The Fractional Riccati Equation and Tractable Pricing under Rough Heston

El Euch and Rosenbaum establish the theoretical bridge between two independently motivated strands of the market microstructure and derivatives literature: the observation that near-critical Hawkes processes with power-law kernels generate rough variance paths in the macroscopic limit, and the need for a tractable characteristic-function formula for option pricing under non-Markovian rough volatility. The paper derives the rough Heston model as the limit in distribution of a rescaled sequence of nearly-critical Hawkes processes — where criticality means that the branching ratio approaches one and the kernel is power-law with exponent α ∈ (1/2, 1) — and shows that the macroscopic variance process V_t satisfies a fractional stochastic Volterra integral equation driven by a correlated Brownian motion. This micro-to-macro derivation is significant because it provides a mechanistic justification for rough volatility: the empirically observed roughness of index variance is a consequence of the near-critical, self-exciting nature of high-frequency order flow, and H = α − 1/2 ∈ (0, 1/2) is determined by the tail exponent of the microscopic excitation kernel. The characteristic function E_Q[e^{iuX_T}], where X_T = log(S_T/S_0) is the log-price at maturity T, is shown to admit a semi-closed-form expression analogous to the classical Heston formula. In the classical Heston model, the characteristic function is exp(g(T)u + ∫_0^T h(T−s)V_s ds) where h solves a linear Riccati ODE; in the rough Heston extension, the same exponential-affine structure is preserved but h now solves a fractional Riccati equation D^α h = F(h) where D^α is the Caputo fractional derivative of order α = H + 1/2. This is the central analytical result: the Volterra-affine structure of the variance process is rigid enough to generate a characteristic function that can be evaluated by numerical integration of the fractional ODE, enabling fast Fourier inversion for European option prices. The Volterra structure of the variance process deserves emphasis: V_t is not a semimartingale and does not have a standard Itô decomposition, yet the conditional characteristic function of X_T given the path of V retains the exponential-affine form in the running variance integral. This preservation of quasi-affine structure under fractional extension is the technical achievement that makes rough Heston analytically tractable at all, and it is what distinguishes it from more general rough volatility models such as the Bergomi model, where no such closed form exists.

The rough Heston variance process is defined via a Volterra integral equation: V_t = V_0 + ∫_0^t K(t−s)[λ(V_s) ds + ν√V_s dW_s] where K(t) = t^{H−1/2} / Γ(H+1/2) is the Riemann–Liouville kernel with H ∈ (0, 1/2), λ(v) = κ(θ − v) is the mean-reversion drift familiar from classical Heston, ν is the vol-of-vol parameter, and W is a Brownian motion with correlation ρ to the price process. The characteristic function takes the exponential-affine form E_Q[e^{iuX_T}] = exp(g(T, u) + ∫_0^T h(T−s, u) V_s ds) where the function h(·, u) solves the fractional Riccati equation (D^α h)(t, u) = F(h(t, u), u) with F(h, u) = −u²/2 − iuρν h + (κ − iuρν) h/κ − ν²h²/2, and D^α denotes the Caputo fractional derivative of order α = H + 1/2 ∈ (1/2, 1). The function g is determined from h by integration. The fractional Riccati equation reduces to the classical Heston Riccati ODE in the limit H → 1/2 (α → 1), providing a smooth interpolation. For the Hawkes-to-rough derivation, the microscopic branching kernel φ(τ) ∝ τ^{−α} is rescaled under a suitable time-space limit to yield the Riemann–Liouville kernel K at the macroscopic level.

The Hawkes-to-rough Heston limit is a rare example of a rigorous micro-to-macro derivation in the financial mathematics literature, and its methodological significance extends well beyond the specific model. The argument proceeds by showing that, under a nearly-critical parameterization of the Hawkes kernel — branching ratio n_ε = 1 − ε, kernel tail exponent fixed at α ∈ (1/2, 1) — the rescaled variance process (1/ε) times the Hawkes intensity converges weakly to the rough Heston variance as ε → 0. This is a functional central limit theorem for Hawkes processes, and it establishes that the macroscopic Hurst index H = α − 1/2 is directly determined by the microscopic tail exponent of order-flow self-excitation. The implication is that roughness is not a phenomenological postulate but a derived consequence of near-criticality, which is itself empirically documented in the Hardiman–Bouchaud–Bercot branching ratio analysis. The stability of affine/Riccati structure under the fractional extension is the key technical finding. Classical affine models retain their exponential-affine characteristic functions because the logarithm of the characteristic function satisfies an ODE whose nonlinearity is at most quadratic in the log-characteristic function; rough Heston retains this quadratic nonlinearity while replacing the integer-order ODE with a fractional ODE of order α. The fractional Riccati equation has to be solved numerically, and its computational cost scales with the history length because the Caputo derivative is a convolution operator, but the cost is manageable with Adams-type ODE solvers or spectral methods. Comparison with the Bergomi model at the calibrated H ≈ 0.1 confirms that rough Heston reproduces the observed term structure of ATM skew more accurately than classical Heston, with the skew decaying as T^{H−1/2} rather than the classical T^{−1/2} rate. The Euler discretization of Volterra processes introduces a weak error of order Δt^H when using standard schemes, which is significantly worse than the Brownian case (order Δt) for small H, and this has practical consequences for any Monte Carlo pricing or hedging simulation under the rough Heston model.

The paper provides the conceptual glue linking the desk's reading across microstructure and derivatives: the Hawkes-to-rough limit establishes that the near-critical branching processes documented in equity order flow (Hardiman et al.) and the rough volatility surfaces documented in options markets (Gatheral et al.) are the same phenomenon viewed at different scales, connected by the Hurst index H. This unified mental model has direct consequences for how the desk should cross-reference its microstructure regime signals with vol surface signals: when the branching ratio estimate n is near unity at the microstructure level, the rough Heston prior assigns higher weight to macro-scale roughness, and conditioning procedures that use H-dependent path statistics should be calibrated with an H derived from the same power-law tail exponent as the estimated Hawkes kernel. The practical implementation of rough Heston as a pricing model for the desk's exotic book requires solving the fractional Riccati equation for each (T, u) pair in the Fourier inversion grid, and the computational overhead relative to classical Heston is a factor of roughly ten to fifty depending on precision targets and the number of maturities. For the desk's purposes the model is most useful not as a live pricing engine but as a calibration benchmark against which surface features are interpreted: a surface that deviates from rough Heston predictions at given H, κ, ν is informative about regime change or about flow-driven dislocations that the model does not capture.

The Hawkes-to-rough limit is an asymptotic result that operates in a specific mathematical limit — nearly-critical kernel, power-law tail, infinite time horizon rescaling — and the quality of the approximation at finite observation frequencies and realistic branching ratios is not analytically controlled. In practice, venue fragmentation, discrete tick sizes, and the finite support of any empirical excitation kernel all introduce deviations from the Volterra-process limit that the theoretical derivation abstracts away. The convergence rate in the functional CLT is not explicitly characterized, and the practical accuracy of the rough Heston model as a description of market dynamics at the intraday scale is therefore a matter of empirical validation rather than theoretical guarantee. The numerical solution of the fractional Riccati equation introduces its own approximation errors, and the standard Adams–Bashforth discretization of the Caputo derivative on a uniform grid requires O(N²) operations for N time steps due to the convolution structure, making high-precision solutions expensive. The Euler discretization of the Volterra process for Monte Carlo simulation has weak error of order Δt^H, which for H = 0.1 means that achieving accuracy comparable to the Brownian case requires time steps roughly ten times finer, substantially increasing simulation cost for any hedge ratio or greeks computation that relies on Monte Carlo.