Rough Volatility with Poisson Jumps

• The paper introduces a hybrid framework combining rough volatility with Poisson-driven jumps to capture volatility clustering and persistent memory.
• It employs stochastic Volterra equations with Hawkes kernel-inspired Poisson measures to model both continuous market behavior and discrete spikes.
• The approach enhances calibration methods to better replicate empirical features like heavy tails, multifractality, and extreme market events.

Rough volatility models with Poisson random measures are a significant extension of the rough volatility paradigm, incorporating jump processes into the stochastic modeling of asset price volatility. The defining characteristic of rough volatility models is the use of stochastic processes with low Hölder regularity—typically, fractional Brownian motion (fBM) with Hurst parameter $H \ll \frac12$—to drive volatility dynamics. The inclusion of Poisson random measures generalizes these models to account for discrete, potentially self-exciting, events (jumps or spikes) that are not explainable by continuous diffusive dynamics alone. These jumps capture microstructural features of order flow and enable the model to replicate observed extremal behavior such as volatility clustering, heavy tails, and volatility “spikes”.

1. Microstructural Foundations and the Role of Poisson Random Measures

The most developed microstructural foundation for rough volatility models driven by Poisson random measures models the arrival of orders (market and limit) via self-exciting point processes, notably Hawkes processes with power law kernels (Horst et al., 21 Dec 2024). In this framework:

• Market orders: Each market order increases the future intensity of market order arrivals by a Hawkes kernel $\phi(t)$, which is typically chosen to have power law decay:

$$\phi(t) = \alpha (1 + t)^{-\alpha - 1}, \quad \alpha \in (1/2, 1).$$

The slow decay of this kernel ensures persistent (long memory) effects and leads to rough dynamics as the scaling limit is taken.

• Limit orders/cancellations: Modeled by a marked Poisson random measure $N_{\text{l}}(ds, dy, dz)$, where each event is assigned a random lifetime $y$. Although these events arrive at higher frequency, their individual impact is temporary but, upon aggregation, produce observable volatility "spikes" and clustered jumps.

In the scaling limit—where both the frequency of events and the orderbook depth scale appropriately—the cumulative impact of these order flows (encoded by the Poisson random measures) yields a non-Markovian, path-dependent stochastic Volterra equation with both continuous (rough) and jump components.

2. Mathematical Formulation and Stochastic Volterra Equations

The limiting variance (volatility) process in such models is governed by a stochastic Volterra equation of the form: $$\begin{aligned} V_*(t) =& \ V_*(0) \left( 1 - F^{\alpha,\gamma}(t) \right) + \frac{a}{b} F^{\alpha,\gamma}(t) \ &+ \int_0^t f^{\alpha,\gamma}(t-s) \frac{\zeta_*^\mathrm{m} \sqrt{\lambda_*^\mathrm{m}/b} \sqrt{V_*(s)}}{b} \, dB(s) \ &+ \int_0^t\int_0^\infty\int_0^{V_*(s)} \left( \int_{(t-s-y)^+}^{t-s} \frac{\zeta_*^\mathrm{l}}{b} f^{\alpha,\gamma}(r)\, dr \right) \widetilde{N}(ds,dy,dz) \end{aligned}$$ Here:

• $f^{\alpha,\gamma}(t)$ is the Mittag-Leffler density reflecting memory effects from the Hawkes kernel.
• The last integral involves the compensated Poisson random measure $\widetilde{N},$ which induces jumps/spikes into the volatility process.
• The drift and diffusion coefficients ($a, b, \zeta_*^\mathrm{m}, \lambda_*^\mathrm{m}$) link the microstructural parameters to the macroscopic volatility.

This formulation unifies long memory from order flow clustering (through rough, Volterra-type kernels) and discontinuous shocks from limit order book dynamics.

3. Existence, Uniqueness, and Characterization of Solutions

A central technical challenge is to establish well-posedness for such stochastic Volterra equations driven by Poisson random measures. Results in (Horst et al., 21 Dec 2024) provide:

• Existence and uniqueness of pathwise solutions by casting the problem as a fixed point in a suitable Banach space with norm

$$\| f \|_{L^\infty_{T,\alpha}} = \sup_{0 < t \leq T} t^{1-\alpha} |f(t)|$$

tailored to the scaling properties of the kernel.

• The Laplace functional of the solution is characterized by a nonlinear fractional Volterra–Riccati equation, including a nonlinear operator $\mathcal{V}$ that captures the aggregation of jumps and their path dependency:

$$\psi^\lambda_g(t) = \lambda f^{\alpha,\gamma}(t) + (g * f^{\alpha,\gamma})(t) - \frac12 \left( \frac{\zeta_*^\mathrm{m} \sqrt{\lambda_*^\mathrm{m}/b} }{b}\right)^2 (|\psi^\lambda_g|^2 * f^{\alpha,\gamma})(t) - (\mathcal{V} \circ \psi^\lambda_g)*f^{\alpha,\gamma}(t)$$

This equation underpins the probabilistic properties of the solution and governs option pricing implications.

4. Statistical Properties and Financial Stylized Facts

This class of models reproduces, via endogenous mechanisms, the main empirical features observed in financial data:

• Rough paths: The slow power-law decay in the Hawkes kernel results in a stochastic convolution kernel in the scaling limit, producing volatility trajectories with low Hölder regularity (roughness), quantified by Hurst parameters $H < \frac12$.
• Jumps/spikes: The marked Poisson random measure produces isolated or clustered spikes in volatility, in line with extreme events observed in markets.
• Volatility clustering and long memory: The combined action of continuous and jump drivers ensure persistence and clustering both in absolute returns and in volatility increments.
• Multifractality: The interaction between continuous rough memory and discrete jump mechanisms gives rise to multifractal scaling behavior and non-Gaussian tails in price returns.

5. Methodological Integration with Classical Rough Volatility and Jump Models

Rough volatility models driven purely by fBM (or rough Volterra equations) capture continuous path roughness but do not account for discontinuities. Standard jump-diffusion models with Poisson random measures (e.g., Merton or Bates models) include jumps but lack realistic memory structure. The integration of Poisson random measures into rough volatility models combines both effects:

	Memory/Roughness	Discontinuities/Jumps
Rough (fBM/Volterra)	Yes	No
Jump-Diffusion	No	Yes
Rough+Poisson RM	Yes (Rough, Volterra kernel)	Yes (Poisson-generated spikes)

This hybrid approach is supported theoretically by the convergence of microscopic Hawkes-driven models (with PRMs) to fractional Heston or rough Bergomi variants with jump components (Horst et al., 21 Dec 2024).

6. Calibration, Numerical Schemes, and Application

Simulating and calibrating such models requires:

• Numerical schemes for stochastic Volterra equations with jumps, often via discretization of both the Volterra convolution components and Poisson integrals.
• Estimation procedures that match both the roughness (memory) and jump features to market data, often targeting observed implied volatility surfaces and higher moments.
• Explicit construction of Monte Carlo samplers capable of handling the path-dependence and the jump structures.

Applied studies demonstrate that rough volatility models with Poisson random measures can reproduce both the steepness of short-maturity volatility skews and the fat-tailed distributions of returns, capturing the joint empirical phenomena unattainable by classical models (Horst et al., 21 Dec 2024).

7. Future Directions and Open Problems

Challenges and directions for further research include:

• Extension to multidimensional settings (e.g., joint modeling of SPX and VIX) and multifactor volatility structures blending several rough and jump components.
• Analytical approximation and tractable pricing formulas for broader classes of options beyond European plain vanilla.
• Higher order asymptotic analysis, large deviation principles, and stability of calibrated parameters under time-varying market microstructure.
• Systematic empirical investigation into the joint estimation of memory exponents, jump intensities, and spike clustering, as well as real-time inference on market microstructure from observed volatility spikes.

These models, by unifying microstructure insights with rough path and jump process techniques, offer a robust, well-posed, and increasingly tractable mathematical framework for the quantitative description of financial volatility in the presence of both persistent memory and discrete random shocks.