On the Structural Foundations of Signature Volatility Models: Existence, Arbitrage, Completeness, and the Hedging-Error Decomposition

Battle Count: 73

The paper provides foundational structural theory for signature volatility models, which are an emerging class in quantitative finance for joint SPX-VIX calibration and path-dependent pricing. The completeness depth and hedging-error decomposition have direct implications for derivative hedging strategies and risk management. However, the paper is purely theoretical with no algorithms, no backtests, and no empirical validation. Its relevance is indirect: it supplies the mathematical guarantees under which existing pricing/hedging algorithms (e.g., Abi Jaber-Gérard Fourier methods) are well-posed. For practitioners, the results inform model selection and quantify structural unhedgeable risk.

The theoretical framework involves infinite-dimensional weighted tensor algebras, shuffle Hopf algebra structures, stochastic exponentials on signature paths, and quotient Hilbert-space projections. Direct implementation would require: (1) finite truncation of signature coordinates, (2) computation of shuffle products and Gram matrices, (3) solution of finite Riccati ODE systems, (4) GKW projection via normal equations. The paper explicitly states it does not provide a constructive hedging algorithm. Numerical implementation of the structural results (e.g., computing N*_S for a given model) is non-trivial and model-dependent.

The paper is self-contained with all proofs provided. It relies on standard results from rough path theory, stochastic integration, and quadratic hedging recalled in appendices. No computational experiments or code are needed for verification. The theoretical results are deterministic and verifiable by reading the proofs. However, the infinite-dimensional analysis and weighted tensor algebra constructions require significant mathematical expertise to verify independently.