Skip to content

The Topology of Risk

Identifying, Measuring, and Controlling Inconsistency in Financial Systems

Ian R. C. Buckley

This book develops a single idea from first principles: financial risk is fundamentally about inconsistency.

A bilateral contract is consistent with another bilateral contract. Three bilateral contracts may be mutually consistent — or they may form a funding loop that cannot be closed. Four consistent triangles may still fail to fit together. Each level of inconsistency has a name, a measure, and a set of instruments that can (or cannot) address it. The three levels are \(H^0\), \(H^1\), and \(H^2\).

No prior knowledge of algebraic topology is assumed. Every concept is introduced through a worked example that a credit risk manager, XVA desk, or regulator would recognise. Definitions follow examples, not the other way round.

Who this book is for

  • Risk managers and XVA desks who want to understand why certain risks cannot be hedged away, not just as a practical observation but as a theorem
  • Regulators and central bankers who want a framework that distinguishes systemic risk (\(H^2\)) from bilateral and triangular risk, and that gives exact measures rather than heuristics
  • Fund managers facing basis risk, wrong-way risk, LDI, and currency overlay problems that standard models handle poorly
  • Quantitative analysts who want the mathematical foundations without category theory or spectral sequences

How to read this book

The chapters build sequentially. Chapters 1–2 establish the geometric language (simplicial complexes, boundary operators). Chapters 3–5 cover \(H^0\) and \(H^1\) — the bilateral and triangular levels that existing tools partially address. Chapters 6–8 build to \(H^2\) and the 2008 crisis.

Each chapter opens with a numerical example, derives the concept from that example, and closes with the formal definition.

Chapters

  1. Double-entry bookkeeping as \(\partial^2 = 0\) — the simplest homology
  2. The balance sheet as a simplicial complex — nodes, edges, triangles
  3. When do bilateral rates fit together? — \(H^0\) and global sections
  4. Funding loops and the first Betti number — \(H^1\) and independent cycles
  5. Netting and clearing — removing edges vs filling triangles; the EN model as \(H^0\)
  6. Basis, convexity, wrong-way risk — the unhedgeable residual as an \(H^1\) class
  7. The hollow tetrahedron — when four consistent triangles still fail to close; \(H^2\)
  8. The 2008 crisis as an \(H^2\) event — correlation desks, AIG, Lehman prime brokerage

Relationship to the papers

The EconIAC papers (available at zenodo.org/communities/econiac) contain the formal theorems and proofs. This book contains the same ideas in a different register: worked examples, financial intuition, and applications. The fastest entry point to the papers is The Topology of Risk: A Primer (13 pages, no prior mathematics required).