Chapter 5: Netting and Clearing
Bilateral netting removes edges. CCP novation fills triangles. Neither removes all loops.
The example
Four dealers: A, B, C, D. Each has a bilateral IRS position with the other three — six edges in total, forming a complete graph \(K_4\). The gross notional is £10bn per edge; £60bn total.
Bilateral netting within each pair: A and B net their positions against each other, reducing their bilateral exposure from £10bn gross to perhaps £1bn net. The edge remains but is smaller. \(\beta_1\) is unchanged: the loops are still there.
CCP novation: A and B each novate their trade to a central counterparty (CCP). Now A has a contract with the CCP, and B has a contract with the CCP. The original A--B edge is replaced by two edges (A--CCP and B--CCP). But this adds the CCP as a new node — a hub connecting all four dealers. The star graph has \(\beta_1 = 0\): no funding loops. The loops have been filled, not just reduced.
This is the topological distinction between netting and clearing.
Removing edges vs filling triangles
| Operation | Topological effect | Effect on \(\beta_1\) |
|---|---|---|
| Bilateral netting | Reduces edge weight | None — loop still exists |
| Multilateral netting | May remove edges | Reduces \(\beta_1\) if loop is broken |
| CCP novation (full) | Replaces cycle with star | \(\beta_1 \to 0\) for cleared products |
| Partial novation | Replaces some legs | \(\beta_1\) reduced but not zero |
The Netting Hierarchy Theorem (Paper 439) makes this precise: bilateral netting \(\subset\) multilateral netting \(\subset\) CCP novation in terms of \(\beta_1\) reduction. Each is strictly more powerful than the previous, but none can reduce \(\beta_2\) to zero.
The \(H^2\) Impossibility Theorem
CCP clearing can reduce \(\beta_1\) to zero for a single product class. But financial institutions trade multiple products: IRS, CDS, FX, repo, equity derivatives. A loop that passes through IRS on one leg and FX on another is not cleared by any single CCP.
The \(H^2\) Impossibility Theorem: no combination of bilateral netting, multilateral netting, or CCP novation by private institutions can reduce \(\beta_2(\Gamma)\) to zero. The irresolvable four-party conflicts (hollow tetrahedra) survive any netting or clearing arrangement that operates product by product.
This is why the 2008 crisis was not resolved by close-out netting: the problem was not bilateral or triangular, it was \(H^2\). Novation closed the loops on individual products; the cross-product hollow tetrahedra remained.
The EN model revisited
Chapter 3 introduced Eisenberg-Noe as a global section problem (\(H^0\)). We can now see its limitation precisely. EN finds the clearing payment vector for a given network of bilateral debts. It operates on the 0-skeleton and 1-skeleton only. It cannot:
- Detect funding loops (\(H^1\)) that amplify before a default is triggered
- Identify hollow tetrahedra (\(H^2\)) that survive close-out
The 2008 EN-style stress tests returned green because the bilateral exposures were manageable. The crisis was driven by \(H^1\) amplification (repo run) and \(H^2\) irresolvability (AIG, Lehman prime brokerage). These were invisible to the tools in use.