Modeling and Forecasting Realized Volatility with Multivariate Fractional Brownian Motion

ArXiv ID: 2504.15985 “View on arXiv”

Authors: Unknown

Abstract

A multivariate fractional Brownian motion (mfBm) with component-wise Hurst exponents is used to model and forecast realized volatility. We investigate the interplay between correlation coefficients and Hurst exponents and propose a novel estimation method for all model parameters, establishing consistency and asymptotic normality of the estimators. Additionally, we develop a time-reversibility test, which is typically not rejected by real volatility data. When the data-generating process is a time-reversible mfBm, we derive optimal forecasting formulae and analyze their properties. A key insight is that an mfBm with different Hurst exponents and non-zero correlations can reduce forecasting errors compared to a one-dimensional model. Consistent with optimal forecasting theory, out-of-sample forecasts using the time-reversible mfBm show improvements over univariate fBm, particularly when the estimated Hurst exponents differ significantly. Empirical results demonstrate that mfBm-based forecasts outperform the (vector) HAR model.

Keywords: Multivariate Fractional Brownian Motion, Realized Volatility, Time Reversibility, Hurst Exponent, Forecasting, Equities / Volatility

Complexity vs Empirical Score

• Math Complexity: 9.5/10
• Empirical Rigor: 7.2/10
• Quadrant: Holy Grail
• Why: The paper relies heavily on advanced stochastic calculus, Gaussian process theory, and estimation with asymptotic normality proofs, indicating high mathematical complexity. It provides out-of-sample forecasting comparisons, a time-reversibility test, and a reference to R code for implementation, demonstrating solid empirical implementation.

  flowchart TD
    A["Research Goal<br>Model & Forecast<br>Realized Volatility"] --> B{"Key Methodology<br>Multivariate fBm mfBm Model"}
    
    B --> C["Data: Multivariate<br>Realized Volatilities"]
    
    C --> D["Parameter Estimation<br>Consistent & Asymptotic Normal Estimators"]
    
    D --> E["Time-Reversibility Test<br>Validates Model Assumption"]
    
    E --> F["Optimal Forecasting<br>Derives Formulae for mfBm"]
    
    F --> G{"Outcomes"}
    
    G --> H["Theoretical Insight<br>MfBm with Varied Hurst Exponents<br>Reduces Forecast Error"]
    
    G --> I["Empirical Result<br>Outperforms Univariate fBm<br>& Vector HAR Model"]