Rough Volatility

The defining parameter of rough volatility. While classical models assume a smooth diffusion with H=0.5, empirical measurements of the Hurst exponent for volatility consistently fall near H ≈ 0.1. This indicates anti-persistence or mean reversion in volatility paths, where upward movements are statistically more likely to be followed by downward corrections. This roughness is not noise but a structural feature of the market's microstructural dynamics.

A foundational model where the log-volatility is driven by a fractional Brownian motion (fBm). The key mechanism is the Mandelbrot-van Ness representation, which introduces long memory into the driving noise. This process provides the mathematical backbone for the rough Bergomi model, allowing the volatility surface to be captured with very few parameters by linking the roughness of the spot variance to the steepness of the at-the-money volatility skew.

A critical empirical phenomenon explained by rough volatility. Traditional stochastic volatility models predict a flat or parabolic at-the-money (ATM) skew for short maturities. In reality, the ATM skew explodes as time to expiry approaches zero, behaving like a power law. Rough volatility models naturally reproduce this power-law decay of the skew, providing a superior fit for short-dated options without requiring extreme parameter values or jump components.

The roughness is not an arbitrary mathematical construct but emerges from market microstructure. Endogenous liquidity dynamics and the Hawkes process nature of order flow create a self-exciting environment where volatility reacts quickly and reverts sharply. This links rough volatility directly to the behavior of high-frequency traders and the limit order book, grounding the model in the actual mechanics of price formation rather than abstract statistical assumptions.

Modern implementations often combine rough volatility kernels with deep learning. Neural Stochastic Differential Equations (Neural SDEs) can parameterize the drift and diffusion of the variance process while maintaining the fractional kernel. This allows for signature-based calibration where the path signature of the price process is used to efficiently estimate the Hurst parameter and other latent variables, bypassing computationally expensive maximum likelihood estimation.

Rough volatility fundamentally alters hedging strategies. The delta hedge in a rough volatility environment requires adjustments for the anticipated rapid mean reversion of volatility. Standard Black-Scholes hedging leaves residual risk that can only be captured by a rough P&L expansion. This has led to the development of rough local volatility and rough Heston models that maintain analytical tractability for pricing while respecting the observed roughness of the underlying time series.

The fundamental driving noise behind rough volatility models. Unlike standard Brownian motion, fBm exhibits long-range dependence and self-similarity governed by the Hurst exponent (H). In rough volatility, H is typically less than 0.5 (often around 0.1), indicating anti-persistent behavior where increments are negatively correlated. This generates paths that are significantly rougher and more jagged than those of a standard Wiener process, accurately mimicking the high-frequency fluctuations observed in the VIX and at-the-money implied volatility.

A statistical measure used to classify the roughness or memory of a time series. For rough volatility calibration, the exponent H is estimated from high-frequency asset data using methods such as:

• Rescaled Range (R/S) Analysis: Measures the range of partial sums of deviations.
• Detrended Fluctuation Analysis (DFA): Removes local trends to avoid spurious detection of long-range correlation.
• Generalized Method of Moments (GMM): Fits the scaling properties of log-volatility increments. Empirical estimates for volatility consistently yield H ≈ 0.1, rejecting standard diffusion models.

A non-Markovian stochastic volatility model that explicitly parameterizes the roughness of volatility. It models the forward variance curve driven by a Riemann-Liouville fractional Brownian motion. Key properties include:

• Explosion of the ATM skew: Reproduces the steep at-the-money implied volatility skew observed in short-dated equity options.
• Power-law term structure: The term structure of the skew decays as T^(H-1/2), a stylized fact impossible to capture with classical Markovian models.
• Simulation: Requires hybrid schemes combining Cholesky decomposition with exact conditional Monte Carlo for the fractional kernel.

A critical challenge in estimating rough volatility from tick data. Observed prices are contaminated by market microstructure noise (bid-ask bounce, discrete price grids), which artificially roughens the path. To isolate the true latent volatility roughness, estimators like the pre-averaging method are used. This technique constructs local averages of returns over short blocks to filter out high-frequency noise before computing realized variance. Without this correction, the Hurst exponent is biased downward, leading to spurious detection of roughness.

A class of point process models that bridge market microstructure and rough volatility. These are self-exciting processes where the intensity of price jumps is driven by a convolution of past events with a power-law kernel. In the scaling limit, Hawkes processes with a slowly decaying kernel converge to rough volatility dynamics. This provides a microstructural foundation for roughness, linking it directly to the clustering of order flow and the long memory of market activity rather than assuming an exogenous mathematical construct.

The primary market observable that validates rough volatility. The at-the-money (ATM) skew is the derivative of implied volatility with respect to strike at the forward price. In rough models, the ATM skew explodes as time-to-expiry T → 0 following a power law T^(H-1/2). This contrasts sharply with classical stochastic volatility models (Heston, SABR) where the skew remains bounded. The steepness of the short-end skew in equity index options provides direct empirical evidence for H significantly less than 0.5.