The Fractional Riccati Equation and Tractable Pricing under Rough Heston

The paper derives the characteristic function of the log-price in the rough Heston model via a limit of nearly-unstable, heavy-tailed Hawkes processes, showing that the classical Heston Riccati ODE is replaced by a fractional Riccati equation. This restores semi-analytic Fourier pricing to a non-Markovian model and, for the desk, supplies a microstructural derivation of roughness from order-flow self-excitation.

The characteristic function solves a fractional Riccati equation of order α = H + 1/2 ∈ (1/2, 1) in place of the classical Heston ODE; the rough Heston variance arises as the macroscopic limit of a Hawkes intensity whose kernel decays as a power law near criticality.

The conceptual bridge — that price-level roughness can emerge from the high-frequency self-exciting structure of trades — is the part we weight most. It links the rough-volatility literature to the Hawkes / reflexivity literature under one generator, rather than leaving them as parallel empirical coincidences.

Lets the desk reason about event windows with a single mental model: elevated endogeneity at the microstructure level and rough macroscopic variance are two views of one state. Near-critical order flow is therefore a prior for unstable implied dynamics.

The fractional Riccati must be solved numerically; calibration is materially heavier than classical Heston, and the Hawkes-to-rough limit is an idealization that abstracts away venue fragmentation.