Rough volatility: evidence from option prices
Rough volatility: evidence from option prices
This paper provides evidence that volatility is rough using option prices rather than historical high-frequency data. The authors analyze at-the-money implied volatilities of S&P500 options with short maturities (one month) and find that the log-volatility process exhibits scaling properties consistent with a fractional Brownian motion with Hurst parameter around 0.32, confirming earlier results from historical data. They also apply the Medvedev-Scaillet correction to extract spot volatility from implied volatilities across maturities, obtaining similar Hurst parameters (0.33–0.34). The estimated Hurst values are slightly larger than those from historical data (typically ~0.1–0.2), which is explained by a smoothing effect due to the remaining time to maturity. Through Monte Carlo simulations and analytical derivations, the authors show that using implied volatilities as proxies for spot volatility introduces an upward bias in Hurst parameter estimation, with the bias increasing with option maturity. The study demonstrates that rough volatility is a robust feature of financial markets, observable even through option-implied measures.
Highlights
- 1Confirms rough volatility using option-implied volatility proxies, not just historical data.
- 2Finds Hurst parameter around 0.32–0.34 from S&P500 at-the-money options, slightly higher than historical estimates.
- 3Demonstrates upward bias in Hurst parameter estimation due to remaining time to maturity of options.
- 4Uses Medvedev-Scaillet correction to derive spot volatility from implied volatility across maturities.
- 5Provides Monte Carlo and analytical evidence for smoothing effect of implied volatility on roughness.
Methods
- MScaling analysis of log-volatility increments via structure function m(q, Δ).
- MMedvedev-Scaillet approximation formula to extract spot volatility from at-the-money implied volatility.
- MMonte Carlo simulation of rough volatility model (fractional Brownian motion) to study bias.
- MAnalytical derivation of second moment of implied variance differences to illustrate smoothing.
Results
- RLog-volatility increments from implied volatilities exhibit monofractal scaling with Hurst parameter ≈ 0.32.
- RMedvedev-Scaillet corrected proxies yield similar Hurst parameters (0.33–0.34), confirming roughness.
- RImplied volatility-based Hurst estimates are upward biased compared to true spot volatility (e.g., 0.27 vs 0.04 for 20-day maturity).
- RGaussian distribution of log-volatility increments is approximately preserved.
- RBias increases with option maturity; 1-day maturity yields accurate Hurst estimate (0.06 vs true 0.04).
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