The Boundary of Hedgeability: Pricing and Hedging in Volterra-Lévy Markets

Abstract

We develop a pricing and hedging framework for markets driven by Volterra-Lévy noise. The operator-energy layer is not pathwise: the derivative is the Banach/Hilbert adjoint D = δ * , defined by duality with stochastic integration rather than by differentiating sample paths. The terminal calculus applies to Volterra functionals V T = T 0 K(T, s) dL s , including fractional kernels. The only semimartingale input used below is the classical Lévy-Itô formula for the frozen-terminal process V T (t) = t 0 K(T, s) dL s ; this does not impose semimartingale structure on the Volterra path t → V t. Scenario-map hedging results for stochastic volatility require either regular kernels or an explicit diagonal regularization when jumps are present. This separation is essential: singular rough kernels and jump shocks are not automatically compatible at the path level. The operator derivative D = δ * , defined by duality with the stochastic integral on a combined Hilbert-Banach energy space, gives a single bookkeeping device for the continuous and jump components. For terminal Volterra functionals, the compensated Poisson integrand splits into a z-linear singular part and a ν-regular Taylor remainder. The latter is the kernel-weighted Leibniz defect. Risk-neutral pricing leads to a two-parameter Volterra risk-neutral constraint (VRNC): the effective drift rate Ψ t =-σ L θ W t + z e θ J t (z)-1 ν(dz).

Keywords: 60H07, 60H05, 60G51, 60G22, 91G20 Volterra-Lévy processes, operator derivative, risk-neutral pricing, rough volatility, hedgeability, Leibniz defect, bracket invariance, calibration invariance, Volterra–Lévy processes, jump diffusion, stochastic volatility, variance swap, Malliavin calculus, fractional Brownian motion, incomplete markets, Lean 4, formal verification