Signature-Linear Volatility and the Survival of the Riccati Structure

The paper specifies instantaneous volatility as a linear functional of the time-extended signature of a Brownian motion, a class general enough to nest Stein–Stein, Bergomi, Heston, and path-dependent variants. The joint characteristic functional of log-price and integrated variance is obtained whenever an infinite-dimensional tensor-algebra-valued Riccati equation is solvable, restoring Fourier pricing and quadratic hedging to genuinely path-dependent volatility.

σ_t = ⟨ℓ, 𝕊(B)_{0,t}⟩ for a linear functional ℓ on the signature 𝕊; the characteristic functional solves an extended (tensor-algebra-valued) Riccati equation, the infinite-dimensional analogue of the classical Heston ODE.

The result is structural: the affine / Riccati machinery that makes Heston tractable survives the move to signature-linear volatility, so a very large path-dependent class remains Fourier-pricable. For a desk this is the unifying statement that path-dependence and tractability are not in opposition.

Provides a principled vocabulary in which the desk's separate volatility models are special cases of one signature-linear family, and in which path-dependent event exposures can in principle be priced and hedged consistently.

Solvability of the infinite-dimensional Riccati is non-trivial and must be truncated in practice; truncation order trades accuracy against an expanding parameter space.