Tail-Safe Stochastic-Control SPX-VIX Hedging: A White-Box Bridge Between AI Sensitivities and Arbitrage-Free Market Dynamics

ArXiv ID: 2510.15937 “View on arXiv”

Authors: Jian’an Zhang

Abstract

We present a white-box, risk-sensitive framework for jointly hedging SPX and VIX exposures under transaction costs and regime shifts. The approach couples an arbitrage-free market teacher with a control layer that enforces safety as constraints. On the market side, we integrate an SSVI-based implied-volatility surface and a Cboe-compliant VIX computation (including wing pruning and 30-day interpolation), and connect prices to dynamics via a clipped, convexity-preserving Dupire local-volatility extractor. On the control side, we pose hedging as a small quadratic program with control-barrier-function (CBF) boxes for inventory, rate, and tail risk; a sufficient-descent execution gate that trades only when risk drop justifies cost; and three targeted tail-safety upgrades: a correlation/expiry-aware VIX weight, guarded no-trade bands, and expiry-aware micro-trade thresholds with cooldown. We prove existence/uniqueness and KKT regularity of the per-step QP, forward invariance of safety sets, one-step risk descent when the gate opens, and no chattering with bounded trade rates. For the dynamics layer, we establish positivity and second-order consistency of the discrete Dupire estimator and give an index-coherence bound linking the teacher VIX to a CIR-style proxy with explicit quadrature and projection errors. In a reproducible synthetic environment mirroring exchange rules and execution frictions, the controller reduces expected shortfall while suppressing nuisance turnover, and the teacher-surface construction keeps index-level residuals small and stable.

Keywords: Hedging, VIX, Control Barrier Functions, Dupire Local Volatility, Quadratic Programming, Equities / Volatility

Complexity vs Empirical Score

• Math Complexity: 8.5/10
• Empirical Rigor: 6.2/10
• Quadrant: Holy Grail
• Why: The paper employs advanced stochastic control, convex optimization (QP with CBFs), and theoretical proofs (existence, invariance, bounds), indicating high math density. It also includes detailed empirical methodology (synthetic environment, exchange rule simulation, reproducibility) and performance metrics (expected shortfall, turnover suppression), demonstrating substantial data/implementation rigor.

  flowchart TD
    %% 1. Research Goal
    G["Research Goal: Joint SPX-VIX Hedging"]
    
    %% 2. Methodology Setup
    M["Methodology: White-Box Teacher-Controller"]
    
    %% 3. Inputs
    D["Data/Inputs: SPX/VIX Surface, Cboe Rules"]
    
    %% 4. Computational Processes
    C["Computations:<br/>1. Teacher: SSVI & Dupire Local Vol<br/>2. Controller: CBF-QP with Safety Gates"]
    
    %% 5. Outcomes
    F["Findings: Reduced Shortfall, Stable Residuals"]

    %% Connections
    G --> M
    M --> D
    D --> C
    C --> F