Paper: arXiv 2504.15985

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.

Complexity vs Empirical Score

  • Math Complexity: 9.5/10
  • Empirical Rigor: 7.2/10
  • Quadrant: Holy Grail — high math complexity, high empirical rigor

Why this score: 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.

Research Flowchart

  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"]