Clearing, Netting, and the Topology of Obligation

"The gold at the centre of the system is not gold at all — it is the promise of orderly netting." — Perry Mehrling, The New Lombard Street

Whether a network of financial obligations clears bilaterally, through a central counterparty, or not at all is not just an operational detail. It changes the topology of the network — which determines what kinds of failure cascade are even possible.

The three levels

Arrangement	Topology	Cohomology	What it eliminates
No netting (gross exposures)	Bare graph	—	nothing
Bilateral netting	Fewer edges	Reduces \(\beta_1\)	Edge-level loops
CCP / multilateral clearing	Filled triangles	Annihilates \(H^1\) generators	Loop as topological class

The distinction matters because topology determines what can go wrong, not just how much is at risk.

Level 0: gross exposures

Start with four institutions \(A, B, C, D\) and a set of gross obligations:

A → B  (A owes B)
B → C  (B owes C)
C → A  (C owes A)
C → D
D → A

\(A \to B\) means A has a gross obligation to B. The arrow runs from obligor to creditor. These are gross exposures before any netting. This is the natural setting for repo and securities-lending markets, where rehypothecation means gross positions can be many times net positions — and a net position of zero can still sit inside a large funding loop that matters for systemic risk.

The network has \(n=4\) nodes, \(m=5\) edges, \(c=1\) component, so \(\beta_1 = m - n + c = 2\): exactly two independent funding loops.

The Eisenberg-Noe model: clearing as an H⁰ problem

The Eisenberg-Noe (2001) model asks: if some institutions cannot pay in full, what is the clearing payment vector \(p^*\) — the actual payments made, respecting seniority and pro-rata rules?

The answer is a fixed point:

\[p_i^* = \min\!\left(\bar{p}_i,\; e_i + \sum_j \frac{L_{ji}}{\bar{p}_j} p_j^*\right)\]

where \(\bar{p}_i\) is the total liability of institution \(i\) and \(e_i\) is its external asset value.

In cohomological language, this is an H⁰ computation. The clearing vector \(p^*\) is the unique global section of the payments sheaf on the 1-skeleton (the graph of bilateral edges). EN uses only the edges — the bilateral exposure matrix \(L_{ij}\). When \(p^*\) exists and is unique, the payments sheaf has a consistent global section: \(H^0\) is well-posed. EN's main theorem (existence and uniqueness of \(p^*\)) is exactly a statement about \(H^0\).

What EN correctly captures: direct, first-order contagion. If \(A\) defaults, \(B\) loses exactly the unpaid portion of \(L_{AB}\). The cascade propagates along edges.

What EN cannot see:

Failure mode	Why EN misses it
Funding freeze via shared repo counterparty	No bilateral edge between MMF and distant dealer — indirect channel is a triangle, not an edge
Basis blow-out (credit/rates correlation)	H¹ of the joint sheaf; no bilateral clearing can unwind it
AIG's CDS book: collateral calls created by the very protection it sold	Conflict cycle across four institutions — H²; EN's iteration does not converge uniquely

EN is the right instrument for its level. The problem in 2008 was applying an \(H^0\) tool to a system whose failure was \(H^1\) (funding loops) and \(H^2\) (cross-desk correlation inconsistency).

Bilateral netting: removing edges

A bilateral netting agreement (e.g. an ISDA Master Agreement between \(A\) and \(B\)) allows \(A\) and \(B\) to replace their gross claims on each other with a single net position:

\[L_{AB}^{\mathrm{net}} = \max(L_{AB} - L_{BA},\; 0), \qquad L_{BA}^{\mathrm{net}} = \max(L_{BA} - L_{AB},\; 0)\]

If both owe each other comparable amounts, \(L_{AB}^{\mathrm{net}} \approx 0\): the edge effectively disappears from the graph.

Topological effect: removes an edge, reduces \(\beta_1\).

If \(A\) and \(C\) net bilaterally in the example above, the edge \(C \to A\) disappears. Loop 1 (\(A \to B \to C \to A\)) collapses — it can no longer be traversed. \(\beta_1\) drops from 2 to 1.

But bilateral netting between \(A\) and \(C\) does nothing about:

The \(C \to D \to A\) loop (Loop 2) — it remains
Any triangle involving \(B\) — still present
Any default correlation between \(B\) and \(C\) — invisible to netting

Bilateral netting reduces edge count. It does not change the dimension of the simplicial complex. The highest non-degenerate cells are still edges (1-simplices). No triangle is filled in.

When do we fill in a triangle?

A triangle \(A\)-\(B\)-\(C\) is filled in (added as a 2-simplex) when \(A\), \(B\), and \(C\) enter a genuinely trilateral arrangement — one where the commitment runs to the triple, not just to each bilateral pair.

The test: does closing out \(A\)'s position automatically and simultaneously affect \(B\)'s and \(C\)'s positions, without any further bilateral negotiation? If yes, the triangle is filled in.

Three examples:

1. CCP novation

A central counterparty (CCP) novates three bilateral trades: \(A\)-\(B\), \(B\)-\(C\), \(C\)-\(A\). After novation, the CCP is buyer to every seller and seller to every buyer. The three bilateral edges are replaced by three edges to a central node (the CCP), and the original triangle \(A\)-\(B\)-\(C\) is filled in as a 2-simplex.

The loop \(A \to B \to C \to A\), which was a free generator of \(H^1\), is now the boundary of the filled triangle. A boundary is not a free cycle — it bounds something. So the loop is annihilated as an \(H^1\) generator. \(\beta_1\) drops by 1.

This is stronger than bilateral netting: CCP novation moves the cycle from \(H^1\) into the image of the boundary map \(\partial_2\). Bilateral netting removes an edge (changes the graph); CCP novation changes the homological class of the cycle.

Concretely: CCP novation gives multilateral default protection that bilateral netting does not. If \(A\) defaults under bilateral netting, \(B\) and \(C\) have separate close-out processes that may not coordinate. Under CCP novation, a single waterfall (initial margin, default fund, CCP equity) covers all three simultaneously.

2. Trilateral repo agreement

Three dealers agree to a tri-party repo with a custodian bank as agent. The custodian holds the collateral and manages substitution rights for all three simultaneously. This fills in the triangle: the custodian's simultaneous management of all three legs makes it a 2-simplex, not three separate bilateral 1-simplices.

3. Cross-margin agreement across three products

An exchange allows margin offsets across equity futures (\(A\)), equity options (\(B\)), and index futures (\(C\)). The offset is computed at the portfolio level — not pairwise. This fills in the product-type triangle and annihilates the basis-risk loop as an \(H^1\) generator.

When do we fill in a tetrahedron?

A tetrahedron \(A\)-\(B\)-\(C\)-\(D\) is filled in (added as a 3-simplex) when all four institutions enter a genuinely quadrilateral arrangement — one where the commitment runs to the quadruple simultaneously.

In practice, this almost never happens in finance. Four-party simultaneous commitments are rare: CCPs clear pairs or triples; cross-margining covers product pairs or triples at a single venue. This is precisely why \(H^2\) obstructions persist in the financial system — because the resolution instruments (bilateral netting, CCPs) only go up to the triangle level.

The hollow tetrahedron is the default state of any four-institution network. The four triangular faces \((A,B,C)\), \((A,B,D)\), \((A,C,D)\), \((B,C,D)\) each carry their own pricing or risk residual — an \(H^1\) class computed on that face. Genuine \(H^2\) is a statement about whether those four face-level residuals are mutually consistent. If they were all computed from a single model, they automatically satisfy

c_BCD − c_ACD + c_ABD − c_ABC = 0

This is the boundary identity \(\delta^2 \circ \delta^1 = 0\) (Pentagon identity in its simplest form). A single source, however many triangles, cannot generate \(H^2\).

The \(H^2\) obstruction appears when the four face residuals come from more than one source — different desks, different models, different jurisdictions — that each produce internally consistent triangles but whose triangle-level residuals fail to close around the tetrahedron.

This is precisely the structure of the 2008 correlation desk failure: each desk's own triangle looked fine; the four overlapping triangles did not close.

The rule of thumb

Fill in when...	Cohomological effect
\(k+1\) parties enter a simultaneous joint arrangement	Adds a \(k\)-simplex
Bilateral netting (2-party)	Removes an edge (reduces \(m\), reduces \(\beta_1\))
CCP novation (3-party simultaneous)	Fills a triangle (annihilates \(H^1\) generator)
Hypothetical 4-party simultaneous resolution	Fills a tetrahedron (would annihilate \(H^2\) generator)
Government bailout of AIG / Lehman	The only actual \(H^2\) resolution in 2008 — no private 4-party mechanism existed

The absence of a filling is not just an operational gap — it is a structural constraint on what resolution mechanisms exist. An \(H^2\) obstruction cannot be resolved by any combination of bilateral agreements or CCPs. It requires an instrument that acts at the tetrahedron level: a central bank, a resolution authority, or (in Paper 426's framework) a \(K^{\text{top}}\) charge that prices the irresolvability before the fact.

What the topology tells the regulator

Question	Classical answer	Topological answer
Is this institution safe?	Net exposure, VaR	\(H^0\): consistent global section of payments sheaf
Is this funding loop dangerous?	Count the cycles	\(H^1\): \(\beta_1\) independent funding loops; netting or CCP reduces this
Is this crisis resolvable bilaterally?	Workout, ISDA close-out	\(H^2 = 0\)? If not, no bilateral resolution exists
What would it take to resolve it?	Government intervention	Fill the tetrahedron: act at the 4-party level

The 2008 crisis was a sequence of \(H^1\) loops (repo, commercial paper) cascading into an \(H^2\) obstruction (AIG, Lehman prime brokerage) that only a government-level instrument could resolve. No amount of bilateral netting or CCP novation of triangles resolves an \(H^2\) obstruction — because the obstruction lives one dimension higher than any available instrument.