Prediction of linear fractional stable motions using codifference, with application to non-Gaussian rough volatility

ArXiv ID: 2507.15437 “View on arXiv”

Authors: Matthieu Garcin, Karl Sawaya, Thomas Valade

Abstract

The linear fractional stable motion (LFSM) extends the fractional Brownian motion (fBm) by considering $α$-stable increments. We propose a method to forecast future increments of the LFSM from past discrete-time observations, using the conditional expectation when $α>1$ or a semimetric projection otherwise. It relies on the codifference, which describes the serial dependence of the process, instead of the covariance. Indeed, covariance is commonly used for predicting an fBm but it is infinite when $α<2$. Some theoretical properties of the method and of its accuracy are studied and both a simulation study and an application to real data confirm the relevance of the approach. The LFSM-based method outperforms the fBm, when forecasting high-frequency FX rates. It also shows a promising performance in the forecast of time series of volatilities, decomposing properly, in the fractal dynamic of rough volatilities, the contribution of the kurtosis of the increments and the contribution of their serial dependence. Moreover, the analysis of hit ratios suggests that, beside independence, persistence, and antipersistence, a fourth regime of serial dependence exists for fractional processes, characterized by a selective memory controlled by a few large increments.

Keywords: Linear Fractional Stable Motion (LFSM), Stable Processes, Time Series Forecasting, High-Frequency FX, Rough Volatility

Complexity vs Empirical Score

• Math Complexity: 8.5/10
• Empirical Rigor: 6.0/10
• Quadrant: Holy Grail
• Why: The paper employs advanced stochastic calculus and stable distribution theory (α-stable processes, spectral measure, codifference), indicating high mathematical complexity. It includes both simulation studies and real-data applications (FX rates, volatility) to validate the forecasting method, demonstrating solid empirical rigor, though the practical implementation details (e.g., coding, data specifics) are not fully elaborated.

  flowchart TD
    A["Research Goal: Forecast LFSM increments<br>from discrete observations"] --> B{"Input: High-Frequency FX &<br>Volatility Time Series"};
    B --> C["Key Methodology:<br>Compute Codifference<br>(Serial Dependence)"];
    C --> D{"Conditional Logic"};
    D -- α > 1 --> E["Process: Conditional Expectation"];
    D -- α ≤ 1 --> F["Process: Semimetric Projection"];
    E & F --> G["Computational Output:<br>Forecasted Increments"];
    G --> H["Key Findings:<br>Outperforms fBm in FX;<br>Isolates Kurtosis & Persistence;<br>Identifies 'Selective Memory' Regime"];
    
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