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

Battle Count: 55

The paper is directly relevant to quantitative trading desks managing option books. It addresses core operational concerns: P&L attribution, hedging, smile corrections, execution costs, liquidity-aware rebalancing, and book-level netting. The framework is designed to sit on top of existing desk models without replacing them, making it practically implementable. The EURUSD and USDTRY case studies demonstrate real-world applicability. The separation of curvature overlays (P&L attribution) from cost/risk overlays (execution costs) provides a disciplined language for incorporating desk features. The coordinate-awareness property ensures comparability across different risk-factor parameterizations used in practice.

The framework requires understanding of differential geometry concepts (affine connections, covariant Hessians, Levi-Civita connections, torsion-free conditions, metric compatibility). Implementation involves: (1) computing baseline Greeks and Hessians; (2) constructing the Greek design matrix and solving weighted ridge least-squares systems for connection coefficients; (3) building trade-space impact matrices from quoted widths and clips; (4) computing hedge response matrices and induced factor-space penalties; (5) optionally computing Levi-Civita connections from metrics for liquidity-adjusted Greeks; (6) handling coordinate transformations for parameterization invariance. The mathematical machinery is sophisticated but the calibration reduces to small linear systems. Practical challenges include ensuring rank conditions on calibration instruments, handling poorly conditioned systems, managing liquidity buckets, and maintaining unit consistency across trade units.

The paper provides a public GitHub repository (https://github.com/Lu-Mengjue/curved-greeks-geometric-layer-option-pnl-adjustments) with scripts and notebooks to reproduce the experiments. Market data dates are specified (May 16, 2022 for EURUSD; May 2, 2023 for USDTRY). All mathematical derivations, calibration procedures, and component formulas are provided in the main text and appendices. However, the case studies are diagnostic rather than broad empirical validation, and some inputs (indicative liquidity parameters, tiered impact functions) are illustrative rather than from a specific proprietary data source.