the RFV paradigm, the obligations L ji ( t ) − L ji ( τ j ) are only partially repaid based on the recovery rate R ∈ [0 , . We assume thisfixed recovery of prior defaulted obligations since defaults usually cause a delay in payments;this delay in the actualisation of payments (due to, e.g., bankruptcy courts) during the periodof interest [0 , T ] prompts us to assume the RFV paradigm rather than a clearing system suchas studied in [10, 24].It remains to consider the value of future interbank coupon payments. Unlike the externalassets x i , these interbank assets L ji are nonmarketable , i.e., there is no market in which todetermine the value the risk of default for these future coupon payments. In fact, each bankonly has direct information about its own assets and liabilities. As such, due to this asymmetry From the perspective of RFV, though we call these obligations coupon payments, each of these obligations willbe treated as a separate zero-coupon bonds with different maturities. f information and without a market to mark these interbank assets, we assume that the banksfollow a historical price accounting rule for valuing these assets. That is, all future couponpayments owed from solvent firms are marked in full until an actualised default event occurs.Once a default event occurs, following the RFV paradigm presented above, all future obligationsare immediately marked down by the recovery rate R ∈ [0 , .Therefore, in total, the value of interbank assets L ji depends on the default time of bank j .If we set the default time of bank j to τ j (to be discussed in greater detail below), interbankassets L ji are marked at time t as: L ji ( T ) if t < τ j , and L ji ( τ j ) + R [ L ji ( T ) − L ji ( τ j )] if t ≥ τ j (equivalently j ∈ A t and j ∈ N \A t respectively).The stylised balance sheet encoding these assets and liabilities at time t ∈ [0 , T ] is displayedin Figure 1a. The realised capital K i ( t ) at time t ∈ [0 , T ] for bank i is, thus, the differencebetween the value of the assets and liabilities. That is, given the set of solvent banks A t ⊆ N at time t , the realised capital for bank i is computed as: K i ( t ) := E [ x i ( T ) | F t ] + (cid:88) j ∈A t L ji ( T ) + (cid:88) j ∈N \A t ((1 − R ) L ji ( τ j ) + RL ji ( T )) − (cid:88) j ∈N L ij ( T ) . (2.1)As in the static framework of [15], bank i is deemed to be in default once its assets are worthless than its total liabilities, i.e., when it has negative realised capital. That is, insolvency forbank i occurs at at the stopping time τ i = inf { t ∈ [0 , T ] | K i ( t ) ≤ } . This is consistent with thenotion that there exists a repurchase agreement market external to the system of banks as brieflymentioned previously; if a bank’s realised capital is positive then it can raise the necessary cash inorder to remain solvent, but it cannot raise sufficient cash to pay its obligations if it has negativerealised capital. The set of solvent firms at time t is, formally, given by A t := { i ∈ N | τ i > t } . Assumption 2.1.
The modelling assumptions expressed above can be summarised thusly:(i) the external assets of each bank follow a stochastic process (which can be correlated to eachother) and (being marketable) are marked-to-market with risk-neutral measure P ;(ii) the interbank assets and liabilities are solely based on contracts written prior to time andhave fixed coupon schedule;(iii) interbank assets (being nonmarketable) are valued using historical price accounting, i.e.,priced at face value prior to a default event and reevaluated with RFV after default; and(iv) defaults occur once the realised capital of a bank drops below zero.These four key modelling assumptions lead to a contagion mechanism in which defaults come asa shock to the system and cause a jump in the capital of any connected institution. The shocks due to default outlined above are realistic since the interbank assets are nonmar-ketable. If, however, banks attempted a counterparty or network valuation adjustment (see, e.g.,[3, 4]) default shocks would still be expected due to the asymmetric and incomplete informationavailable to the different banks.
Under the setting summarised in Assumption 2.1, the realised capitalfor bank i can be considered as an equilibrium setting that depends directly on the defaultingevents of all other banks. Explicitly, this is provided by (2.1) where the set of solvent firms A t ( K ) and default times τ ( K ) are considered directly as functions of the wealths process. The existenceof a clearing solution to this equilibrium problem is provided in the following proposition. Proposition 2.2 (Clearing capital) . There exists a greatest and least clearing capital K ↑ ≥ K ↓ (component-wise and a.s. for every time t ) to the network clearing problem defined by (2.1) .Proof. First, note that the fixed point problem for the capital process K always maps into thecomplete lattice (with component-wise and almost sure ordering for every time t ) (cid:81) t ∈ [0 ,T ] D t for D t := E [ x ( T ) | F t ] + (cid:88) j ∈N [ R, × L j · ( T ) − (cid:88) j ∈N L · j ( T ) . econd, note that capital process K only depends on itself through the default times τ ( K ) (andthe set of solvent firms A t ( K ) = { i ∈ N | τ i ( K i ) > t } ). Clearly, by definition, as the capitalprocess K decreases the default times τ ( K ) do not increase. Further, as banks default, theentire system’s wealth drops as well since R ≤ . With this, we are able to complete this proofthrough an application of Tarski’s fixed point theorem.As is typical in the literature, we will primarily focus on the greatest clearing solution K ↑ .Briefly, we will discuss how to use a fictitious default algorithm to compute this clearing solutionforward in time. Such an algorithm assumes that at time t ∈ [0 , T ] , any bank that was solventprior to t ( A t (cid:57) ) is assumed a priori to still be solvent; this is the best case scenario for all banksdue to the downward stresses from a default. Solvency ( K i ( t ) > ) of all banks is then checkedunder this scenario; if no banks default we can move forward in time, otherwise any new defaultsmay cause a domino effect of further defaults. In the case of defaults, we update the balancesheet of all solvent firms to determine if this shock causes a cascade of failures. This sequentialtesting for new defaults and updating the balance sheets continue until no new defaults occur.In practice this algorithm is run using an event finding algorithm to determine the time of theinitial default, at that time the cascading defaults are determined until the system re-stabilisesat a new set of solvent institutions, and the stochastic processes evolve normally until the nextdefault event. This is demonstrated in Figure 1b where the insolvency of one bank causes anotherbank to default as well. If desired, the least clearing solution K ↓ could be found analogouslywith a fictitious solvency algorithm instead. Recall that L ij ( T ) denotes the liabilities for the full period [0 , T ] . To have a simple model forrepayment, we will assume that there is a non-decreasing function ψ ( T, · ) : [0 , T ] → [0 , ∞ ) andconstant relative liabilities λ ij ≥ such that the liabilities of bank i owed for the remainingperiod [ t, T ] are given by L ij ( T ) − L ij ( t ) = ψ ( T, t ) λ ij (3.1)with L ij (0) = 0 , for i, j ∈ N , where λ ij = 0 if i = j or i = 0 . For example, we could consider alinear repayment schedule so that ψ ( T, t ) = T − t .Let us write λ ext i := λ i for the external liabilities. By using (3.1) and reorganising the sumsin equation (2.1) for the capital K i ( t ) , we find that it satisfies K i ( t ) = E [ x i ( T ) | F t ] − ψ ( T, (cid:0) λ ext i + (cid:88) j ∈N [ λ ij − λ ji ] (cid:1) − (1 − R ) (cid:88) j ∈N \A t ψ ( T, τ j ) λ ji , (3.2)where we recall that N is the full set of banks while A t is the set of solvent banks at time t .This coupled system is the focus of the rest of the paper, where we will further assume that thestochastic processes x i ( t ) are given by geometric Brownian motions dx i ( t ) = x i ( t )[ µ i ( t ) dt + σ i ( t )] dW i ( t ) with W i ( t ) = (cid:112) − ρ B i ( t ) + ρB ( t ) , (3.3)for independent Brownian motions B , . . . , B n , and continuous functions µ i , σ i . As has been observed in several empirical studies, inter-bank networks often display pronounced core-periphery features with negligible periphery-to-periphery interactions [8, 14, 27]. This type of network structure for the relative liabilities λ ij can be conveniently captured by the framework we introduce next. Naturally, our framework isan idealisation of reality, but it leads to a tractable model, and we stress that interbank liabilitiesare not fully observable in practice so approximations will always be involved.In short, one may reasonably hope to explain the heterogeneity of the relative liabilities byonly a low number of underlying characteristics, say k (cid:28) n , for a system with n banks. Perhapsa little surprisingly at first, we can have a rich and practically relevant model already with k = 1 . ndeed, similarly to [14], we can take as our one characteristic a global score for the ‘coreness’of each bank, so that the relative liabilities are given by λ ij = u i v j for i (cid:54) = j (and λ ij = 0 for i = j ), where u i ≥ is a score for ‘how core’ bank i is in terms of borrowing and v j ≥ isa score for ‘how core’ bank j is in terms of lending. In practice, these scores u i and v j could,e.g., be obtained by generating samples of the partially observed liabilities and then minimisingthe squared off-diagonal deviations (cid:80) i (cid:54) = j ( λ ij − u i v j ) .Depending on the task at hand, we may want a higher level of granularity than a global scorefor ‘coreness’. As discussed in [8] financial networks typically have a ‘tiering’ order, whereby top-tier banks lend to each other and lower-tier banks, while lower-tier banks do not lend toeach other but instead lend to top-tier banks. This suggests an organisation of the liabilitynetwork according to (i) what tier and subgroup thereof the ‘borrower’ belongs to, and (ii) howthe lending of the ‘lender’ is spread out according to this. Generalising the earlier -dim corenessscore, we propose to let (i) be summarised by a k -dim score u i = ( u i , . . . , u ik ) along k givencharacteristics, and (ii) by a corresponding k -dim score v i = ( v i , . . . , v ik ) such that λ ij = u i · v j = k (cid:88) l =1 u il v jl for i (cid:54) = j, and λ ij = 0 for i = j. (3.4)To illustrate, k = 4 gives a simple way of splitting the system into four subgroups, where,e.g., u i = (0 , u i , , says that bank i belongs to the second grouping of top-tier banks with thescore u i giving its relative importance, while u j = (0 , , u j , says that bank j belongs to thefirst grouping of lower-tier banks with relative importance u j , and so on; see Figure 2. Figure 2:
The first picture shows the general structure of the relative liabilities ( λ ij ) for the simple k = 4 exampledescribed above. The second picture gives a concrete realisation of this with no heterogeneity within groups. In place of (3.4) serving as a model for the network structure, we note that one could alsotreat it as a form of principle component analysis of the network, similarly to the use of spectraldecompositions for core-periphery detection [9] and related financial contagion models [1, 26].
As in real-world systems, we assume net-lenders in theinterbank still have strictly positive net liabilities overall, i.e., λ ext i + (cid:80) j ∈N [ λ ij − λ ji ] > . Thus, K i ( t ) < E [ x i ( T ) | F t ] in (3.2), simply saying that the capital is strictly less than the value ofexternal assets, and so we can introduce the logarithmic distances-to-default X i ( t ) := log (cid:16) E [ x i ( T ) | F t ] E [ x i ( T ) | F t ] − K i ( t ) (cid:17) , for i ∈ N . (3.5)This allows us to describe the health of the financial system in the following way. Proposition 3.1 (Particle system) . Assuming (3.1) , (3.3) and (3.4) , the greatest clearing cap-ital solution to (3.2) corresponds to X in (3.5) being the unique càdlàg solution to dX i ( t ) = − σ i ( s ) ds + σ i ( s ) dW i ( t ) − dF i ( t ) , τ i = inf { t ≥ X i ( t ) ≤ } ,F i ( t ) = log (cid:16) − R Λ i k (cid:88) l =1 v il (cid:90) t ψ ( T, s ) ψ ( T, d L nl,i ( s ) (cid:17) , L nl,i ( t ) = n (cid:88) j (cid:54) = i u jl t ≥ τ j X i (0) = log (cid:16) x i (0) ψ ( T, i (cid:17) + (cid:90) T µ i ( t ) dt, Λ i = λ ext i + k (cid:88) l =1 n (cid:88) j (cid:54) = i ( v il u jl − u il v jl ) , (3.6) ith default set D t := { i : τ i = t } = { i : X i ( t (cid:57) ) − Θ n ( t ; ∆ L n ( t ); i ) ≤ } and corresponding jumpsizes ∆ L nl,i ( t ) = Ξ nl ( t, ∆ L n ) − u il i ∈D t given by the following ‘cascade condition’ ∆ L nv ( t ) = lim m → n ∆ n, ( m ) t,v , ∆ n, (0) t,v := Ξ( t ; 0 , v ) , ∆ n, ( m ) t,v := Ξ n ( t, ∆ n, ( m − t,v , v ) , Ξ n ( t ; f, v ) := k (cid:88) l =1 v l Ξ nl ( t ; f ) , Ξ nl ( t ; f ) := n (cid:88) j =1 u jl { X j ( t (cid:57) ) ∈ [0 , Θ n ( t ; f ; j )] , t ≤ τ j } , Θ n ( t ; f ; j ) := log (cid:16) − R Λ j (cid:90) t (cid:57) ψ ( T, s ) ψ ( T, d L nv j ( s ) + 1 − R Λ j ψ ( T, s ) ψ ( T, f ( v j ) (cid:17) − F j ( t (cid:57) ) , (3.7) where L nv ( t ) := (cid:80) kl =1 v l ( L nl,i ( t ) + u il t ≥ τ i ) which only depends on v ∈ R k and not on i .Proof. Let us begin by observing that the system can undergo at most n jumps, and in be-tween these jumps the system has diffusive dynamics, so the uniqueness follows easily since wehave a specific rule for the jump sizes in (3.7). Given the dynamics (3.3), we can compute E [ x i ( T ) | F t ] = x i ( t ) e (cid:82) Tt µ i ( s ) ds , and using also λ ij = u i · v j for i (cid:54) = j from (3.4), we thus deducethat (3.2) can be rewritten as a coupled system K i ( t ) = x i ( t ) e (cid:82) Tt µ i ( s ) ds − ψ ( T, i − (cid:90) t (1 − R ) ψ ( T, s ) d L ni ( s ) L ni ( t ) := n (cid:88) j =1 λ ji t ≥ τ j = k (cid:88) l =1 v il (cid:88) j (cid:54) = i u jl t ≥ τ j , τ i = inf { t ≥ K i ( t ) < } , (3.8)where we stress that L ni ( t ) = L nv i ( t ) − u i · v i t ≥ τ i . Applying the transformation (3.5), this givesus the dynamics (3.6). From here, we observe that (3.8) gives the greatest clearing capital K ↑ from Proposition 2.2 precisely when the τ i ’s are determined as follows: with A t (cid:57) denoting therandom set of indices i ≤ n such that t ≤ τ i for each i ∈ A t (cid:57) (i.e., bank i was solvent strictlybefore time t ), these banks all remain solvent at time t except if X i ( t (cid:57) ) = 0 for some i ∈ A t (cid:57) ,in which case τ i = t , and we then need the smallest possible losses from contagion. To identifythe corresponding defaults, notice first of all that the banks D t := { j ∈ A t (cid:57) : X j ( t (cid:57) ) = 0 } arenecessarily in default precisely from time t onwards (i.e., τ j = t ). Next, with ∆ n, (0) t, · , Ξ n , and Θ n defined in (3.7), we get ∆ n, (0) t,v i = Ξ n ( t ; 0 , v i ) = k (cid:88) l =1 v l (cid:88) j ∈D t u jl , for i = 1 , . . . , n, and, supposing D t (cid:54) = ∅ , we then see that the banks D t ⊂ A t (cid:57) \D t are necessarily also in defaultfrom time t , where i ∈ D t if and only if i ∈ A t (cid:57) \D t and X i ( t (cid:57) ) − Θ n ( t ; ∆ n, (0) t, · ; i ) ≤ . Supposingfurther that D t (cid:54) = ∅ , we have ∆ n, (1) t,v i := Ξ n ( t ; ∆ n, (0) t, · , v i ) = k (cid:88) l =1 v il (cid:88) j ∈D t ∪D t u jl , for i = 1 , . . . , n, and we now see that the banks D t ⊂ A t (cid:57) \D t must also be in default, due to contagion from D t ,where i ∈ D t if and only if i ∈ A t (cid:57) \D t and X i ( t (cid:57) ) − Θ n ( t ; ∆ n, (1) t, · , i ) ≤ . For any m ≤ n , wedefine D mt analogously, and we note ∆ n, ( m ) t,v i ≥ ∆ n, ( m − t,v i for all i ≤ n , since v i · u j ≥ , so thesequence is increasing in m . Furthermore, we must have D ¯ mt = ∅ for some ¯ m ≤ n − and then ∆ n, ( m +1) t, · = ∆ n, ( m ) t, · for m = ¯ m, . . . , n , so we have a well-defined limit lim m → n ∆ n, ( m ) t,v i = ∆ n, ( ¯ m ) t,v i .By the above, we can conclude that the default set D t has the desired representation, andthat the corresponding jump size of each L nv i ( t ) is precisely ∆ L nv i ( t ) = lim m → n ∆ n, ( m ) t,v i , just as ∆ L nl,i ( t ) is given by Ξ nl ( t, ∆ L n ( t )) after adjusting for u il if i ∈ D t , and so we are done.When bank j defaults, it contributes u jl to each of the k contagion processes L nl,i . Any apriori solvent bank i (cid:54) = j then feels an aggregate stress proportional to (cid:80) kl =1 v il L nl,i according toits exposure ( v i , . . . , v ik ) to the k characteristics determining the network of liabilities. .3 The mean-field model Given a financial system with n banks, we artificially constructlarger and larger systems of size N = pn for p ≥ with the same network structure. Specifically,we assume the pairs { u i , v i } ni =1 are drawn from some distribution ˆ (cid:36) and then we define λ Nij := nN ˆ u i · ˆ v j = nN k (cid:88) l =1 ˆ u il ˆ v jl N ≥ , (3.9)where the (ˆ u i , ˆ v i ) ’s are i.i.d. samples from ˆ (cid:36) . We are thus creating p = N/n copies of each bankfrom the original system, each with their liability positions scaled down by /p = n/N .Since the true network is not fully observable, we can think of this as choosing our approx-imate network structure (3.4) by fixing a distribution ˆ (cid:36) , e.g., based on generating samples ofthe true network. If one prefers a simple ‘multitype’ structure with homogeneity within groups,one can instead start from (3.4) and simply let ˆ (cid:36) be the corresponding empirical measure. Figure 3:
Limiting ˆ (cid:36) for the particular four-group network structure in Figure 2. In the simplest case with no heterogeneity within groups, the ˆ (cid:36) i ’s are simply given by the point masses δ u ,v , δ u ,v , δ u ,v , and δ u m ,v m . . Let ˆ (cid:36) be a probability measure on R k × R k with the above interpretation, and restrict toheterogeneous parameters of the form f i = f u i ,v i for f i = λ ext i , µ i , σ i . Sending N → ∞ in the N -bank particle systems (3.6) with (3.9), we are then led to formulate the mean-field problem dX u,v ( t ) = − σ u,v ( t ) dt + σ u,v ( t ) (cid:0)(cid:112) − ρ dB ( t ) + ρdB ( t ) (cid:1) − dF u,v ( t ) ,F u,v ( t ) = log (cid:16) − R Λ u,v k (cid:88) l =1 v l (cid:90) t ψ ( T, s ) ψ ( T, d L l ( s ) (cid:17) , L l ( t ) = (cid:90) R k × R k u l P ( t ≥ τ u,v | B ) d ˆ (cid:36) ( u, v ) , τ u,v = inf { t ≥ X u,v ( t ) ≤ } ,X u,v (0) = log (cid:16) x u,v (0) ψ ( T, u,v (cid:17) + (cid:90) T µ u,v ( t ) dt, Λ u,v = λ ext u,v + k (cid:88) l =1 ( v l E [ u l ] − u l E [ v l ]) , (3.10)where B, B are independent Brownian motions. In this mean-field formulation, the hetero-geneity is now modelled by the distribution ˆ (cid:36) of the pairs ( u, v ) . Formally, an ‘infinitesimal’bank indexed by ( u, v ) has liabilities proportional to u · ˆ v = (cid:80) kl =1 u l ˆ v l towards infinitesimalbanks indexed by (ˆ u, ˆ v ) , where each u l ˆ v l gives the exposure to the k characteristics determin-ing the network structure. The value P ( t ≥ τ u,v | B ) can be interpreted as the proportion ofinfinitesimal banks indexed by ( u, v ) that have defaulted by time t . Aggregating the contagion,an infinitesimal bank indexed by (ˆ u, ˆ v ) has thus felt an accumulated stress proportional to ˆ v l L l from its exposure to the l ’th characteristic and a stress proportional to (cid:80) kl =1 ˆ v l L l overall.Under suitable assumptions, we demonstrate in a separate work [11] that weak limit pointsof the particle systems (3.6) are indeed solutions to the McKean–Vlasov problem (3.10). Herewe will instead concentrate on deriving an intuitive condition (3.11) ruling out singularities inthe dynamics of (3.10). We note in passing that this then allows us to show uniqueness of (3.10)given (3.11) in [11], and so we have a unique mean-field limit in this case.While the finite system (3.6) has contagion occurring as jumps dictated by the cascadecondition (3.7), one may hope for a smoother mean-field problem. Indeed, it turns out we havea simple criterion for the mean-field to evolve continuously, meaning that contagion events aresmoothed out in time. Intuitively, the default of an infinitesimal bank indexed by ( u, v ) causesa stress proportional to u · ˆ v ≥ for banks indexed by (ˆ u, ˆ v ) , so if the density of infinitesimalbanks indexed by ( u, v ) is sufficiently inversely proportional to this, the overall effect should becontrolled. Before stating the result, we write ˆ (cid:36) = Law( u , v ) and let S ( u ) , S ( v ) denote thesupport of the random variables u , v . From here on we assume S ( u ) , S ( v ) are compact with u · v ≥ for all v ∈ S ( v ) and u ∈ S ( u ) (as in the finite system for u = u i and v = v j ). heorem 3.2 (Continuous mean-field) . Let X u,v (0) in (3.10) have a density V ( ·| u, v ) . If (cid:107) V ( ·| u, v ) (cid:107) ∞ < Λ u,v − R { u · ˆ v : ˆ v ∈ S ( v ) s . t . u · ˆ v > } ∀ ( u, v ) ∈ S ( u , v ) , (3.11) then any a priori càdlàg solution to (3.10) is continuous in time (here / max ∅ = + ∞ ).Proof. Fix t ≥ . With Ξ and Θ defined in (3.13) of Proposition 3.3 below, we can estimate Ξ( t ; f, v ) ≤ k (cid:88) l =1 v l (cid:90) R k × R k ˆ u l P (cid:0) X ˆ u, ˆ v ( t (cid:57) ) ∈ [0 , Θ( t ; f, ˆ v )] , t ≤ τ | B (cid:1) d ˆ (cid:36) (ˆ u, ˆ v ) ≤ max ˜ v ∈ S ( v ) f (˜ v ) (cid:90) R k × R k v · ˆ u − R Λ ˆ u, ˆ v (cid:107) V ( ·| ˆ u, ˆ v ) (cid:107) ∞ d ˆ (cid:36) (ˆ u, ˆ v ) < max ˜ v ∈ S ( v ) f (˜ v ) for all v ∈ S ( v ) , by (3.11). Now let L v ( t ) := (cid:80) kl =1 v l L l ( t ) and take f ( v ) := ∆ L v ( t ) . Then itfollows from the above bound and (3.12) in Proposition 3.3 below that max v ∈ S ( v ) ∆ L v ( t ) = max v ∈ S ( v ) Ξ( t ; ∆ L , v ) < max v ∈ S ( v ) ∆ L v ( t ) , where we have used the compactness of S ( v ) along with continuity in v . Therefore, we deduce ∆ L · ( t ) ≡ , for any t ≥ , showing that the dynamics in (3.10) are continuous.The proof of the previous theorem relied on the following observation. Proposition 3.3 (Jump size constraint) . Setting L v ( t ) := (cid:80) kl =1 v l L l ( t ) , any càdlàg solution tothe mean-field problem (3.10) satisfies the fixed point constraint ∆ L v ( t ) = Ξ( t ; ∆ L , v ) for all v ∈ S ( v ) , (3.12) for any t ∈ [0 , T ] , where Ξ( t ; f, v ) := (cid:80) kl =1 v l Ξ l ( t ; f ) with Ξ l ( t ; f ) := (cid:90) u l P (cid:0) X u,v ( t (cid:57) ) ∈ [0 , Θ( t ; f, u, v )] , t ≤ τ u,v | B (cid:1) d ˆ (cid:36) ( u, v )Θ( t ; f, u, v ) := log (cid:16) − R Λ u,v (cid:90) t (cid:57) ψ ( T, s ) ψ ( T, d L v ( s ) + 1 − R Λ u,v ψ ( T, s ) ψ ( T, f ( v ) (cid:17) − F u,v ( t (cid:57) ) (3.13) Proof.
Let L v be as in the statement. Clearly, the càdlàgness entails t (cid:55)→ L v ( t ) is càdlàg, andthe dynamics further imply that X u,v has a jump-discontinuity at time t if and only if this isthe case for L v . By assumption v · ˆ u ≥ for all ˆ u ∈ S ( u ) and v ∈ S ( v ) , so we get ∆ L v ( t ) = (cid:90) R k × R k v · ˆ u (cid:0) P ( t ≤ τ | B ) − lim s ↑ t P ( s ≤ τ ˆ u, ˆ v | B ) (cid:1) d ˆ (cid:36) (ˆ u, ˆ v ) (3.14)for all v ∈ S ( v ) . Furthermore, by the càdlàgness, the dynamics (3.10) give that any jumpsatisfies ∆ X ˆ u, ˆ v ( t ) = Θ( t ; ∆ L , ˆ u, ˆ v ) , where Θ is as defined in (3.13), and hence we have P ( t = τ ˆ u, ˆ v | B ) = P ( X ˆ u, ˆ v ( t (cid:57) ) ∈ [0 , Θ( t ; ∆ L , ˆ u, ˆ v )] , t ≤ τ ˆ u, ˆ v | B ) , for all ˆ u, ˆ v ∈ S ( u , v ) . Combining this with (3.14), we therefore arrive at ∆ L v ( t ) = k (cid:88) l =1 v l (cid:90) R k × R k ˆ u P ( X ˆ u, ˆ v ( t (cid:57) ) ∈ [0 , Θ( t ; ∆ L , ˆ u, ˆ v )] , t ≤ τ ˆ u, ˆ v | B ) d ˆ (cid:36) (ˆ u, ˆ v ) for all v ∈ S ( v ) . Noting that the right-hand side is precisely Ξ( t ; ∆ L , v ) for Ξ as defined in(3.13), this proves the desired constraint (3.12), and so we are done. he closest to (3.10) in the literature is the McKean–Vlasov problem studied in [23], whose‘multitype’ network structure is a special case of the framework introduced here. We emphasise,however, that our treatment differs markedly from [23], by focusing on the financial underpin-nings and connecting the mean-field problem to a finite particle system. Moreover, we notethat [23] is focused on criteria for the dynamics to undergo a jump, while we obtain the abovecriterion for continuity. Finally, concerning jumps, our cascade condition sheds new light on[23] in terms of how to characterise jumps in the mean-field problem, as we discuss next.In the absence of (3.11), it is less clear if (3.10) is well-posed, as zero may no longer be theonly solution to the jump size constraint (3.12) and there could be multiple non-zero solutions.Thus, we need to identify which jump size satisfying (3.12) is selected in the limit by the cascadecondition (3.7). With Ξ and Ξ l from (3.13), we conjecture that (3.7) selects solutions to (3.10)with jumps ∆ L l = Ξ l ( t ; ∆ L ) given by the following mean-field cascade condition ∆ L v ( t ) = lim ε ↓ lim m ↑∞ ∆ ( m,ε ) t,v , ∆ ( m,ε ) t,v = Ξ( t ; ε + ∆ ( m − ,ε ) t, · , v ) , ∆ (0 ,ε ) t,v = Ξ( t ; ε, v ) . (3.15)Unlike in (3.7), Ξ( t ; 0 , · ) is always zero, which explains the need for an ε perturbation. Note alsothat (3.15) is indeed well-defined, since ∆ ( m,ε ) t,v forms a bounded sequence increasing as m ↑ ∞ and decreasing as ε ↓ ; crucially, dominated convergence shows that ∆ L given by (3.15) satisfiesthe jump size constraint (3.12). The iterative structure of (3.15) is a particularly nice featuremaking it easy to implement numerically, as we have done in Figure 4 below, illustrating (3.10)and (3.15). A further analysis of the correct jump sizes will form part of [11]. Figure 4:
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