[Paper Review] Volatility of Volatility and Leverage Effect from Options

Carsten Chong, Viktor Todorov|arXiv (Cornell University)|May 6, 2023

Stochastic processes and financial applications41 references4 citations

TL;DR

This paper proposes model-free, nonparametric estimators for spot volatility of volatility and leverage effect using high-frequency short-dated options. By integrating option prices to recover the conditional characteristic function and estimating spot volatility nonparametrically, the method achieves faster convergence rates than return-based estimators, with bias correction via autocovariance and feasible inference under unknown error sources.

ABSTRACT

We propose model-free (nonparametric) estimators of the volatility of volatility and leverage effect using high-frequency observations of short-dated options. At each point in time, we integrate available options into estimates of the conditional characteristic function of the price increment until the options' expiration and we use these estimates to recover spot volatility. Our volatility of volatility estimator is then formed from the sample variance and first-order autocovariance of the spot volatility increments, with the latter correcting for the bias in the former due to option observation errors. The leverage effect estimator is the sample covariance between price increments and the estimated volatility increments. The rate of convergence of the estimators depends on the diffusive innovations in the latent volatility process as well as on the observation error in the options with strikes in the vicinity of the current spot price. Feasible inference is developed in a way that does not require prior knowledge of the source of estimation error that is asymptotically dominating.

Motivation & Objective

• To develop model-free estimators for spot volatility of volatility and leverage effect using high-frequency options data.
• To overcome the slow convergence rates of return-based estimators by leveraging short-dated options for direct volatility observation.
• To correct bias in volatility of volatility estimation using first-order autocovariance of spot volatility increments.
• To enable feasible inference without prior knowledge of the dominant estimation error source.
• To provide diagnostic tools for stochastic volatility modeling and asset pricing analysis involving volatility risk factors.

Proposed method

• Estimate the conditional characteristic function of price increments using high-frequency options with short time-to-maturity.
• Recover spot volatility nonparametrically from the estimated characteristic function, exploiting the shrinking time-to-maturity of options.
• Construct the volatility of volatility estimator as the sample variance of spot volatility increments, corrected by first-order autocovariance to reduce bias from option observation errors.
• Estimate the leverage effect as the sample covariance between price returns and estimated volatility increments.
• Use a truncation scheme and asymptotic analysis to ensure robustness under unknown dominant error sources in estimation.
• Apply non-asymptotic bounds and martingale techniques to derive convergence rates under general assumptions on microstructure noise and volatility dynamics.

Experimental results

Research questions

• RQ1 Can nonparametric estimators for spot volatility of volatility and leverage effect be constructed using only high-frequency options data?
• RQ2 How does the use of short-dated options improve estimation efficiency compared to return-based methods?
• RQ3 What is the impact of option observation errors on volatility of volatility estimation, and how can they be corrected?
• RQ4 What are the attainable convergence rates for these estimators under different error structures?
• RQ5 Can feasible inference be conducted without prior knowledge of the dominant estimation error source?

Key findings

• The proposed estimators achieve faster convergence rates than return-based methods, with the best attainable rate of $ n^{1/4} $ under no microstructure noise and $ n^{1/8} $ under noisy returns.
• The use of short-dated options enables direct observation of latent volatility, significantly improving estimation efficiency compared to return-based local volatility estimators.
• Bias in the volatility of volatility estimator is effectively corrected by incorporating the first-order autocovariance of spot volatility increments.
• Feasible inference is developed without requiring prior knowledge of the dominant estimation error source, ensuring robustness in practice.
• The convergence rates depend on both diffusive innovations in the latent volatility process and the observation error in options with strikes near the current spot price.
• Theoretical bounds and asymptotic results are established under general assumptions, including càdlàg volatility processes and bounded characteristic function coefficients.