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

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).

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

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).
Paper Structure (13 sections, 1 theorem, 113 equations, 5 figures, 4 tables)

This paper contains 13 sections, 1 theorem, 113 equations, 5 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Related Work
  3. Notation, Units and Coordinate Invariance
  4. Vanna--Volga as Curvature: Market-Implied Connection
  5. Coordinate Invariance and Geodesics
  6. Liquidity Curves the Greeks
  7. The Constant-$\Lambda$ Regime
  8. Example: Knock-in barrier options with FX underlying
  9. Further Applications and Extensions
  10. Conclusion and Future Research
  11. Levi--Civita interpretation and metric reconstruction
  12. Connection coefficients under coordinate changes
  13. Component formulas for $g_\ell$ and its derivatives

Key Result

corollary 1

For any increment $\delta x\in T_xX$ and its representation $\delta y$ in another chart, the quadratic predictor is an intrinsic scalar, i.e. it equals $V_\alpha\,\delta y^\alpha+\tfrac{1}{2}\,\widetilde{H}_{\alpha\beta}\,\delta y^\alpha\delta y^\beta$.

Figures (5)

  • Figure 1: EURUSD UIC barrier P&L increment reported in USD with $K = 0.98$, $T = 1.0$, $B=1.01$ and weekly reconstruction frequency.
  • Figure 2: Histogram of EURUSD UIC barrier P&L increment with $K = 0.98$, $T = 1.0$, $B=1.01$ and daily reconstruction frequency.
  • Figure 3: Impact function of the hedging set $\mathcal{H}$ with units $\mathrm{\texttt{TRY}}/\mathrm{\texttt{USD}}^2$. The x-axis is trade size $|q|$ in USD millions.
  • Figure 4: Information of weekly hedging of a USDTRY UIC Barrier option (notional $10MM$USD) using hedging set $\mathcal{H}$ and weekly rebalancing. Parameters of UIC option: $K = 38.0$, $T = 2.0$, $B=40.0$.
  • Figure 5: Histograms of absolute hedge trades $|q|$ in USD millions executed at each weekly rebalancing date for the instruments in $\mathcal{H}$, for a $10$MM USD notional USDTRY UIC barrier option.

Theorems & Definitions (1)

  • corollary 1: Coordinate invariance of the quadratic predictor