Estimating the roughness exponent of stochastic volatility from discrete observations of the integrated variance

ArXiv ID: 2307.02582 “View on arXiv”

Authors: Unknown

Abstract

We consider the problem of estimating the roughness of the volatility process in a stochastic volatility model that arises as a nonlinear function of fractional Brownian motion with drift. To this end, we introduce a new estimator that measures the so-called roughness exponent of a continuous trajectory, based on discrete observations of its antiderivative. The estimator has a very simple form and can be computed with great efficiency on large data sets. It is not derived from distributional assumptions but from strictly pathwise considerations. We provide conditions on the underlying trajectory under which our estimator converges in a strictly pathwise sense. Then we verify that these conditions are satisfied by almost every sample path of fractional Brownian motion (with drift). As a consequence, we obtain strong consistency theorems in the context of a large class of rough volatility models, such as the rough fractional volatility model and the rough Bergomi model. We also demonstrate that our estimator is robust with respect to proxy errors between the integrated and realized variance, and that it can be applied to estimate the roughness exponent directly from the price trajectory. Numerical simulations show that our estimation procedure performs well after passing to a scale-invariant modification of our estimator.

Keywords: Rough Volatility, Fractional Brownian Motion, Roughness Exponent, Stochastic Volatility, Time Series Estimation

Complexity vs Empirical Score

• Math Complexity: 8.5/10
• Empirical Rigor: 5.0/10
• Quadrant: Holy Grail
• Why: The paper employs advanced stochastic analysis and fractional calculus to derive strong consistency results for a pathwise estimator, indicating high math complexity. It demonstrates robustness to proxy errors and includes a simulation study, showing significant empirical validation, though it lacks backtest-ready implementation details.

  flowchart TD
    Start["Research Goal<br/>Estimate roughness exponent h<br/>from discrete observations of<br/>integrated variance (IV)"] --> Method["Methodology<br/>New pathwise estimator<br/>simple & efficient form"]
    Method --> Data["Data / Inputs<br/>Discrete observations of IV<br/>∫₀ᵗ σ²_u du (proxy for realized variance)"]
    Data --> Comp["Computational Process<br/>1. Compute log-IV increments<br/>2. Scale-invariant modification<br/>3. Linear regression on log-lags<br/>4. Estimate slope → -h"]
    Comp --> Outcomes["Key Findings / Outcomes<br/>• Strict pathwise convergence<br/>• Strong consistency for fBM/vol models<br/>• Robust to proxy errors<br/>• Applicable to price data<br/>• Numerical simulations validate"]