Hurst Exponent

Volatility time series modeling by single-qubit quantum circuit learning ArXiv ID: 2512.10584 “View on arXiv” Authors: Tetsuya Takaishi Abstract We employ single-qubit quantum circuit learning (QCL) to model the dynamics of volatility time series. To assess its effectiveness, we generate synthetic data using the Rational GARCH model, which is specifically designed to capture volatility asymmetry. Our results show that QCL-based volatility predictions preserve the negative return-volatility correlation, a hallmark of asymmetric volatility dynamics. Moreover, analysis of the Hurst exponent and multifractal characteristics indicates that the predicted series, like the original synthetic data, exhibits anti-persistent behavior and retains its multifractal structure. ...

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. ...

The Omniscient, yet Lazy, Investor ArXiv ID: 2510.24467 “View on arXiv” Authors: Stanisław M. S. Halkiewicz Abstract We formalize the paradox of an omniscient yet lazy investor - a perfectly informed agent who trades infrequently due to execution or computational frictions. Starting from a deterministic geometric construction, we derive a closed-form expected profit function linking trading frequency, execution cost, and path roughness. We prove existence and uniqueness of the optimal trading frequency and show that this optimum can be interpreted through the fractal dimension of the price path. A stochastic extension under fractional Brownian motion provides analytical expressions for the optimal interval and comparative statics with respect to the Hurst exponent. Empirical illustrations on equity data confirm the theoretical scaling behavior. ...

Multi Scale Analysis of Nifty 50 Return Characteristics Valuation Dynamics and Market Complexity 1990 to 2024 ArXiv ID: 2509.00697 “View on arXiv” Authors: Chandradew Sharma Abstract This study presents a unified, distribution-aware, and complexity-informed framework for understanding equity return dynamics in the Indian market, using 34 years (1990 to 2024) of Nifty 50 index data. Addressing a key gap in the literature, we demonstrate that the price to earnings ratio, as a valuation metric, may probabilistically map return distributions across investment horizons spanning from days to decades. Return profiles exhibit strong asymmetry. One-year returns show a 74 percent probability of gain, with a modal return of 10.67 percent and a reward-to-risk ratio exceeding 5. Over long horizons, modal CAGRs surpass 13 percent, while worst-case returns remain negative for up to ten years, defining a historical trapping period. This horizon shortens to six years in the post-1999 period, reflecting growing market resilience. Conditional analysis of the P/E ratio reveals regime-dependent outcomes. Low valuations (P/E less than 13) historically show zero probability of loss across all horizons, while high valuations (P/E greater than 27) correspond to unstable returns and extended breakeven periods. To uncover deeper structure, we apply tools from complexity science. Entropy, Hurst exponents, and Lyapunov indicators reveal weak persistence, long memory, and low-dimensional chaos. Information-theoretic metrics, including mutual information and transfer entropy, confirm a directional and predictive influence of valuation on future returns. These findings offer actionable insights for asset allocation, downside risk management, and long-term investment strategy in emerging markets. Our framework bridges valuation, conditional distributions, and nonlinear dynamics in a rigorous and practically relevant manner. ...

Why is the volatility of single stocks so much rougher than that of the S&P500? ArXiv ID: 2505.02678 “View on arXiv” Authors: Othmane Zarhali, Cecilia Aubrun, Emmanuel Bacry, Jean-Philippe Bouchaud, Jean-François Muzy Abstract The Nested factor model was introduced by Chicheportiche et al. to represent non-linear correlations between stocks. Stock returns are explained by a standard factor model, but the (log)-volatilities of factors and residuals are themselves decomposed into factor modes, with a common dominant volatility mode affecting both market and sector factors but also residuals. Here, we consider the case of a single factor where the only dominant log-volatility mode is rough, with a Hurst exponent $H \simeq 0.11$ and the log-volatility residuals are ‘‘super-rough’’ or ‘‘multifractal’’, with $H \simeq 0$. We demonstrate that such a construction naturally accounts for the somewhat surprising stylized fact reported by Wu et al. , where it has been observed that the Hurst exponents of stock indexes are large compared to those of individual stocks. We propose a statistical procedure to estimate the Hurst factor exponent from the stock returns dynamics together with theoretical guarantees of its consistency. We demonstrate the effectiveness of our approach through numerical experiments and apply it to daily stock data from the S&P500 index. The estimated roughness exponents for both the factor and idiosyncratic components validate the assumptions underlying our model. ...

Impact of the COVID-19 pandemic on the financial market efficiency of price returns, absolute returns, and volatility increment: Evidence from stock and cryptocurrency markets ArXiv ID: 2504.18960 “View on arXiv” Authors: Tetsuya Takaishi Abstract This study examines the impact of the coronavirus disease 2019 (COVID-19) pandemic on market efficiency by analyzing three time series – price returns, absolute returns, and volatility increments – in stock (Deutscher Aktienindex, Nikkei 225, Shanghai Stock Exchange (SSE), and Volatility Index) and cryptocurrency (Bitcoin and Ethereum) markets. The effect is found to vary by asset class and market. In the stock market, while the pandemic did not influence the Hurst exponent of volatility increments, it affected that of returns and absolute returns (except in the SSE, where returns remained unaffected). In the cryptocurrency market, the pandemic did not alter the Hurst exponent for any time series but influenced the strength of multifractality in returns and absolute returns. Some Hurst exponent time series exhibited a gradual decline over time, complicating the assessment of pandemic-related effects. Consequently, segmented analyses by pandemic periods may erroneously suggest an impact, warranting caution in period-based studies. ...

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. ...

Integrating the implied regularity into implied volatility models: A study on free arbitrage model ArXiv ID: 2502.07518 “View on arXiv” Authors: Unknown Abstract Implied volatility IV is a key metric in financial markets, reflecting market expectations of future price fluctuations. Research has explored IV’s relationship with moneyness, focusing on its connection to the implied Hurst exponent H. Our study reveals that H approaches 1/2 when moneyness equals 1, marking a critical point in market efficiency expectations. We developed an IV model that integrates H to capture these dynamics more effectively. This model considers the interaction between H and the underlying-to-strike price ratio S/K, crucial for capturing IV variations based on moneyness. Using Optuna optimization across multiple indexes, the model outperformed SABR and fSABR in accuracy. This approach provides a more detailed representation of market expectations and IV-H dynamics, improving options pricing and volatility forecasting while enhancing theoretical and pratcical financial analysis. ...

Multiscale Markowitz ArXiv ID: 2411.13792 “View on arXiv” Authors: Unknown Abstract Traditional Markowitz portfolio optimization constrains daily portfolio variance to a target value, optimising returns, Sharpe or variance within this constraint. However, this approach overlooks the relationship between variance at different time scales, typically described by $σ(Δt) \propto (Δt)^{“H”}$ where $H$ is the Hurst exponent, most of the time assumed to be (\frac{“1”}{“2”}). This paper introduces a multifrequency optimization framework that allows investors to specify target portfolio variance across a range of frequencies, characterized by a target Hurst exponent $H_{“target”}$, or optimize the portfolio at multiple time scales. By incorporating this scaling behavior, we enable a more nuanced and comprehensive risk management strategy that aligns with investor preferences at various time scales. This approach effectively manages portfolio risk across multiple frequencies and adapts to different market conditions, providing a robust tool for dynamic asset allocation. This overcomes some of the traditional limitations of Markowitz, when it comes to dealing with crashes, regime changes, volatility clustering or multifractality in markets. We illustrate this concept with a toy example and discuss the practical implementation for assets with varying scaling behaviors. ...

Detecting Structural breakpoints in natural gas and electricity wholesale prices via Bayesian ensemble approach, in the era of energy prices turmoil of 2022 period: the cases of ten European markets ArXiv ID: 2410.07224 “View on arXiv” Authors: Unknown Abstract We investigate the impact of several critical events associated with the Russo Ukrainian war, started officially on 24 February 2022 with the Russian invasion of Ukraine, on ten European electricity markets, two natural gas markets (the European reference trading hub TTF and N.Y. NGNMX market) and how these markets interact to each other and with USDRUB exchange rate, a financial market. We analyze the reactions of these markets, manifested as breakpoints attributed to these critical events, and their interaction, by using a set of three tools that can shed light on different aspects of this complex situation. We combine the concepts of market efficiency, measured by quantifying the Efficient market hypothesis (EMH) via rolling Hurst exponent, with structural breakpoints occurred in the time series of gas, electricity and financial markets, the detection of which is possible by using a Bayesian ensemble approach, the Bayesian Estimator of Abrupt change, Seasonal change and Trend (BEAST), a powerful tool that can effectively detect structural breakpoints, trends, seasonalities and sudden abrupt changes in time series. The results show that the analyzed markets have exhibited different modes of reactions to the critical events, both in respect of number, nature, and time of occurrence (leading, lagging, concurrent with dates of critical events) of breakpoints as well as of the dynamic behavior of their trend components. ...