Covered Interest Parity as an H¹ Condition

"The cross-currency basis should be zero. It isn't. And it hasn't been since 2008. The question is not whether the deviation is real — it is why it is structurally impossible to arbitrage away."

Covered interest parity (CIP) is one of the most tested conditions in international finance. It says: the cost of borrowing in one currency and swapping into another via the forward market should equal the cost of borrowing directly in the second currency. Before 2008, it held to within a few basis points. Since 2008, persistent deviations of 20–100bp have been documented in every major currency pair, correlated with regulatory capital costs and quarter-end balance-sheet pressure.

The cohomological framework gives a precise answer to why these deviations persist — and why they cannot be arbitraged away even when every participant knows they exist.

The CIA square

The covered interest arbitrage trade involves four quantities arranged in a rectangle:

         S(t)
  USD ————————————> GBP
   |                 |
   | P_USD(t,T)      | P_GBP(t,T)
   |                 |
  USD ————————————> GBP
         F(t,T)⁻¹

$S(t)$: spot rate (USD per GBP today)
$F(t,T)$: forward rate (USD per GBP at time $T$)
$P_{\text{USD}}(t,T)$: USD discount factor (price of $1 at $T$ in USD)
$P_{\text{GBP}}(t,T)$: GBP discount factor (price of £1 at $T$ in GBP)

The CIP condition is that going around this square returns you to where you started:

\[F(t,T) = S(t) \cdot \frac{P_{\text{USD}}(t,T)}{P_{\text{GBP}}(t,T)}\]

or equivalently, the cross-currency basis $\phi(t,T)$ is zero:

\[\phi(t,T) = \ln F(t,T) - \ln S(t) - \ln P_{\text{USD}}(t,T) + \ln P_{\text{GBP}}(t,T) = 0\]

CIP is a holonomy condition (H⁰ and H¹)

In the language of the framework:

H⁰: each rate taken alone is internally consistent. The spot rate is a well-defined price; the forward rate is a well-defined price; each discount factor is a valid yield curve. No single rate is "wrong."

H¹ = 0 ↔ CIP holds. The four rates fit together around the square — the loop $\text{USD} \to \text{GBP} \to \text{GBP}_T \to \text{USD}_T \to \text{USD}$ returns a holonomy of exactly 1. The cross-currency basis is zero. The square is the boundary of a filled-in 2-simplex — the no-arbitrage surface exists, and the loop is trivial in H¹.

H¹ ≠ 0 ↔ CIP fails. The loop does not close. The cross-currency basis $\phi(t,T) \neq 0$ is a non-trivial H¹ class of the currency sheaf. It is not measurement error, not a bid-ask spread, not a model artefact. It is a topological obstruction: a cycle that cannot be made into a boundary without adding a 2-simplex.

The Crown Jewel Theorem (Paper 295) states this precisely:

CIP holds if and only if the connection on the currency bundle is flat. CIP deviation = curvature = non-trivial H¹ class.

Why can't you arbitrage it away?

Classical arbitrage theory says: if $\phi \neq 0$, trade the loop and pocket the basis. Post-2008, this does not work — and the cohomological framework explains exactly why.

Filling in the 2-simplex (making the CIA loop into a boundary) requires taking on a balance-sheet commitment: you must simultaneously hold a spot FX position, a forward FX position, and two interest rate positions for the full tenor $T$. Each of these consumes regulatory capital under Basel III (leverage ratio, LCR/NSFR). The cost of filling the simplex is the regulatory capital cost of the four-leg trade.

The minimum achievable CIP deviation is therefore not zero — it is the minimum curvature consistent with the balance-sheet constraints:

\[|\phi^*(t,T)| \geq f(\Lambda, \text{LCR}, \text{NSFR})\]

where $\Lambda$ is the Basel III leverage ratio requirement. Du, Tepper, and Verdelhan (2018) document empirically that CIP deviations are systematically correlated with leverage ratio pressure, quarter-end balance-sheet windows, and dealer identity — exactly the pattern predicted by a curvature floor set by regulatory capital costs.

The topological reading: post-2008 regulation made the 2-simplex expensive to create. The H¹ class that used to be zero (cheap to trivialise) is now persistently non-zero because the instrument that would trivialise it — the fully funded four-leg CIA trade — costs more to hold than it earns.

Central bank swap lines as topological surgery

A Federal Reserve swap line with (say) the ECB adds a new direct edge between USD and EUR at the central bank level, bypassing the private dealer network. In homological terms, this is topological surgery: a new edge is added to the graph, which can fill in the CIA triangle at the central bank level even when the private market cannot.

The swap-line theorem (Paper 295) states: a swap line between two central banks partially cancels the CIP deviation but cannot fully eliminate it, because each path in the private network is still subject to balance-sheet constraints. The central bank fills one triangle; the private network still carries the residual curvature on all other paths.

This is why swap lines stabilise FX markets in crises (2008, 2020) without permanently eliminating the CIP basis: they reduce the H¹ class on the central-bank triangle but leave the rest of the currency sheaf's curvature unchanged.

The multi-currency case: triangular arbitrage as H¹

With three currencies — USD, GBP, EUR — there are three CIA squares (one per currency pair) and one triangular arbitrage condition:

\[S(\text{USD/GBP}) \cdot S(\text{GBP/EUR}) \cdot S(\text{EUR/USD}) = 1\]

The three CIA squares are edges in the currency complex; the triangular arbitrage condition is the face — it is a 2-simplex. When spot markets are liquid, this face is always filled: triangular spot arbitrage is fast, costless, and keeps the triangle closed. The spot triangle contributes zero to H¹.

The forward triangle is different. Three CIA forward conditions $\phi_{\text{USD/GBP}}(T)$, $\phi_{\text{GBP/EUR}}(T)$, $\phi_{\text{USD/EUR}}(T)$ each carry their own curvature. Even if each pairwise basis is non-zero, they must satisfy a consistency condition around the triangle — a lower-dimensional version of the Pentagon identity. If they do not, there is H¹ at the triangle level: the three forward bases are mutually inconsistent, not just individually non-zero.

This is the multi-currency basis inconsistency that appeared in USD/EUR/JPY triangles during the 2008 and 2020 crises: each pairwise basis looked explicable, but the three bases together failed to close the triangle.

What this means for a rates desk

Observable	Cohomological meaning	Instrument
Cross-currency basis $\phi(t,T)$	H¹ class of the currency sheaf	Cross-currency basis swap
CIP holds	H¹ = 0; connection is flat	No basis swap needed
Post-2008 persistent basis	H¹ ≠ 0; regulatory capital floors curvature	Basis swap; FX swap
Central bank swap line	Topological surgery; adds edge at CB level	N/A (CB instrument)
Triangular arbitrage closed	Forward triangle is a filled 2-simplex	Three-way FX package
Multi-currency basis inconsistency	H¹ at triangle level; three bases don't close	Three-way basis package
XVA on CIA trade	Cost of filling the 2-simplex	CVA/FVA/MVA charges

The cross-currency basis is not noise — it is a signal about the topological structure of the currency network. A desk that models it as a spread to be fitted is working at H⁰. A desk that models it as a holonomy class is working at H¹ — and can compute the policy gradient $\partial \phi / \partial \Lambda$ to answer: how much would the basis tighten if the leverage ratio were relaxed by 1%?

Observable	Cohomological meaning	Instrument
Cross-currency basis \(\phi(t,T)\)	H¹ class of the currency sheaf	Cross-currency basis swap
CIP holds	H¹ = 0; connection is flat	No basis swap needed
Post-2008 persistent basis	H¹ ≠ 0; regulatory capital floors curvature	Basis swap; FX swap
Central bank swap line	Topological surgery; adds edge at CB level	N/A (CB instrument)
Triangular arbitrage closed	Forward triangle is a filled 2-simplex	Three-way FX package
Multi-currency basis inconsistency	H¹ at triangle level; three bases don't close	Three-way basis package
XVA on CIA trade	Cost of filling the 2-simplex	CVA/FVA/MVA charges