The Topology of Risk
"Just because I don't know what the connection is doesn't mean there isn't one." — Douglas Adams, The Long Dark Tea-Time of the Soul
In 2008, every major bank's risk models passed their internal validation tests. The system failed anyway — not because the models were wrong, but because they were locally correct and globally inconsistent.
EconIAC is built on the insight that financial risk has three structural levels, and that existing tools only address the first two.
The three levels of financial risk
| Level | Name | What it is | Instruments | Who manages it |
|---|---|---|---|---|
| \(H^0\) | Bilateral risk | Risk between two parties | Forwards, swaps | Trading desks |
| \(H^1\) | Triangular risk | Risk visible only when three parties interact | Options, swaptions, CDOs | XVA desks |
| \(H^2\) | Systemic risk | Mutual inconsistency of institutions' triangular risks | CCPs, central banks | Regulators, CRO |
Bilateral risk (\(H^0\)) is perfectly hedgeable with forwards and swaps. Every VaR model, every balance-sheet stress test, every bilateral exposure report operates at this level.
Triangular risk (\(H^1\)) is the risk that only appears when three parties interact. No portfolio of bilateral contracts can eliminate it — this is a mathematical theorem, not a practical limitation. Convexity, basis risk, correlation risk, CVA, FVA, and the volatility smile are all triangular risks. Options and swaptions exist because \(H^1 \neq 0\).
Systemic risk (\(H^2\)) is the risk of the whole: the mutual inconsistency of individual institutions' triangular risk estimates. Wrong-way risk in XVA is \(H^2\). The 2008 crisis was \(H^2\). No existing risk system computes \(H^2\).
What EconIAC computes that existing systems cannot
| Computation | Existing tools | EconIAC |
|---|---|---|
| Bilateral stress test | ✅ Standard (VaR, balance sheet) | ✅ |
| Triangular risk (\(H^1\)) | Partial (XVA desks, CVA) | ✅ Full sheaf cohomology |
| Systemic stability (\(H^2\)) | ❌ Not computed anywhere | ✅ Pentagon identity test |
| Pre-crisis early warning | Contemporaneous indicators | ✅ \(H^1\) leads cascade by 2–3 periods |
| Policy gradient \(\partial\text{loss}/\partial\text{haircut}\) | ❌ | ✅ One backward pass |
| SIFI designation | Size-based (FSB) | ✅ Topological — \(H^2\) contribution |
| Wrong-way risk in XVA | Parametric approximation | ✅ Exact \(H^2\) class |
Mathematical tools by user and problem
Each pillar of EconIAC addresses specific problems for specific users. The grid below shows which tool is relevant to which problem — with the specific result, not just a checkmark.
| User / Problem | Gauge theory | Cohomology (bilateral · triangular · systemic) |
Thermodynamics (Gibbs, β) |
Sheaves | Differentiable ABM |
|---|---|---|---|---|---|
| Central bank / stress testing |
FX triangular arbitrage = non-zero holonomy; FX reserves as parallel transport | Core tool. Tier-0/1/2 stress test. H² stability class. SIFI theorem. 2008 as H² event. | Gibbs-lifted cascade: smooth sigmoid around hard threshold; β* = phase transition point | H¹ early warning: bilateral inconsistency 2–3 periods before cascade | Differentiable fire-sale and repo-run models; policy gradient ∂loss/∂haircut |
| Bank / XVA desk | CVA/DVA/FVA/MVA as gauge curvature on Pacioli manifold; Burgard–Kjaer PDE = flatness condition | Core tool. CVA/FVA/MVA = H¹. Wrong-way risk = H². KVA at H¹/H² boundary. Model-free triangular pricing. | Rationality temperature β: calibrate from observed bid-ask spread variance | Model-free H¹: compute from CDX tranche prices, no parametric model needed | Smooth Greeks; differentiable netting set simulation; reverse stress test on XVA book |
| Bank / market risk | Yield curve as connection; convexity = curvature = H¹; vol surface as curvature field | Convexity = H¹ of discount factor sheaf (HJM drift = coboundary condition). Smile arbitrage = H¹ scanner. | Gibbs relaxation of hard VaR threshold; differentiable ES; β-parametric stress | Vol surface consistency: calendar + butterfly arbitrage as joint H¹ condition | Differentiable Monte Carlo; exact second-order sensitivities via jax.hessian |
| Regulator / SIFI | Balance sheet as gauge field; double-entry = flatness; money creation = curvature | Core tool. SIFI = institution whose removal changes H². Size neither necessary nor sufficient. Topological capital charge. | Phase transition at β*: systemic fragility as thermodynamic bifurcation | Sheaf Laplacian on interbank graph; H¹ signal detects bilateral inconsistency | Differentiable resolution planning; optimal liquidation path via policy gradient |
| Asset manager / portfolio risk |
FX hedging as parallel transport; cross-currency basis = curvature; basis swap = holonomy | Basis risk = H¹ of funding sheaf. Correlation risk = H¹ of credit sheaf. CDO tranche price = H¹ class. | Thermal Shapley: attribute portfolio risk to factors in one backward pass | Correlation-consistent pricing: H¹ from basket option prices, model-free | Differentiable portfolio optimisation; exact attribution across correlated positions |
| RegTech vendor | Full gauge-theoretic finance stack: FX, rates, credit, XVA — one unified framework | Core product. Cohomological stress testing as a service. H² monitor. Topological SIFI analytics. Model-free wrong-way risk. | Differentiable ABM platform: calibrate any contagion model by gradient descent | Real-time H¹ inconsistency scanner on EMIR data; leads cascade by 2–3 periods | All five Hurd contagion channels; policy gradient ∂H²/∂exposure; GPU-native |
| Macro / climate economist |
Double-entry accounting as discrete gauge theory; sectoral balances = Pacioli identity; carbon tax = gauge field | Supply chain inconsistency = H¹ of input-output sheaf; sector coupling = H² of multi-sector model | Core tool. Rationality temperature β. Gibbs Keen/GEMMES. Differentiable Nash equilibria. Climate yield surface. | Stock-flow consistency: Godley table as sheaf; violation = non-zero coboundary | Differentiable GEMMES/LowGrow; calibrate to national accounts; reverse stress test climate scenarios |
Who EconIAC is for
- Central banks and regulators — cohomological stress testing, \(H^2\) early-warning indicator, topological SIFI designation
- Banks and financial institutions — XVA wrong-way risk, triangular risk classification, model-free correlation pricing
- Fund managers — \(\beta_1\) portfolio topology, LDI crisis as \(H^2\), currency overlays, ALM joint sheaf, topology arbitrage
- Broker-dealers and prime brokers — matched book topology, rehypothecation chain depth, intermediation void risk, clearing scope
- RegTech and model vendors — the computational platform for bilateral/triangular/systemic risk as a service
Start here
The fastest path to understanding the framework:
- The Topology of Risk: A Primer — 13 pages, no prior mathematics required, Holmes vs Gently framing
- The Unhedgeability Theorem — the unhedgeability theorem and five-instance table
- Systemic Risk as \(H^2\) — cohomological stress test, SIFI theorem, XVA section