Option Greeks — delta, gamma, vanna, volga — are derivatives of the option price with respect to risk factors. But the risk factors (spot, volatility, time) admit multiple parametrizations, and the second-order Greeks depend on the coordinate choice. Gamma computed in spot coordinates differs from gamma computed in log-spot coordinates. The P&L prediction changes depending on how you measure the risk.
The paper replaces the Hessian with a covariant Hessian defined by an affine connection — the same mathematical structure that describes curvature in general relativity. The connection absorbs the coordinate dependence: first-order Greeks (delta, vega) remain unchanged, but the quadratic terms become coordinate-invariant by construction.
The connection is calibrated, not derived from first principles. Trading desks already apply ad hoc adjustments — Vanna-Volga corrections, execution cost penalties — that modify the second-order P&L. The geometric framework absorbs these adjustments into the connection, making them systematic rather than case-by-case. Different connections correspond to different risk management philosophies, but all produce coordinate-invariant predictions.
The FX barrier examples (EURUSD, USDTRY) demonstrate that the corrections are quantitatively significant for exotic options, where the second-order terms dominate the P&L and the coordinate choice matters most.
Finance as geometry. The portfolio sits on a manifold of risk factors, and the P&L prediction is a Taylor expansion that should be covariant. The standard Greeks are the Christoffel-symbol-free version — correct only in special coordinates. The curved Greeks work everywhere.