Proof-Carrying No-Arbitrage Surfaces: Constructive PCA-Smolyak Meets Chain-Consistent Diffusion with c-EMOT Certificates

ArXiv ID: 2511.09175 “View on arXiv”

Authors: Jian’an Zhang

Abstract

We study the construction of SPX–VIX (multi\textendash product) option surfaces that are simultaneously free of static arbitrage and dynamically chain\textendash consistent across maturities. Our method unifies \emph{“constructive”} PCA–Smolyak approximation and a \emph{“chain\textendash consistent”} diffusion model with a tri\textendash marginal, martingale\textendash constrained entropic OT (c\textendash EMOT) bridge on a single yardstick $\LtwoW$. We provide \emph{“computable certificates”} with explicit constant dependence: a strong\textendash convexity lower bound $\muhat$ controlled by the whitened kernel Gram’s $λ_{"\min"}$, the entropic strength $\varepsilon$, and a martingale\textendash moment radius; solver correctness via $\KKT$ and geometric decay $\rgeo$; and a $1$-Lipschitz metric projection guaranteeing Dupire/Greeks stability. Finally, we report an end\textendash to\textendash end \emph{“log\textendash additive”} risk bound $\RiskTotal$ and a \emph{“Gate\textendash V2”} decision protocol that uses tolerance bands (from $α$\textendash mixing concentration) and tail\textendash robust summaries, under which all tests \emph{“pass”}: for example $\KKT=\CTwoKKT\ (\le 4!!\times!10^{"-2"})$, $\rgeo=\CTworgeo\ (\le 1.05)$, empirical Lipschitz $\CThreelipemp!\le!1.01$, and Dupire nonincrease certificate $=\texttt{“True”}$.

Keywords: Static arbitrage-free surfaces, Entropic optimal transport (c-EMOT), Proximal gradient methods, Dupire equation, Smolyak approximation, Equity derivatives (SPX-VIX)

Complexity vs Empirical Score

• Math Complexity: 9.2/10
• Empirical Rigor: 8.5/10
• Quadrant: Holy Grail
• Why: The paper employs advanced mathematics including multi-marginal entropic optimal transport, PCA-Smolyak approximation, and functional analysis, while providing detailed empirical metrics (KKT residuals, Lipschitz bounds, Dupire certificates) and an end-to-end risk bound.

  flowchart TD
    A["Research Goal:<br/>Construct Arbitrage-Free & Chain-Consistent<br/>SPX-VIX Surfaces"] --> B["Key Methodology:<br/>Unify PCA-Smolyak &<br/>c-EMOT Transport"]
    B --> C["Data/Inputs:<br/>SPX/VIX Market Prices &<br/>Yardstick L2W Distance"]
    C --> D["Computational Process:<br/>Solve Proximal Gradients<br/>w/ Martingale Constraints"]
    D --> E["Key Findings/Outcomes"]
    E --> F["Certificates:<br/>KKT ≤ 4e-2, r_geo ≤ 1.05"]
    E --> G["Stability:<br/>Lipschitz ≤ 1.01 &<br/>Dupire Non-Increase"]
    E --> H["Protocol:<br/>Gate-V2 Decision Passes<br/>(Log-Additive Risk Bound)"]