Arbitrage-Free Option Price Surfaces via Chebyshev Tensor Bases and a Hamiltonian Fog Post-Fit

Paper: arXiv 2512.01967

Authors: Robert Jenkinson Alvarez

Abstract

We study the construction of arbitrage-free option price surfaces from noisy bid-ask quotes across strike and maturity. Our starting point is a Chebyshev representation of the call price surface on a warped log-moneyness/maturity rectangle, together with linear sampling and no-arbitrage operators acting on a collocation grid. Static no-arbitrage requirements are enforced as linear inequalities, while the surface is fitted directly to prices via a coverage-seeking quadratic objective that trades off squared band misfit against spectral and transport-inspired regularisation of the Chebyshev coefficients. This yields a strictly convex quadratic program in the modal coefficients, solvable at practical scales with off-the-shelf solvers (OSQP). On top of the global backbone, we introduce a local post-fit layer based on a discrete fog of risk-neutral densities on a three-dimensional lattice (m,t,u) and an associated Hamiltonian-type energy. On each patch of the (m,t) plane, the fog variables are coupled to a nodal price field obtained from the baseline surface, yielding a joint convex optimisation problem that reweights noisy quotes and applies noise-aware local corrections while preserving global static no-arbitrage and locality. The method is designed such that for equity options panels, the combined procedure achieves high inside-spread coverage in stable regimes (in calm years, 98-99% of quotes are priced inside the bid-ask intervals) and low rates of static no-arbitrage violations (below 1%). In stressed periods, the fog layer provides a mechanism for controlled leakage outside the band: when local quotes are mutually inconsistent or unusually noisy, the optimiser allocates fog mass outside the bid-ask tube and justifies small out-of-band deviations of the post-fit surface, while preserving a globally arbitrage-free and well-regularised description of the option surface.

Complexity vs Empirical Score

• Math Complexity: 8.5/10
• Empirical Rigor: 6.0/10
• Quadrant: Holy Grail — high math complexity, high empirical rigor

Why this score: The paper introduces a highly novel mathematical framework combining warped Chebyshev tensor bases and a local 3D Hamiltonian fog layer to fit arbitrage-free option surfaces via convex QPs. While the mathematical derivation and optimization setup are extensive and mathematically rigorous, empirical validation is presented largely as aggregated summary metrics rather than fully documented backtests with attached open code.

Research Flowchart

  flowchart TD
    A[Goal: Construct Arbitrage-Free Option Price Surfaces from Noisy Quotes] --> B[Input Data: Option Bid-Ask Quotes across Strike & Maturity]
    
    B --> C[Global Fit: Chebyshev Surface & Linear No-Arbitrage Inequalities]
    
    C --> D[Global Optimization: Solve Convex QP via OSQP with Coverage & Regularization]
    
    D --> E[Local Post-Fit Layer: Discrete Fog of Risk-Neutral Densities & Hamiltonian Energy]
    
    E --> F[Outcomes: High Inside-Spread Coverage 98-99% & Ultra-Low Arbitrage Violations <1%]
    E --> G[Outcomes: Robust Controlled Leakage During Stressed Market Regimes]

Cited References (26)

1. Arbitrage-Free SVI Volatility Surfaces (2012)
2. Arbitrage-free smoothing of the implied volatility surface (2009)
3. Volatility Interpolation (2011)
4. Prices of State-Contingent Claims Implicit in Option Prices (1978)
5. Implied Binomial Trees (1994)
6. Pricing with a Smile (1994)
7. Riding on a Smile (1994)
8. Calibrating Volatility Surfaces Via Relative-Entropy Minimization (1997)

0 of 26 cited works are in the corpus; resolved via arXiv source bundles, Semantic Scholar and OpenAlex. Corpus-wide: 2518 linked papers.