Curved Greeks: A Geometric Layer for Option P&L Adjustments

TL;DR

This paper introduces a geometric, model-agnostic framework for option P&L adjustments that ensures quadratic risk measures are invariant under different parameterizations, improving hedge accuracy and liquidity management.

Contribution

It proposes a covariant Hessian using an affine connection to make quadratic P&L predictions coordinate-invariant, with practical calibration and application to FX case studies.

Findings

Invariant quadratic P&L predictor demonstrated in FX case studies

Calibration via small linear systems is efficient and well-posed

Framework enables local liquidity-aware sensitivities and rebalancing strategies

Abstract

Short-horizon option book management relies on P&L expansions in a small set of risk factors. In practice, the quadratic term and common desk adjustments (smile corrections, execution cost add-ons) depend on the chosen factor coordinates, so predicted second-order P&L can change when moving between spot, forward, and log-forward parameterizations. We propose a local, model-agnostic framework that makes the quadratic term coordinate invariant. The usual Hessian is replaced by a covariant Hessian defined by an affine connection, yielding an invariant quadratic predictor. The connection is calibrated to match a desk target for quadratic P&L (Vanna-Volga for smile effects or, in principle, a local fit to realized P&L) while leaving first-order hedge Greeks unchanged. Execution frictions enter through a quadratic cost model for hedge trades. Combined with hedge ratios, this induces an equivalent quadratic penalty on factor moves, makes portfolio netting of costs explicit, and provides local liquidity-aware second-order sensitivities and rebalancing directions. Calibration reduces to small linear systems with clear identifiability conditions. Two FX barrier case studies (EURUSD, USDTRY) illustrate the workflow, and we briefly sketch extensions to other quadratic penalties (risk normalization, scenario/gap terms, and xVA/capital add-ons).