Rough Stochastic Volatility Models

• Rough stochastic volatility is a financial modeling framework where volatility follows an irregular, rough path driven by fractional Brownian motion (H<1/2).
• The model utilizes Brownian semistationary processes and fractional kernels to capture both micro-scale roughness and long-range memory in market data.
• Advanced numerical methods, including Markovian approximations and FFT-based sampling, are essential for simulating these non-Markovian volatility dynamics.

Rough stochastic volatility refers to a paradigm in financial modeling where the instantaneous volatility process exhibits lower regularity—i.e., “roughness”—than Brownian motion due to being driven by fractional Brownian motion (fBM) or related Gaussian Volterra processes with Hurst parameter $H<1/2$. Empirical studies of financial time series, both high-frequency returns and option-implied volatilities, have repeatedly demonstrated that volatility fluctuations display scaling, autocorrelation decay, and path properties consistent with a rough, rather than classical semimartingale, structure. This shift from Markovian, diffusive stochastic volatility models to pathwise rough models has profound consequences for option pricing, risk management, statistical estimation, and numerical simulation.

1. Mathematical Foundations and Modeling Frameworks

In rough stochastic volatility models, the volatility process $v_t$ (or log-volatility $X_t$) is driven by a noise source with scaling index determined by the Hurst parameter $H<1/2$:

• The prototypical dynamics are given by log-volatility following fBM:

$d\log v_t = \kappa_t dt + \eta dW^H_t,$

where $W^H$ is a fBM with covariance

$$\mathbb{E}[W^H_t W^H_s] = \frac{1}{2}(t^{2H} + s^{2H} - |t-s|^{2H}),$$

and $\eta$ determines the volatility-of-volatility scale (Fukasawa et al., 2019).

A more flexible and general construction, capturing both roughness and long memory or persistence, employs the Brownian semistationary (BSS) process:

$X_t = \int_{-\infty}^t g(t-s) v_s dW_s,$

with $g$ a kernel function such that for short lags $g(x)\sim x^\alpha L_0(x)$, $\alpha\in(-1/2,1/2)$ (the roughness index), and for large lags $g(x)\sim e^{-\lambda x}x^{-\gamma}L_1(x)$, with $\gamma, \lambda\geq 0$, to adjust persistence and possible long memory (Bennedsen et al., 2016). When $\alpha < 0$, paths of $X$ are rougher than Brownian motion, and the model allows one to decouple the micro-scale roughness from the long-term memory or decay of autocorrelations.

The resulting process $v_t = \xi \exp(X_t)$ then inherits both small-scale roughness ($1-\mathrm{Corr}(v_t, v_{t+h})\sim c|h|^{2\alpha+1}$ as $h\to 0$) and long-range properties ($\rho_X(h)$ decays polynomially or exponentially as $|h|\to \infty$ depending on $\gamma,\lambda$). This flexibility is crucial: in empirical studies, estimated $\alpha$ typically lies between $-0.3$ and $-0.4$, indicating highly irregular sample paths for log-volatility, while the decay of autocorrelation may indicate statistical long memory or strong persistence (Bennedsen et al., 2016, Mouti, 2023).

2. Empirical Evidence and Estimation Techniques

Scaling Tests and Estimation:

The roughness of volatility is evidenced by scaling relationships of the form

$$\mathbb{E}\left[\left|\log v_{t+\Delta} - \log v_t\right|^q\right] \sim K_q \Delta^{qH},$$

for a wide range of assets and over several years of data. Using realized volatility, range-based proxies (e.g., Parkinson, Garman–Klass estimators), or option-implied volatilities (corrected for time-to-maturity bias), fitted $H$ values are repeatedly found below $0.1$, sometimes as low as $0.02$, depending on estimation method and asset (Livieri et al., 2017, Fukasawa et al., 2019, Mouti, 2023).

Measurement Noise Correction:

Direct estimation of $H$ is complicated by access only to noisy proxies of latent volatility. Both quasi-likelihood (Whittle) estimators (Fukasawa et al., 2019) and GMM methods (Bolko et al., 2020) explicitly model the measurement error, analytically correct the autocovariance structure, and optimize moment conditions accordingly. Failure to account for this error systematically biases the estimated $H$ downward (the "illusive roughness" phenomenon). Simulation studies and empirical work confirm that bias-corrected methods recover $H$ values consistent with rough paths (i.e., $H \sim 0.05$) across major equity indices.

Filtering:

State-space approaches to inference with rough volatility face the challenge of the non-Markovian, non-semimartingale properties of fBM. Approximation of fBM as a superposition of Markovian Ornstein–Uhlenbeck (OU) processes via the Mandelbrot–Van Ness representation enables the use of particle filtering and parameter learning for rough volatility models (Damian et al., 2023).

3. Option Pricing, Skew, and Asymptotics

Implied Volatility Surface:

Rough volatility models naturally reproduce the empirically observed power-law blow-up in the short-maturity-at-the-money skew of the implied volatility surface:

$$|\partial_k \sigma_{\mathrm{iv}}(k, T)| \sim T^{H-1/2}, \quad T \to 0,$$

where $\sigma_{\mathrm{iv}}$ is the Black–Scholes implied volatility and $k$ is log-moneyness (Bayer et al., 2017, Bayer et al., 2017, Bayer et al., 2020). For $H$ close to zero, this predicts an extreme short-term skew, in contrast to the flat behavior in classical (Markovian) models (Livieri et al., 2017, Mouti, 2023).

High-order moderate deviation expansions, via sharp large deviation principles and Laplace/Wiener–Azencott techniques, yield explicit asymptotic formulas of the form:

$$-\log c(k_t, t) = \frac{I''(0)k^2}{2}t^{-2H + 2\beta} + \frac{I'''(0)k^3}{6}t^{-2H+3\beta} + \dots,$$

for strikes $k_t = k t^{1/2 - H + \beta}$ with $0 < \beta < H$ (Bayer et al., 2017).

Rough Term Structure Exponent:

The characteristic term structure exponent $\theta(H)$ controls the leading order in the implied volatility expansion. In fast mean reverting rough volatility,

• For $H<1/2$, the effective option price correction is independent of $H$ (matching the Markov case), but in slow-mean-reverting or strong fluctuation regimes, $\theta(H) = H+1/2$ governs implied volatility term-structure decay (Garnier et al., 2017).

Affine and Extended Models:

The rough Hawkes Heston model (Bondi et al., 2022) and log-modulated rough models (Bayer et al., 2020) further account for jump clustering and allow “super-rough” ($H \rightarrow 0$) behavior, respectively, while preserving analytic tractability (via Riccati–Volterra equations and explicit expansions).

4. Numerical Methods and Markovian Approximations

Simulation Challenges:

Rough volatility models are non-Markovian and non-semimartingale, precluding standard simulation schemes. Multiple approaches have emerged:

• Cholesky and hybrid FFT-based sampling for fBM increments, balancing exactness with computational cost (Matas et al., 2021).
• Multi-factor (Markovian) approximations: Fractional kernels $K(t)\sim t^{H-1/2}$ are approximated by finite sums of exponentials, so that the non-Markovian model is embedded in an augmented Markovian state space, enabling efficient simulation and calibration (Jaber et al., 2018, Bayer et al., 2021). For instance, in the rough Heston model the fractional Riccati equation governing the characteristic function is replaced by an $n$-dimensional system of ODEs, with super-polynomial convergence in the mode number $n$.
• Regularity structures and RPDEs: Recent advances adapt Hairer's regularity structures to rigorously define and numerically compute options under rough volatility and local volatility models, using rough-path-lifted Feynman–Kac representations and finite-difference schemes for rough partial differential equations (Bayer et al., 2017, Bank et al., 2023).

Variance Reduction and Pricing:

Variance reduction techniques such as "turbocharging," control-variate mixing, and regression-based least squares Monte Carlo adapted to infinite-dimensional Markovian representations (e.g., for VIX option pricing) allow high-precision pricing despite the rough path features (Guerreiro et al., 2021, Matas et al., 2021).

5. Practical Applications and Forecasting Performance

Empirical Outperformance:

Forecasting experiments using realized volatility measures and a large panel of asset data show that models that capture both roughness and long memory—especially BSS-based and Cauchy models—consistently outperform standard benchmarks such as random walk, ARMA, HAR, ARFIMA, and even classical stochastic volatility and RFSV models (Bennedsen et al., 2016, Mouti, 2023). Out-of-sample quasi-likelihood losses, as well as other statistical performance measures, confirm the predictive advantage due to proper modeling of both micro-scale roughness (e.g., with $\alpha\approx -0.3$ or $H\ll 1/2$) and macro-scale persistence or long memory.

Interest Rate Markets and Rough Term Structures:

Rough stochastic volatility has been incorporated into interest-rate modeling, such as in the rough SABR Forward Market Model (Adachi et al., 30 Sep 2025), which extends the classical SABR and FMM via rough volatility drivers. Short-maturity swaption implied volatility expansions reflect the power-law structure in the term structure and persistent skew, with roughness-induced correction terms of order $t^{H-1/2}$, and rigorous justification of common industry “freezing approximations.”

6. Open Problems, Limitations, and Future Directions

Estimation and Model Selection:

Significant open challenges include the development of formal statistical tests for distinguishing roughness and hypotheses $H<1/2$, as well as efficiently handling model selection when volatility is unobservable and measurement error or microstructure noise contaminate proxies (Fukasawa et al., 2019, Bolko et al., 2020).

Theory and Numerics:

While the strong convergence rate of numerical schemes for rough models is well understood (order $H$), weak error estimates exhibit phase transitions depending on $H$ and the correlation between price and volatility drivers (rate $n^{-(3H+1/2)}$ below $H=1/6$, saturating at $n^{-1}$ for larger $H$) (Friz et al., 2022). Further research into efficient and robust simulation, as well as adaptive regularization and renormalization methods, remains ongoing (Bayer et al., 2017, Bank et al., 2023).

Market Microstructure and Mechanisms:

Microstructure invariance and the relationship to Hawkes-type self-exciting jumps, as in the rough Hawkes Heston model, provide promising directions for joint modeling of price and volatility events and for reconciling spot and derivative market data features (Bondi et al., 2022). The debate over the origin of roughness—intrinsic to the volatility process or artifact of microstructure noise—appears resolved in favor of intrinsic roughness when using robust range-based proxies and cross-market observations (Mouti, 2023).

Risk Management and Hedging:

Integration of rough stochastic volatility into pricing, hedging, and dynamic risk management procedures—especially for path-dependent and pathwise-sensitive derivatives—remains an active area, with robust numerical methods (e.g., those based on Markovianization, regularity structures, and particle filtering) being essential for industrial applicability.

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Summary Table: Empirical Inference of Roughness

Estimation Method	Typical $H$ Found	Data Source
Realized volatility scaling	$H \sim 0.02-0.1$	Intraday returns (Fukasawa et al., 2019, Mouti, 2023)
Option-implied volatilities	$H \sim 0.3$ (biased high)	SPX options (Livieri et al., 2017)
Range-based estimators (e.g., GK, Parkinson)	$H < 0.1$	Daily OHLC prices (Mouti, 2023)
GMM/Whittle (bias-corrected)	$H \sim 0.024-0.05$	High-freq index data (Bolko et al., 2020)

The table highlights that, across independent estimation frameworks and data sources, volatility exhibits intrinsic roughness with low Hurst exponents—implying sample paths orders of magnitude less regular than Brownian motion. This property fundamentally underpins the success of rough stochastic volatility models in both empirical explanation and financial engineering.