Paper: arXiv 2603.14438

Authors: Pedro Pablo Pérez Velasco, Mengjue Lu, Daniel Arrieta

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).

Complexity vs Empirical Score

  • Math Complexity: 7.0/10
  • Empirical Rigor: 4.5/10
  • Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: The paper introduces an innovative differential geometric approach to option P&L attribution by replacing the standard Hessian with a covariant Hessian. While mathematically elegant and highly original, the empirical validation is limited to illustrative FX barrier case studies rather than exhaustive backtests against historical data.

Research Flowchart

  flowchart TD
    A[Goal: Coordinate-Invariant Option P&L & Liquidity Adjustments] --> B[Inputs: Spot/Vol Data, Desk Targets & Execution Frictions]
    B --> C[Construct Covariant Hessian via Affine Connection]
    C --> D[Calibrate Connection to Target P&L & Add Quadratic Trade Costs]
    D --> E[FX Barrier Case Studies: EURUSD & USDTRY]
    E --> F[Outcomes: Invariant P&L, Portfolio Netting & Rebalancing Directions]

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0 of 30 cited works are in the corpus; resolved via arXiv source bundles, Semantic Scholar and OpenAlex. Corpus-wide: 2518 linked papers.