From Market to Portfolio Chirality

<p><span>In a companion paper (Ambarish, Kreuser and Seigel, 2026 — hereafter AKS-I) we established that financial markets are intrinsically chiral: the implied volatility surface Σ(k,T) is a curved Riemannian manifold that cannot be superimposed on its mirror image. The chirality index χ(T) = (∂Σ/∂k)/(∂²Σ/∂k²) quantifies this asymmetry at each maturity T, estimated from two SPX datasets spanning Q1 2026 (61 trading days, 427 maturity-day observations) and 2019–2023. We develop the relationship between market-level chirality and the chirality of individual instruments and portfolios across the full maturity term structure. Market chirality is the ambient Riemannian geometry in which all instruments live; instrument chirality is mediated through the position’s vanna Λ = ∂²V/∂S∂Σ, expressed as ξ(X,T) = χ(T)·(Λ/Γ) per unit gamma, with ρ(S,Σ) entering separately as the market transmission parameter linking chirality to realised hedging P&amp;L. We define a portfolio chirality index Ψ (the vanna-weighted average of χ(T) across all maturities) and establish a chirality irreducibility theorem: chirality cannot be eliminated from any options portfolio by delta hedging alone. Ψ determines the direction of the expected asymmetry in realised delta-hedging P&amp;L between up- and down-market days, while the signed aggregate exposure Ξₚ determines its magnitude. Instrument and portfolio chirality data are presented at a consistent five-maturity grid (1M, 3M, 6M, 12M, 24M) throughout. Detailed notation is in Annexure A.</span></p>