Is Volatility Rough ?

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Is Volatility Rough ?

Masaaki Fukasawa
Tetsuya Takabatake
Rebecca Westphal
Published on 5/13/2019
Equities
Stocks
Volatility effect
Machine learning

This paper critically examines the 'rough volatility' hypothesis, which claims that log-volatility follows a fractional Brownian motion with Hurst parameter H<0.5. The authors first demonstrate via simulations that the popular scaling method of Gatheral et al. (2014) systematically produces H≈0.1 even when the true volatility is not rough (e.g., an Ornstein-Uhlenbeck process), because the method ignores the estimation error in realized volatility proxies. To address this, they develop a novel quasi-likelihood estimator for the parameters (H, η) of a fractional stochastic volatility model. The estimator incorporates the measurement error of realized variance via a central limit theorem approximation (i.i.d. normal with variance 2/m), and uses a Whittle-type likelihood for the approximate Gaussian increments of log-volatility. The authors prove consistency under high-frequency asymptotics where both the sampling interval δ→0 and the number of intraday observations m→∞, with conditions ensuring that the measurement error vanishes sufficiently fast. Simulation studies confirm good finite-sample performance. Applying the estimator to daily realized volatility data from the Oxford-Man Realized Library for major stock indices (e.g., S&P 500) yields H estimates around 0.08–0.09, even smaller than the previously reported 0.1, thus providing stronger evidence for the rough volatility hypothesis. The paper highlights the importance of accounting for measurement error in volatility proxy data when estimating the smoothness of latent volatility.

Highlights

  • 1Shows that the scaling method in Gatheral et al. (2014) tends to suggest roughness (H≈0.1) even when volatility is not rough, due to estimation error in latent volatility.
  • 2Proposes a novel quasi-likelihood estimator for the Hurst parameter H and volatility-of-volatility η in a fractional stochastic volatility model, accounting for realized volatility measurement error.
  • 3Proves consistency of the estimator under high-frequency asymptotics with double limits (δ→0, m→∞) and conditions on sampling frequency and number of intraday observations.
  • 4Applies the estimator to major stock indices and finds H even smaller than 0.1, supporting the 'rough volatility' hypothesis.

Methods

  • M
    Quasi-likelihood estimation combining normal approximation of log-realized volatility error (from central limit theorem), local Gaussian approximation of log-realized variance increments, and Whittle-type approximation for self-similar Gaussian processes.
  • M
    Reparametrization (ν = η δ^H) to handle singularity in the Fisher information matrix under high-frequency asymptotics.
  • M
    Simulation study using fractional Ornstein-Uhlenbeck processes to assess finite-sample performance of the estimator.

Results

  • R
    The scaling method of Gatheral et al. (2014) yields H≈0.1 for simulated data from a non-rough volatility model (e.g., Ornstein-Uhlenbeck) when using 5-minute realized volatility, indicating the method is biased by measurement error.
  • R
    The proposed adapted Whittle estimator is consistent under conditions (H.1)-(H.3) and performs well in finite samples for various H values.
  • R
    Empirical estimates for S&P 500 and other indices yield H around 0.08–0.09, even rougher than previously reported H≈0.1.
  • R
    The estimator successfully separates the effects of volatility roughness and measurement error, unlike the scaling method.
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