Rough Hawkes Heston Model

• The Rough Hawkes Heston Model is a stochastic volatility framework that unifies microscopic Hawkes processes with macroscopic rough Heston dynamics.
• It employs an affine Volterra structure for tractable calibration, fast Fourier-based pricing, and efficient simulation of SPX and VIX options.
• The model captures key market phenomena such as leverage effect, rough volatility, and jump clustering by deriving long-memory order flow dynamics.

The Rough Hawkes Heston Model is a stochastic volatility framework unifying microscopic price formation via Hawkes processes with macroscopic rough Heston volatility dynamics. It captures prominent empirical features of financial markets, including leverage effect, rough volatility, jump clustering, and microstructural feedback, by deriving the macroscopic limit of high-frequency, self- and cross-exciting order flow. Distinctive for its microstructural foundations and its tractable affine Volterra structure, the model delivers parsimonious yet flexible joint calibration to SPX and VIX options and admits efficient simulation and derivative pricing techniques.

1. Microstructural Foundations: From Hawkes Processes to Rough Heston Dynamics

The origin of the model lies in bi-dimensional Hawkes-type point-process models of tick-by-tick order flow. In this microscopic setup, buy and sell market orders are modeled as coupled counting processes $(N^{T,+}, N^{T,-})$ with stochastic intensities

$\lambda^T_t = \mu_T \begin{pmatrix} 1\1 \end{pmatrix} + \int_0^t \phi^T(t-s)\, dN^T_s,$

where the kernel matrix $\phi^T(\cdot)$ encodes high-frequency market characteristics: no-arbitrage, bid–ask asymmetry ($\beta > 1$), high endogeneity ($a_T \uparrow 1$ as $T \to \infty$), and metaorders (power-law heavy-tailed kernels). Crucially, when the leading eigen-kernel $\lambda_1(t)=\varphi_1(t)+\beta\varphi_2(t)$ is heavy-tailed, i.e., $\lambda_1(t)\sim t^{-1-\alpha}$ with $\alpha \in (1/2,1)$, the collective dynamics of market orders induce order flow with long memory (Omar et al., 2016).

Upon appropriate time-rescaling, the aggregated price and intensity processes converge in law to the rough Heston stochastic volatility model. The Hurst index of the limiting volatility process is given by $H = \alpha - \frac12 \ll 1/2$, reproducing the empirically observed "roughness" of volatility trajectories found in high-frequency markets (Omar et al., 2016, Horst et al., 2023, Wang et al., 24 Mar 2025).

2. Stochastic Volatility Law: Macroscopic Model Specification

The macroscopic limiting dynamics take the form: $$\begin{aligned} dS_t &= C\,\sqrt{V_t}\,dW^1_t, \ V_t &= V_0 + \int_0^t K(t-s)\,\kappa(\theta - V_s)\,ds + \int_0^t K(t-s) \,\sigma \sqrt{V_s}\,dW^2_s, \end{aligned}$$ where $K(t) = \frac{t^{H-1/2}}{\Gamma(H+1/2)}$ for $H = \alpha - 1/2$, and $W^1, W^2$ are correlated Brownian motions with negative correlation induced by microstructural bid–ask asymmetry ($\rho = (1-\beta)/\sqrt{2(1+\beta^2)}$), generating the leverage effect (Omar et al., 2016). The non-local, power-law Volterra kernel establishes the roughness of volatility trajectories, as volatility exhibits Hölder-regularity strictly less than $1/2$ (Bondi et al., 2022, Wang et al., 24 Mar 2025).

In enhanced formulations, such as the rough Hawkes Heston stochastic volatility model, the spot variance process is itself a rough Hawkes-type intensity process. The variance equation becomes: $V_t = g_0(t) + \int_0^t K(t-s)\,dZ_s,$ with driving semimartingale $$dZ_t = b V_t dt + \sqrt{c} \sigma_t dW_{2,t} + \int z\,\tilde\mu(dt, dz)$$, where the jump intensity couples to $V_t$ itself (Bondi et al., 2022).

3. Affine Volterra Structure and Characteristic Functions

The rough Hawkes Heston model is a member of the affine Volterra class. This structure yields semi-explicit expressions for the joint Fourier–Laplace transforms of $(X_t, V_t)$ via deterministic Riccati–Volterra equations: $$\begin{cases} \psi_1(t) = u + \int_0^t K(t-s) F(\psi_1(s), \psi_2(s)) ds, \ \psi_2(t) = w + \int_0^t K(t-s) G(\psi_1(s), \psi_2(s)) ds, \end{cases}$$ where $F, G$ encode diffusion, mean-reversion, leverage and jump terms (Bondi et al., 2022, Gatheral et al., 2018).

For the classical rough Heston model, the log-price characteristic function is given by: $$\phi(u, t) = \exp\left\{ \theta \lambda \int_0^t h(u, s) ds + V_0 I^{1-\alpha} h(u, t) \right\},$$ with $h(u, t)$ satisfying a fractional Riccati equation: $$D^\alpha h(u, t) = \frac{1}{2}(-u^2 - i u) + \lambda (i u \rho \nu - 1) h(u, t) + \frac{1}{2} (\lambda \nu)^2 h(u, t)^2,$$ where $D^\alpha$ denotes the Riemann–Liouville fractional derivative (Euch et al., 2016, Gatheral et al., 2018).

The affine structure enables fast Fourier-based computation for derivatives pricing under the model, including both vanilla and VIX-linked products (Bondi et al., 2022).

4. Simulation and Numerical Implementation

Efficient numerical schemes for pricing and simulation under the rough Hawkes Heston model utilize multi-factor (sum-of-exponentials) approximations of the Volterra kernel, converting the fractional Riccati–Volterra system to a finite-dimensional system amenable to ODE solvers or quadrature (Bondi et al., 2022). The convergence of such schemes is rigorously controlled: for approximate kernel $K_n$ and corresponding solution $\psi_{i,n}$,

$$\sup_{t \leq T} |\psi_i(t) - \psi_{i,n}(t)| = O\left( \int_0^T |K_n(s) - K(s)| ds \right).$$

Alternative simulation techniques are available via discretizations rooted in microstructure: nearly-unstable, heavy-tailed bivariate INAR($\infty$) processes tightly approximate the rough Heston system, providing efficient and unbiased Monte Carlo paths for both vanilla and exotic derivatives. Empirical efficiency and accuracy for path-dependent options (e.g., Asian, lookback, barrier payoffs) are competitive with or superior to classical Euler–Maruyama methods for the Volterra SDE (Wang et al., 24 Mar 2025).

5. Calibration and Empirical Performance

Joint calibration to cross-asset volatility surfaces (SPX options, VIX options) is achieved via least-squares minimization of the implied volatility error, using the affine transform techniques and the Riccati–Volterra system for fast pricing. Empirically, the model accommodates both the "exploding" at-the-money skew for short-maturity SPX options—governed by the power $\alpha$ in the kernel ($\text{skew} \sim T^{\alpha-3/2}$), matching market data for $\alpha \approx 0.5$—and the shifted, flattened VIX smile, an effect captured by manipulating the roughness parameter and jump clustering properties (Bondi et al., 2022). The best-fit parameters in empirical studies are interpretable and parsimonious (7 parameters in the standard specification), reflecting kernel roughness, mean reversion, volatility-of-volatility, leverage, jump clustering, and initial variance.

6. Extensions: Zumbach Effect and Super-Heston Models

Quadratic and nonlinear extensions of Hawkes-based microstructure further generalize limiting rough volatility models. With quadratic feedback, as in models incorporating the Zumbach effect (feedback from past return trends to future volatility), the limiting process features an explicit quadratic term in volatility dynamics: $$V_t = \mu + H_t + Z_t^2, \quad Z_t = \sqrt{\gamma} \int_0^t k(t-s) \sqrt{V_s}\,dB_s,$$ and, in heavy-tailed near-unstable regimes, this term persists in the convolution kernel, yielding "super-Heston" rough volatility models with fatter volatility tails and persistent feedback (Dandapani et al., 2019).

7. Significance and Theoretical Implications

The Rough Hawkes Heston Model provides a fully rigorous microstructural origin for rough volatility and leverage effect observed in financial markets, thus grounding macroscopic volatility features in high-frequency trading phenomena, order flow endogeneity, and large metaorders. The convergence from microscopic Hawkes (or INAR($\infty$)) processes to rough Heston dynamics is established via functional limit theorems, tightly controlling paths and moments and mapping all macroscopic parameters to microstructural interpretable quantities (Omar et al., 2016, Horst et al., 2023, Wang et al., 24 Mar 2025).

Affine Volterra tractability, robust cross-asset calibration, and efficient numerical schemes position the model as a central framework in contemporary rough volatility modeling. Its architecture admits natural extensions to multiple assets, alternative jump laws, and time-varying or multifactor roughness. Integrating feedback mechanisms, bid-ask effects, and self-exciting jumps yields comprehensive characterization of observed market phenomena, providing both theoretical understanding and practical tools for risk management and derivative valuation.