Quantum Heston Stochastic Volatility: A Semiclassical Toeplitz–GKSL Framework

We construct a bounded Quantum Heston framework for Heston-type stochastic volatility using coherent-state Toeplitz quantization and GKSL dynamics. The construction realizes pricing through a positive noncommutative semigroup whose leading Husimi shadow is a truncated classical phase-space generator. On compact subsets of the positive-variance region, where the truncation is inactive, this shadow recovers a jump–Heston–CIR financial generator with risk-neutral normalization. Nonnegative variance and jump-intensity symbols are mapped to positive bounded operators, and ordered Weyl–Lindblad translation channels recover the drift, diffusion, and finite-jump components in the semiclassical limit. The small-jump limit gives the diffusive Heston generator. Pricing is defined by a Toeplitz–GKSL trace functional; affine jump–Heston transforms and Fourier formulas serve as comparison formulas for the semiclassical layer. The first θ -dependent correction is tied to Toeplitz normalization, Lindblad ordering, variance truncation, smooth splitting, and risk-neutral drift adjustment. We interpret θ as a market-resolution scale for joint localization of log-forward price and normalized variance, while the fitted Quantum Heston coefficient records the signed orientation of the selected first-order correction. The empirical section reports SPX and NDX residual analysis with R , δ -sensitivity summaries. Under additional tilted-spectral stability assumptions, the framework gives conditional long-maturity tail-rate comparisons.