Financial risk models care about where the price ends up. A stock at $100 today and $100 tomorrow is “no change.” But the path matters — $100→$150→$100 and $100→$50→$100 are very different experiences. Margin calls, stop-losses, and portfolio rebalancing all happen along the path, not at the terminal point. The entire apparatus of Value at Risk, Expected Shortfall, and regulatory capital requirements was built for terminal distributions and struggles with path dependence.
Bloch (arXiv: 2603.24154) introduces the SigSwap — a financial instrument and framework built on rough path signatures to capture path-dependent risk geometrically.
The path signature is a sequence of iterated integrals of a path. It is a complete characterization: two paths with the same signature are the same path (up to reparameterization). The signature decomposes complex path behavior into a basis of “path features” — lead-lag structure, oscillation patterns, directional persistence. Each signature component is a linear risk factor.
This enables Signature Expected Shortfall: a risk metric that captures toxic path geometries (flash crashes, liquidity spirals, lead-lag dynamics) that traditional metrics miss. The Temporal Exposure Profile uses anticipatory learning on the signature to detect geometric decoupling before it becomes realized volatility.
The through-claim: risk has a geometry, and the path signature is its coordinate system. Traditional risk measures project the path onto its endpoint and measure the result. The SigSwap preserves the full geometric content of the path and decomposes it into linear factors. The conversion from “unmodellable path risk” to “linear factor exposure” is not an approximation — it's a change of basis. The risk was always geometric; the models weren't.
Bloch, 2603.24154. Quantitative finance / risk management / rough paths / path signatures / financial regulation.