Realized Volatility

Multifractality and sample size influence on Bitcoin volatility patterns ArXiv ID: 2511.03314 “View on arXiv” Authors: Tetsuya Takaishi Abstract The finite sample effect on the Hurst exponent (HE) of realized volatility time series is examined using Bitcoin data. This study finds that the HE decreases as the sampling period $Δ$ increases and a simple finite sample ansatz closely fits the HE data. We obtain values of the HE as $Δ\rightarrow 0$, which are smaller than 1/2, indicating rough volatility. The relative error is found to be $1%$ for the widely used five-minute realized volatility. Performing a multifractal analysis, we find the multifractality in the realized volatility time series, smaller than that of the price-return time series. ...

Probabilistic Forecasting Cryptocurrencies Volatility: From Point to Quantile Forecasts ArXiv ID: 2508.15922 “View on arXiv” Authors: Grzegorz Dudek, Witold Orzeszko, Piotr Fiszeder Abstract Cryptocurrency markets are characterized by extreme volatility, making accurate forecasts essential for effective risk management and informed trading strategies. Traditional deterministic (point) forecasting methods are inadequate for capturing the full spectrum of potential volatility outcomes, underscoring the importance of probabilistic approaches. To address this limitation, this paper introduces probabilistic forecasting methods that leverage point forecasts from a wide range of base models, including statistical (HAR, GARCH, ARFIMA) and machine learning (e.g. LASSO, SVR, MLP, Random Forest, LSTM) algorithms, to estimate conditional quantiles of cryptocurrency realized variance. To the best of our knowledge, this is the first study in the literature to propose and systematically evaluate probabilistic forecasts of variance in cryptocurrency markets based on predictions derived from multiple base models. Our empirical results for Bitcoin demonstrate that the Quantile Estimation through Residual Simulation (QRS) method, particularly when applied to linear base models operating on log-transformed realized volatility data, consistently outperforms more sophisticated alternatives. Additionally, we highlight the robustness of the probabilistic stacking framework, providing comprehensive insights into uncertainty and risk inherent in cryptocurrency volatility forecasting. This research fills a significant gap in the literature, contributing practical probabilistic forecasting methodologies tailored specifically to cryptocurrency markets. ...

Multifractality in Bitcoin Realised Volatility: Implications for Rough Volatility Modelling ArXiv ID: 2507.00575 “View on arXiv” Authors: Milan Pontiggia Abstract We assess the applicability of rough volatility models to Bitcoin realized volatility using the normalised p-variation framework of Cont and Das (2024). Applying this model-free estimator to high-frequency Bitcoin data from 2017 to 2024 across multiple sampling resolutions, we find that the normalised statistic remains strictly negative, precluding the estimation of a valid roughness index. Stationarity tests and robustness checks reveal no significant evidence of non-stationarity or structural breaks as explanatory factors. Instead, convergent evidence from three complementary diagnostics, namely Multifractal Detrended Fluctuation Analysis, log-log moment scaling, and wavelet leaders, reveals a multifractal structure in Bitcoin volatility. This behaviour violates the homogeneity assumptions underlying rough volatility estimation and accounts for the estimator’s systematic failure. These findings suggest that while rough volatility models perform well in traditional markets, they are structurally misaligned with the empirical features of Bitcoin volatility. ...

Predicting Realized Variance Out of Sample: Can Anything Beat The Benchmark? ArXiv ID: 2506.07928 “View on arXiv” Authors: Austin Pollok Abstract The discrepancy between realized volatility and the market’s view of volatility has been known to predict individual equity options at the monthly horizon. It is not clear how this predictability depends on a forecast’s ability to predict firm-level volatility. We consider this phenomenon at the daily frequency using high-dimensional machine learning models, as well as low-dimensional factor models. We find that marginal improvements to standard forecast error measurements can lead to economically significant gains in portfolio performance. This makes a case for re-imagining the way we train models that are used to construct portfolios. ...

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. ...

Realized Volatility Forecasting for New Issues and Spin-Offs using Multi-Source Transfer Learning ArXiv ID: 2503.12648 “View on arXiv” Authors: Unknown Abstract Forecasting the volatility of financial assets is essential for various financial applications. This paper addresses the challenging task of forecasting the volatility of financial assets with limited historical data, such as new issues or spin-offs, by proposing a multi-source transfer learning approach. Specifically, we exploit complementary source data of assets with a substantial historical data record by selecting source time series instances that are most similar to the limited target data of the new issue/spin-off. Based on these instances and the target data, we estimate linear and non-linear realized volatility models and compare their forecasting performance to forecasts of models trained exclusively on the target data, and models trained on the entire source and target data. The results show that our transfer learning approach outperforms the alternative models and that the integration of complementary data is also beneficial immediately after the initial trading day of the new issue/spin-off. ...

Multivariate Rough Volatility ArXiv ID: 2412.14353 “View on arXiv” Authors: Unknown Abstract Motivated by empirical evidence from the joint behavior of realized volatility time series, we propose to model the joint dynamics of log-volatilities using a multivariate fractional Ornstein-Uhlenbeck process. This model is a multivariate version of the Rough Fractional Stochastic Volatility model proposed in Gatheral, Jaisson, and Rosenbaum, Quant. Finance, 2018. It allows for different Hurst exponents in the different marginal components and non trivial interdependencies. We discuss the main features of the model and propose a Generalised Method of Moments estimator that jointly identifies its parameters. We derive the asymptotic theory of the estimator and perform a simulation study that confirms the asymptotic theory in finite sample. We carry out an extensive empirical investigation on all realized volatility time series covering the entire span of about two decades in the Oxford-Man realized library. Our analysis shows that these time series are strongly correlated and can exhibit asymmetries in their empirical cross-covariance function, accurately captured by our model. These asymmetries lead to spillover effects, which we derive analytically within our model and compute based on empirical estimates of model parameters. Moreover, in accordance with the existing literature, we observe behaviors close to non-stationarity and rough trajectories. ...

Geometric Deep Learning for Realized Covariance Matrix Forecasting ArXiv ID: 2412.09517 “View on arXiv” Authors: Unknown Abstract Traditional methods employed in matrix volatility forecasting often overlook the inherent Riemannian manifold structure of symmetric positive definite matrices, treating them as elements of Euclidean space, which can lead to suboptimal predictive performance. Moreover, they often struggle to handle high-dimensional matrices. In this paper, we propose a novel approach for forecasting realized covariance matrices of asset returns using a Riemannian-geometry-aware deep learning framework. In this way, we account for the geometric properties of the covariance matrices, including possible non-linear dynamics and efficient handling of high-dimensionality. Moreover, building upon a Fréchet sample mean of realized covariance matrices, we are able to extend the HAR model to the matrix-variate. We demonstrate the efficacy of our approach using daily realized covariance matrices for the 50 most capitalized companies in the S&P 500 index, showing that our method outperforms traditional approaches in terms of predictive accuracy. ...

Kullback-Leibler cluster entropy to quantify volatility correlation and risk diversity ArXiv ID: 2409.10543 “View on arXiv” Authors: Unknown Abstract The Kullback-Leibler cluster entropy $\mathcal{“D_{C”}}[“P | Q”] $ is evaluated for the empirical and model probability distributions $P$ and $Q$ of the clusters formed in the realized volatility time series of five assets (SP&500, NASDAQ, DJIA, DAX, FTSEMIB). The Kullback-Leibler functional $\mathcal{“D_{C”}}[“P | Q”] $ provides complementary perspectives about the stochastic volatility process compared to the Shannon functional $\mathcal{“S_{C”}}[“P”]$. While $\mathcal{“D_{C”}}[“P | Q”] $ is maximum at the short time scales, $\mathcal{“S_{C”}}[“P”]$ is maximum at the large time scales leading to complementary optimization criteria tracing back respectively to the maximum and minimum relative entropy evolution principles. The realized volatility is modelled as a time-dependent fractional stochastic process characterized by power-law decaying distributions with positive correlation ($H>1/2$). As a case study, a multiperiod portfolio built on diversity indexes derived from the Kullback-Leibler entropy measure of the realized volatility. The portfolio is robust and exhibits better performances over the horizon periods. A comparison with the portfolio built either according to the uniform distribution or in the framework of the Markowitz theory is also reported. ...

Rough volatility: evidence from range volatility estimators ArXiv ID: 2312.01426 “View on arXiv” Authors: Unknown Abstract In Gatheral et al. 2018, first posted in 2014, volatility is characterized by fractional behavior with a Hurst exponent $H < 0.5$, challenging traditional views of volatility dynamics. Gatheral et al. demonstrated this using realized volatility measurements. Our study extends this analysis by employing range-based proxies to confirm their findings across a broader dataset and non-standard assets. Notably, we address the concern that rough volatility might be an artifact of microstructure noise in high-frequency return data. Our results reveal that log-volatility, estimated via range-based methods, behaves akin to fractional Brownian motion with an even lower $H$, below $0.1$. We also affirm the efficacy of the rough fractional stochastic volatility model (RFSV), finding that its predictive capability surpasses that of AR, HAR, and GARCH models in most scenarios. This work substantiates the intrinsic nature of rough volatility, independent of the microstructure noise often present in high-frequency financial data. ...