[論文レビュー] Realized range-based estimation of integrated variance

Kim Christensen, Mark Podolskij|SSRN Electronic Journal|Jan 28, 2026

Economic Policies and Impacts被引用数 18

ひとこと要約

realized range-based variance (RRV) for estimating integrated variance of continuous semimartingales, showing consistency, a mixed-Gaussian CLT, and substantial efficiency gains over realized variance (RV); provides bias handling for discretely observed data and non-trading effects, with an empirical TAQ application.

ABSTRACT

We provide a set of probabilistic laws for estimating the quadratic variation of continuous semimartingales with realized range-based variance -- a statistic that replaces every squared return of realized variance with a normalized squared range. If the entire sample path of the process is available, and under a set of weak conditions, our statistic is consistent and has a mixed Gaussian limit, whose precision is five times greater than that of realized variance. In practice, of course, inference is drawn from discrete data and true ranges are unobserved, leading to downward bias. We solve this problem to get a consistent, mixed normal estimator, irrespective of non-trading effects. This estimator has varying degrees of efficiency over realized variance, depending on how many observations that are used to construct the high-low. The methodology is applied to TAQ data and compared with realized variance. Our findings suggest that the empirical path of quadratic variation is also estimated better with the realized range-based variance.

研究の動機と目的

• Motivate the need for more efficient volatility estimation under microstructure limitations (noise and sampling limits).
• Propose a non-parametric, range-based estimator of integrated variance that uses intraday high-low information.
• Develop theoretical guarantees: consistency and a mixed Gaussian central limit theorem for RRV under mild conditions.
• Address practical issues: bias from discretely observed high-low ranges and non-trading effects, including an estimator with finite-m-sample corrections.
• Provide an empirical evaluation using high-frequency TAQ data (GM) to compare RRV with realized variance.

提案手法

• Define realized range-based variance (RRV) as RRV^Δ = (1/λ2) sum_{i=1}^n s_{p_{iΔ,Δ}}^2 using intraday price ranges.
• Derive that RRV^Δ consistently estimates IV and has a mixed Gaussian limit: sqrt(n)(RRV^Δ - IV) -> MN(0, Λ IQ).
• Introduce a discretization-adjusted version RRV_m^Δ with m high-frequency observations per interval to correct for downward bias (λ2,m).
• Establish feasible inference via RRQ (realized range-based quarticity) to estimate IQ in the CLT.
• Provide stable convergence results and a delta-method based log-transform to improve finite-sample performance.
• Conduct Monte Carlo simulations under stochastic volatility to illustrate finite-sample properties and the distributional theory.
• Apply the method to GM TAQ data to compare RRV with RV and assess practical performance.

実験結果

リサーチクエスチョン

• RQ1 Can the realized range-based variance (RRV) consistently estimate integrated variance (IV) for continuous semimartingales under realistic sampling structures?
• RQ2 What is the asymptotic distribution of RRV (and its discretized variants) and how does it depend on the underlying volatility process?
• RQ3 How does the efficiency of RRV compare with realized variance (RV) across intraday sampling schemes and discretization levels?
• RQ4 How to correct for bias due to discretely observed high-low ranges and non-trading effects in empirical data?
• RQ5 Is it feasible to perform inference using RRV with real high-frequency data (e.g., TAQ) and irregular sampling?

主な発見

• RRV^Δ consistently converges to IV as n → ∞ under mild conditions on the price process.
• sqrt(n)(RRV^Δ - IV) converges in the stable sense to a mixed normal limit with variance Λ IQ, where Λ ≈ 0.4 and IQ is the integrated quarticity.
• RRV^Δ is substantially more efficient than RV^Δ, with approximately one-fifth the sampling error in the asymptotic variance factor.
• With finite samples and m > 1, RRV_m^Δ remains consistent and exhibits a CLT with variance factor Λ_c derived from λ_{2,c}, λ_{4,c} (moments of the Brownian range).
• The feasible inference using RRQ^Δ (realized range-based quarticity) provides a consistent estimator of IQ for constructing standard errors and confidence bands.
• Monte Carlo results show improved finite-sample behavior for RRV^Δ, especially when using log-based transformations for t-statistics.
• Empirical TAQ application to General Motors demonstrates practical benefits of RRV in comparison to RV under high-frequency data constraints.