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

Abstract: We establish four structural results for signature volatility models. First, we prove global existence and uniqueness of strong solutions to the signature SDE $dS_t = S_t \langle \ell, \widehat{W}t \rangle \, dB_t$ on the weighted tensor algebra $T_w$, identifying the admissibility class through a summability condition H1 and an exponential-integrability condition H3 for the square-integrable stochastic-exponential construction. Second, we establish the asset-pricing part on the natural filtration of the prolonged signature and separate it from transform non-explosion: H3 makes the reference-measure stochastic exponential a true martingale, hence yields NFLVR, while global solvability of the associated infinite-dimensional Riccati equation is the additional condition equivalent to absence of explosion for finite signature transforms. Third, we characterise market completeness on the price filtration via the density of the truncated signature span $\mathrm{span}{\langle e_I, \widehat{W}_T \rangle : |I| \leq N}$ inside $L^2(\mathcal{F}^S_T, \mathbb{Q})$, and identify the minimal such $N$, the price-filtration completeness depth. Fourth, we derive the hedging-error decomposition $X = \mathbb{E}\mathbb{Q}[X] + \int_0^T H_s \, dS_s + \varepsilon_T$ for square-integrable payoffs, with residual expanded through the Gram projection of signature components beyond the completeness depth and bounded by a model-dependent projection error. The four results are tied by an architectural identity: the admissible weighted tensor algebra on which the stochastic exponential is a true martingale and finite signature transforms do not explode is the natural valuation cell of a signature SDE. The proofs are self-contained except for standard results from rough path theory, stochastic integration, and quadratic hedging, recalled in the appendices.