Paper: arXiv 2509.20015

Authors: Sergio Bianchi, Daniele Angelini

Abstract

We introduce a novel distribution-based estimator for the Hurst parameter of log-volatility, leveraging the Kolmogorov-Smirnov statistic to assess the scaling behavior of entire distributions rather than individual moments. To address the temporal dependence of financial volatility, we propose a random permutation procedure that effectively removes serial correlation while preserving marginal distributions, enabling the rigorous application of the KS framework to dependent data. We establish the asymptotic variance of the estimator, useful for inference and confidence interval construction. From a computational standpoint, we show that derivative-free optimization methods, particularly Brent’s method and the Nelder-Mead simplex, achieve substantial efficiency gains relative to grid search while maintaining estimation accuracy. Empirical analysis of the CBOE VIX index and the 5-minute realized volatility of the S&P 500 reveals a statistically significant hierarchy of roughness, with implied volatility smoother than realized volatility. Both measures, however, exhibit Hurst exponents well below one-half, reinforcing the rough volatility paradigm and highlighting the open challenge of disentangling local roughness from long-memory effects in fractional modeling.

Complexity vs Empirical Score

  • Math Complexity: 8.5/10
  • Empirical Rigor: 7.0/10
  • Quadrant: Holy Grail — high math complexity, high empirical rigor

Why this score: The paper introduces a novel estimator with asymptotic theory (Propositions 2.1-2.3, Proposition 2.6) and a computational optimization section (Brent’s method, Nelder-Mead), indicating high mathematical density. The empirical analysis uses real financial data (CBOE VIX and S&P 500 realized volatility) with a specific statistical procedure (random permutation) and reports statistically significant results, demonstrating solid data/implementation rigor.

Research Flowchart

  flowchart TD
  A["Research Goal: Estimate Hurst Parameter for Log-Volatility<br>using Distribution-Based Roughness Analysis"] --> B["Data Input: S&P 500 5-min Realized Volatility & CBOE VIX"]
  B --> C["Methodology: Novel KS Statistic Estimator &<br>Random Permutation for Dependent Data"]
  C --> D["Computational Process: Derivative-Free Optimization<br>(Brent & Nelder-Mead)"]
  D --> E["Key Findings:<br>1. Significant Hierarchy: VIX (Smoother) < Realized Volatility<br>2. Both Hurst exponents < 0.5 (Rough Volatility Paradigm)"]

Cited References (546)

  1. The wavelet-based synthesis for the fractional Brownian motion proposed by F. Sellan and Y. Meyer: Remarks and fast implementation (1996)
  2. Ondelettes et turbulences—multir'esolutions, algorithmes de d'ecompositions, invariance d’'echelle et signauz de pression. (1997)
  3. Wavelet analysis of long-range-dependent traffic (1998)
  4. Asset pricing with liquidity risk (2005)
  5. Discrete variations of the fractional B rownian motion in the presence of outliers and an additive noise (2010)
  6. Daily Closing Inside Spreads and Trading Volumes Around Earnings Announcements (2002)
  7. Asset pricing and the bid-ask spread (1986)
  8. The Effects of Beta, Bid-Ask Spread, Residual Risk, and Size on Stock Returns (1989)
and 538 more
  1. Illiquidity and stock returns: Cross-section and time-series effects (2002)
  2. Liquidity and Asset Prices (2005)
  3. The illiquidity premium: International evidence
  4. The Pricing of Illiquidity as a Characteristic and as Risk (2015)
  5. Return Volatility and Trading Volume: An Information Flow Interpretation of Stochastic Volatility (1996)
  6. Answering the Skeptics: Yes, Standard Volatility Models Do Provide Accurate Forecasts (1998)
  7. Possible long-range dependence in fractional random fields (1999)
  8. Nonlinear biases in the roughness of a f ractional s tochastic r egularity m odel (2023)
  9. Market information of the fractional stochastic regularity model — → 2409.07159
  10. Kolmogorov– S mirnov estimation of self-similarity in long-range dependent fractional processes (2025)
  11. Ondelettes, Multifractales et Turbulence. De l’ADN auz Croissances Cristallines (1995)
  12. Random cascades on wavelet dyadic trees (1998)
  13. The relationship between stock return volatility and trading volume: the case of the Philippines (2008)
  14. Generalized multifractional B rownian motion: definition and preliminary results (1999)
  15. The generalized multifractional B rownian motion can be multifractal (2000)
  16. The Generalized Multifractional B rownian Motion (2000)
  17. The covariance structure of multifractional B rownian motion, with application to long range dependence (2000)
  18. Wavelet construction of Generalized Multifractional processes (2007)
  19. Stochastic Volatility and Multifractional Brownian Motion (2012)
  20. Continuous G aussian multifractional processes with random pointwise H "o lder regularity (2013)
  21. A new Multifractional Process with Random Exponent (2018)
  22. Moving Average Multifractional Processes with Random Exponent: lower Bounds for local obscillations (2022)
  23. Uniformly and strongly consistent estimation for the random Hurst function of a multifractional process (2023)
  24. Necessary and sufficient conditions for Hölder continuity of Gaussian processes (2014)
  25. Trading location and liquidity: An analysis of U.S. dealer and agency markets for common stocks. (1996)
  26. The Global Financial Crisis and the Efficient Market Hypothesis: What Have We Learned? (2009)
  27. Multifractality of self-affine fractals (1991)
  28. Generators of long-range dependence processes: a survey
  29. Critical phenomena in the collapse of quadrupolar and hexadecapolar gravitational waves (2023)
  30. Pricing under rough volatility (2016)
  31. Do markets overreact: International evidence (1999)
  32. Parameter estimation for fractional O rnstein– U hlenbeck processes: non-ergodic case (2011)
  33. Identification of the Hurst Index of a Step Fractional Brownian Motion (2000)
  34. Fractional Processes as Models in Stochastic Finance (2011)
  35. Decoupling the Short- and Long-Term Behavior of Stochastic Volatility (2021) — link
  36. Statistics for Long Memory Processes (1994)
  37. Financial Modelling by Multiscale Fractional Brownian Motion (2005)
  38. Modelling NASDAQ Series by Sparse Multifractional Brownian Motion (2012)
  39. Local estimation of the Hurst index of multifractional Brownian motion by Increment Ratio Statistic method (2013)
  40. Overfitting of Hurst estimators for multifractional Brownian motion: A fitting test advocating simple models (2018)
  41. Price Volatility, Trading Volume, and Market Depth: Evidence from Futures Markets (1993)
  42. The informative role of trading volume in an expanding spot and futures market (2016)
  43. A New Distribution-Based Test of Self-similarity (2004) — link
  44. Modeling stock prices by multifractional Brownian motion: an improved estimation of the pointwise regularity (2013)
  45. A Dynamical Assessment of Stock Market (In)Efficiency (2013)
  46. Self-Similarity Parameter Estimation for k-dimensional Processes (2013)
  47. Multifractional processes in finance (2014)
  48. Asset Price Modeling: From Fractional to Multifractional Processes (2015)
  49. Efficient Markets and Behavioral Finance: a Comprehensive multifractional Model (2015)
  50. Fractal stock markets: International evidence of dynamical (in)efficiency (2017)
  51. Rough volatility via the L amperti transform (2023)
  52. The pricing of options and corporate liabilities (1973)
  53. Market statistics and technical analysis, the role of volume (1994)
  54. Information, volatility, volume and maturity: an investigation of stock index futures (1990)
  55. A GMM approach to estimate the roughness of stochastic volatility (2022)
  56. Volume, Volatility, and Public News Announcements (2018)
  57. Examining the relationship between stock return volatility and trading volume: new evidence from an emerging economy (2015)
  58. A Demon Of Our Own Design (2007)
  59. Theory of Financial Risk and Derivative Pricing. From Statistical Physics to Risk Management (2000)
  60. Multiscaling and rough volatility: An empirical investigation (2022)
  61. Market microstructure and asset pricing: On the compensation for illiquidity in stock returns (1996)
  62. A statistical test of market efficiency based on information theory (2023)
  63. Fractal properties, information theory, and market efficiency (2024)
  64. Scaling analysis of stock markets (2014)
  65. Measuring multiscaling in financial time-series (2016)
  66. Non-equilibrium turbulence scalings and self-similarity in turbulent planar jets (2019)
  67. Multifractality in Asset Returns: Theory and Evidence (2002)
  68. The Hurst exponent over time: testing the assertion that emerging markets are becoming more efficient (2004)
  69. Trading Volume and Serial Correlation in Stock Returns* (1993)
  70. Fuzzy clustering of time series with time-varying memory (2023)
  71. Simulation of Multifractional Brownian Motion (1998)
  72. Handbook of methods of applied statistics (1967)
  73. An examination of the complementary volume-volatility information theories (2008)
  74. Revisiting the empirical linkages between stock returns and trading volume (2012)
  75. Goodness-of-fit tests with dependent observations (2011)
  76. Weighted K olmogorov- S mirnov test: Accounting for the tails (2012) — link
  77. Commonality in liquidity (2001)
  78. Market Liquidity and Trading Activity (2001)
  79. An Empirical Analysis of Stock and Bond Market Liquidity (2005)
  80. Theory-Based Illiquidity and Asset Pricing (2009)
  81. Causality in quantiles and dynamic stock return–volume relations (2009)
  82. The bivariate GARCH approach to investigating the relation between stock returns, trading volume, and return volatility (2012)
  83. Exploiting stochastic dominance to generate abnormal stock returns (2014)
  84. Inf 'e rence statistique pour les mouvements B rowniens fractionnaires et multifractionnaires (2000)
  85. Simulation and identification of the fractional B rownian motion: A bibliographical and comparative study (2000)
  86. Estimating the parameters of a fractional B rownian motion by discrete variations of its Sample Paths (2001)
  87. Identification of multifractional B rownian motion (2005)
  88. Hurst exponent estimation of locally self-similar G aussian processes using sample quantiles (2008)
  89. From Self-Similarity to Local Self-Similarity: the Estimation Problem (1999)
  90. Bayesian Estimation of the Multifractality Parameter for Image Texture Using a W hittle Approximation (2015)
  91. Long memory in continuous-time stochastic volatility models (1998)
  92. Scaling in Stock Market Data: Stable Laws and Beyond (1997)
  93. Empirical properties of asset returns: stylized facts and statistical issues (2001)
  94. Long range dependence in financial markets (2005)
  95. Rough volatility: fact or artefact? (2024)
  96. Quadratic variation along refining partitions: constructions and examples (2022)
  97. Quadratic variation and quadratic roughness (2023)
  98. A Model of Asset Trading under the Assumption of Sequential Information Arrival (1976)
  99. Multifractional Stochastic Volatility Models (2014)
  100. Do Behavioral Biases Affect Prices? (2005)
  101. Elements of information theory (1991)
  102. Structural causes of the global financial crisis: a critical assessment of the ‘new financial architecture’ (2009)
  103. The K olmogorov- S mirnov, C ramer- V on M ises tests (1957)
  104. Intraday trading volume and return volatility of the DJIA stocks: A note (2003)
  105. Stochastic Analysis of the Fractional Brownian Motion (1999)
  106. Liquidity Risk Premia in Corporate Bond Markets (2012)
  107. Scaling behaviors in differently developed markets (2003)
  108. Multi-scaling in finance (2007)
  109. A long memory property of stock market returns and a new model (1993)
  110. The Brownian movement and stochastic equations (1942)
  111. Fractal Market Hypothesis and Markov Regime Switching Model: A Possible Synthesis and Integration (2018)
  112. Imprints of log-periodic self-similarity in the stock market (1999)
  113. Lack of robustness in two tests of normality against autocorrelation in sample data (1992)
  114. Selfsimilar Processes (2002)
  115. Hurst exponent and prediction based on weak-form efficient market hypothesis of stock markets (2008)
  116. The Stochastic Dependence of Security Price Changes and Transaction Volumes: Implications for the Mixture-of-Distributions Hypothesis (1976)
  117. Roughening Heston (2019)
  118. FRACTAL GEOMETRY OF FINANCIAL TIME SERIES (1995)
  119. Large deviation and self-similarity analysis of graphs: DAX stock prices (1995)
  120. Market efficiency, long-term returns, and behavioral finance (1998)
  121. Measuring the self-similarity exponent in L 'evy stable processes of financial time series (2013)
  122. The modified mixture of distributions model: a revisit (2006)
  123. Asymptotics for rough stochastic volatility models (2017)
  124. Self-similarity of price fluctuations and market dynamics (2013)
  125. Is Volatility Rough? (2019) — link
  126. Volatility has to be rough (2021) — link
  127. Hedging under rough volatility (2021) — link
  128. Consistent estimation for fractional stochastic volatility model under high-frequency asymptotics (2022)
  129. Program Trading and Price Movement: Evidence from the October 1987 Market Crash (1989)
  130. Trading volumes, volatility and spreads in foreign exchange markets: evidence from emerging market countries (2000)
  131. Stock Prices and Volume (1992)
  132. Estimation of time-dependent Hurst exponents with variational smoothing and application to forecasting foreign exchange rates (2017)
  133. H urst Exponents and Delampertized Fractional B rownian Motions (2019)
  134. Fractal analysis of the multifractality of foreign exchange rates (2019)
  135. Forecasting with fractional B rownian motion: a financial perspective (2022)
  136. Long versus short time scales: the rough dilemma and beyond (2022)
  137. Complexity measure, kernel density estimation, bandwidth selection, and the efficient market hypothesis (2024)
  138. Arbitrage-free SVI volatility surfaces (2014)
  139. Volatility is rough (2014)
  140. Volatility is rough (2018)
  141. Estimating serial correlation and self-similarity in financial time series—A diversification approach with applications to high frequency data (2015)
  142. The High Volume Return Premium (2001)
  143. Trading activity, realized volatility and jumps (2010)
  144. Buy rough, sell smooth (2020) — link
  145. Trading volume, return variability and short-term momentum (2018)
  146. Spectral Analysis of New York Stock Market prices (1963)
  147. Asymmetric anomalous diffusion: an efficient way to detect memory in time series (2001)
  148. Trading fractional B rownian motion (2019)
  149. High-frequency trading with fractional B rownian motion (2021)
  150. Forecasting value at risk allowing for time variation in the variance and kurtosis of portfolio returns (2002)
  151. The intraday relationship between volume and volatility in LIFFE futures markets (1999)
  152. Managing smile risk (2002)
  153. Cross-Security Tests of the Mixture of Distributions Hypothesis (1986)
  154. Transaction Data Tests of the Mixture of Distributions Hypothesis (1987)
  155. Differences of opinion make a horse race (1993)
  156. Liquidity and execution costs in equity markets (1988)
  157. Common factors in prices, order flows, and liquidity. (2001)
  158. Regression of spectral estimators with fractionally integrated time series (1993)
  159. Checking the Independence of Two Covariance-Stationary Time Series: A Univariate Residual Cross-Correlation Approach (1976)
  160. From $N$ Parameter Fractional Brownian Motions to $N$ Parameter Multifractional Brownian Motions (2006)
  161. A closed-form solution for options with stochastic volatility with applications to bond and currency options (1993)
  162. Testing for Linear and Nonlinear Granger Causality in the Stock Price-Volume Relation (1994)
  163. Approach to an irregular time series on the basis of the fractal theory (1988)
  164. An example of infinite dimensional quasi-helix (2003)
  165. A Unified Theory of Underreaction, Momentum Trading, and Overreaction in Asset Markets (1999)
  166. The impact of 2008 and 2010 financial crises on the Hurst exponents of international stock markets: implications for efficiency and contagion (2014)
  167. Fractional M artingales and C haracterization of the F ractional B rownian m otion (2009)
  168. One-factor interest-rate models and the valuation of interest-rate derivative securities (1993)
  169. Long-term storage capacity of reservoirs (1951)
  170. Long-term storage. An experimental study (1965)
  171. Volume and skewness in international equity markets (2008)
  172. Some Lessons from the Financial Market Crisis* (2009)
  173. Variations quadratiques et estimation de l’exposant de H "older local d’un processus gaussien (1997)
  174. An equilibrium model of asset trading with sequential information arrival (1981)
  175. Information dissemination and portfolio choice (1983)
  176. Modelling of self-similar teletraffic for simulation (2002)
  177. Prospect Theory: An Analysis of Decision under Risk (1979)
  178. Trading intensity and the volume-volatility relationship on the Tunis Stock Exchange (2018)
  179. On fractional Ornstein-Uhlenbeck processes (2011)
  180. Fractional Ornstein-Uhlenbeck Processes, Publication 1338 (2015)
  181. Tree roots as self-similar branching structures: axis differentiation and segment tapering in coarse roots of three boreal forest tree species (2010)
  182. Investment implications of the Fractal Market Hypothesis (2019)
  183. The Relation between Price Changes and Trading Volume: A Survey (1987)
  184. COSTLY SHORT SALES AND THE CORRELATION OF RETURNS WITH VOLUME (1988)
  185. Intraday relationships between volatility in S&P500 futures prices and volatility in the S&P500 index (1990)
  186. Volume and Volatility Surrounding Quarterly Redesignation of the Lead S&P 500 Futures Contract (2001)
  187. Estimating the fractal dimension of a locally selfsimilar G aussian process using increments (1997)
  188. Return-volume dynamics in futures markets (1998)
  189. Foundations of the theory of probability (1950)
  190. The relationship between trading activity and stock market volatility: Does the volume threshold matter? (2019)
  191. Fractal markets hypothesis and the global financial crisis: scaling, investment horizons and liquidity (2012)
  192. Fractal markets hypothesis and the global financial crisis: wavelet power evidence (2013)
  193. Effect of random permutations applied to random sequences and related applications (2002)
  194. Heteroskedasticity in Stock Return Data: Volume versus GARCH Effects (1990)
  195. Testing for differences between two distributions in the presence of serial correlation using the Kolmogorov–Smirnov and Kuiper’s tests (2021)
  196. Characterization of turbulence stability through the identification of multifractional Brownian motions (2013)
  197. The dynamic relationship between stock returns and trading volume: Domestic and cross-country evidence (2002)
  198. Fads, Martingales, and Market Efficiency* (1990)
  199. Multi-fractional generalized Cauchy process and its application to teletraffic (2020) — link
  200. Some fractional and multifractional Gaussian processes: A brief introduction (2015)
  201. Long memory revisit in Chinese stock markets: Based on GARCH-class models and multiscale analysis (2013)
  202. Rough volatility: Evidence from option prices (2018) — link
  203. Regularity of multifractional moving average processes with random H urst exponent (2021)
  204. Fractal and multifractal analysis: A review (2009)
  205. An algorithm for the characterization of time-series based on local regularity (2007)
  206. An Empirical Study of Volatility and Trading Volume Dynamics Using High-Frequency Data (2010)
  207. An empirical study of volatility and trading volume of heterogeneity of traders using high-frequency data (2011)
  208. Does volume provide information? Evidence from the Irish Stock Market (2005)
  209. Testing the mixture-of-distributions hypothesis using “realized” volatility (2003)
  210. Multifractal Models in Finance: Their Origin, Properties and Applications (forthcoming)
  211. Robust R / S analysis of long run serial correlation (1979)
  212. Computer experiments with fractional G aussian noises: Part 1, averages and variances (1969)
  213. Fractals and Scaling in Finance. Discontinuity, Concentration, Risk Selecta Volume E (1997)
  214. Duration, volume and volatility impact of trades (2005)
  215. An Introduction to Econophysics. Correlations and Complexity in Finance (2004)
  216. Scale invariance and criticality in financial markets (2003)
  217. A Composite Index for Measuring Stock Market Inefficiency (2022)
  218. Theory of rational option pricing (1973)
  219. A Simple Model of Capital Market Equilibrium with Incomplete Information (1987)
  220. Identifying periods of market inefficiency for return predictability (2017)
  221. Dynamical generalized Hurst exponent as a tool to monitor unstable periods in financial time series (2012)
  222. Price and trading volume reaction surrounding earnings announcements: a closer examination (1981)
  223. Wavelets and multifractal formalism for singular signals: Application to turbulence data (1991)
  224. Modelling fluctuations of financial time series: from cascade process to stochastic volatility model (2000)
  225. A table of integrals of the error functions (1969)
  226. Linear estimation of self-similar processes via L amperti’s transformation (2000)
  227. Kolmogorov– S mirnov test for spatially correlated data (2009)
  228. Tables for computing bivariate normal probabilities (1956)
  229. A table of normal integrals: A table (1980)
  230. Surprising information, the MDH, and the relationship between volatility and trading volume (2010)
  231. The extreme value method for estimating the variance of the rate of return (1980)
  232. A new method for estimating the parameter of fractional B rownian motion (1994)
  233. Multifractional B rownian Motion: Definition and Preliminary Results (1995)
  234. Mosaic organization of DNA nucleotides (1994)
  235. From rough to multifractal volatility: The log S-fBM model (2022)
  236. A general class of multifractional processes and stock price informativeness (2018) — link
  237. The level and quality of Value-at-Risk disclosure by commercial banks (2010)
  238. Fractal market analysis – Applying chaos theory to investment and analysis (1994)
  239. Integration questions related to fractional B rownian motion (2000)
  240. CED model for asset returns and fractal market hypothesis (1999)
  241. Comparison of H urst exponent estimates in hydrometeorological time series (1999)
  242. An analysis of U.S. stock price behavior using wavelets (1995)
  243. Not all estimators are born equal: The empirical properties of some estimators of long memory (2013)
  244. Continuous martingales and Brownian motion (1999)
  245. Texture Estimation for Abnormal Tissue Segmentation in Brain MRI (2016)
  246. Multifractal Processes (2002)
  247. The informational efficiency and the financial crashes (2008)
  248. The Dependence of Prices and Volume (1978)
  249. Estimating variance from high, low and closing prices (1991)
  250. Things We Think We Know, in Lecture Notes in Finance Options - 45 Years since the Publication of the Black-Scholes-Merton Model (2023)
  251. Stable non- G aussian random processes: stochastic models with infinite variance (1994)
  252. Contemporaneous and Causal Relationship between Returns and Volumes: Evidence from Nifty Futures (2019)
  253. Why Does Stock Market Volatility Change Over Time? (1989)
  254. Time-varying long range dependence in market returns of FEAS members (2013)
  255. Volume, Volatility, and the Dispersion of Beliefs (1993)
  256. A mathematical theory of communication (1948)
  257. Quantitative MR Image Analysis for Brain Tumor (2018)
  258. Behavioral Capital Asset Pricing Theory (1994)
  259. THE BEHAVIORAL LIFE-CYCLE HYPOTHESIS (1988)
  260. Behavioral Portfolio Theory (2000)
  261. Inefficient Markets: An Introduction to Behavioral Finance (2000)
  262. Measuring market efficiency: The S hannon entropy of high-frequency financial time series (2022)
  263. Scaling in the Norwegian stock market (2000)
  264. Asymmetric information, volatility components and the volume–volatility relationship for the CAC 40 stocks (2016)
  265. An empirical analysis of the stock price-volume relationship (1988)
  266. On the estimation of the discrepancy between empirical curves of distribution for two independent samples (1939)
  267. Arbitrage Asymmetry and the Idiosyncratic Volatility Puzzle (2015)
  268. New Estimators of the Hurst Index for Fractional Brownian Motion (2014)
  269. Rough volatility of Bitcoin (2020)
  270. ARFBF model for non stationary random fields and application in HRTEM images (2015)
  271. Research on State Transition Model Based on Stock Market Volume-Price Distribution (2018)
  272. Risk Finance and Asset Pricing (2010)
  273. Self-similar processes and related ultraviolet and infrared catastrophes (1981)
  274. Estimators for Long-Range Dependence: an empirical study (1995)
  275. On estimating the intensity of long-range dependence in finite and infinite variance time series (1998)
  276. The Price Variability-Volume Relationship on Speculative Markets (1983)
  277. Testing for long-range dependence in the presence of shifting means or a slowly declining trend, using a variance-type estimator (1997)
  278. The interpretation and estimation of effective sample size (1984)
  279. The Efficient Market Hypothesis: Review of Specialized Literature and Empirical Research (2015)
  280. Self-organized criticality in phylogenetic-like tree growths (1995)
  281. Divergence of opinion in complete markets (1985)
  282. Differences of opinion in financial markets (1989)
  283. Introduction to the multifractal analysis of images (1998)
  284. Local H "older regularity-based modeling of RR intervals (2008)
  285. A Model of Competitive Stock Trading Volume (1994)
  286. Trading volume, bid–ask spread, and price volatility in futures markets (2000)
  287. On some generalization of fractional Brownian motions (2006)
  288. The Relationship between Volatility and Trading Volume in the Chinese Stock Market: A Volatility Decomposition Perspective (2012)
  289. Price volatility, trading volume, and market depth: evidence from the Japanese stock index futures market (2001)
  290. Modification of the Kolmogorov-Smirnov Statistic for Use with Correlated Data (1978)
  291. Fractal market hypothesis and two power-laws (2000)
  292. Simulation of stationary G aussian processes in [0, 1]d (1994)
  293. Simulation of multifractional Brownian motion (1998)
  294. An Investigation of Transactions Data for NYSE Stocks (1985)
  295. Time and dynamic volume$-$volatility relation (2006)
  296. Scaling, self-similarity and multifractality in FX markets (2003)
  297. Algorithm AS 76: An integral useful in calculating non-central t and bivariate normal probabilities (1974)
  298. The fBm-driven O rnstein- U hlenbeck process: Probability density function and anomalous diffusion (2012)
  299. Generalized fractional B rownian motion (2017)
  300. New dynamics between volume and volatility (2019)
  301. Intraday futures patterns and volume–volatility relationships: the German evidence (2014)
  302. A comparison of maximum likelihood and absolute moments for the estimation of Hurst exponents in a stationary framework (2022)
  303. Wavelet-based estimation of the self-similarity parameter of long-range dependent signals (2009)
  304. Estimation of roughness parameters: A pplication to rough volatility modeling (2019)
  305. Learning the roughness of stochastic volatility from option prices (2021)
  306. Deep calibration of rough stochastic volatility models (2021)
  307. Kolmogorov- S mirnov E stimation of S elf- S imilarity in L ong- R ange D ependent F ractional P rocesses, arXiv, 2502.07545 (2025) — link
  308. Two-sample D voretzky– K iefer– W olfowitz Inequalities (2012)
  309. Rough volatility: Evidence from option prices (2018)
  310. Consistent estimation for fractional stochastic volatility model under high-frequency asymptotics (2022) — link
  311. Roughness of the Implied Volatility, arXiv, 2207.04930 (2022) — link
  312. Smoothing and differentiation of data by simplified least squares procedures. (1964)
  313. On the theory of filter amplifiers (1930)
  314. Risk and return: Long-run relations, fractional cointegration, and return predictability (2013)
  315. The relation between implied and realized volatility (1998)
  316. Rough volatility via the Lamperti transform (2023) — link
  317. Testing the efficient market hypothesis in Latin American stock markets (2020) — link
  318. Introducing fractal dimension algorithms to calculate the H urst exponent of financial time series (2012)
  319. Fractional motions (2013) — link
  320. Anomalous diffusion: fractional Brownian motion vs fractional Ito motion (2022)
  321. Power B rownian motion (2023) — link
  322. A note on geometric method-based procedures to calculate the H urst exponent (2012) — link
  323. Genetic Algorithms in Search, Optimization, and Machine Learning (1989)
  324. Optimization by simulated annealing (1983)
  325. Recent approaches to global optimization problems through particle swarm optimization (2002)
  326. `` D irect Search’’ Solution of Numerical and Statistical Problems (1961)
  327. Algorithms for M inimization W ithout D erivatives (1973)
  328. A simplex method for function minimization (1965)
  329. Fair Volatility: A Framework for Reconceptualizing Financial Risk (2025)
  330. Fair Volatility: A Framework for Reconceptualizing Financial Risk — → 2509.18837
  331. Decoupling the short- and long-term behavior of stochastic volatility (2022)
  332. Long memory processes and fractional integration in econometrics (1996)
  333. The distribution of realized exchange rate volatility (2001)
  334. Modeling and forecasting realized volatility (2003)
  335. Long memory, realized volatility and heterogeneous autoregressive models (2019)
  336. Forecasting volatility in financial markets: A review (2003)
  337. On the short-time behavior of the implied volatility for jump-diffusion models with stochastic volatility (2007)
  338. L é vy P rocesses and I nfinitely D ivisible D istributions (1999)
  339. Roughness in VIX Index and in Realized Volatility: Rolling Window Estimation by Randomized K olmogorov- S mirnov Distribution (2025)
  340. The relationship between stock return volatility and trading volume: the case of the Philippines (2008)
  341. Asset pricing with liquidity risk (2005)
  342. Daily Closing Inside Spreads and Trading Volumes Around Earnings Announcements (2002)
  343. Asset pricing and the bid-ask spread (1986)
  344. The Effects of Beta, Bid-Ask Spread, Residual Risk, and Size on Stock Returns (1989)
  345. Illiquidity and stock returns: Cross-section and time-series effects (2002)
  346. Liquidity and Asset Prices (2005)
  347. The illiquidity premium: International evidence
  348. The Pricing of Illiquidity as a Characteristic and as Risk (2015)
  349. Possible long-range dependence in fractional random fields (1999)
  350. Answering the Skeptics: Yes, Standard Volatility Models Do Provide Accurate Forecasts (1998)
  351. Random cascades on wavelet dyadic trees (1998)
  352. Ondelettes, Multifractales et Turbulence. De l’ADN auz Croissances Cristallines (1995)
  353. Wavelet construction of Generalized Multifractional processes (2007)
  354. Continuous Gaussian Multifractional Processes with Random Pointwise Hölder Regularity (2013)
  355. A new Multifractional Process with Random Exponent (2018)
  356. The generalized multifractional Brownian motion can be multifractal (2000)
  357. The covariance structure of multifractional Brownian motion, with application to long range dependence (2000)
  358. The Generalized Multifractional Brownian Motion (2000)
  359. Stochastic Volatility and Multifractional Brownian Motion (2012)
  360. Multifractality of self-affine fractals (1991)
  361. Generators of long-range dependence processes: a survey
  362. Risks for the long-run: A potential resolution of asset pricing puzzles. (2004)
  363. Local H "older regularity-based modeling of RR intervals (2008)
  364. Fractional Processes as Models in Stochastic Finance (2011)
  365. The Global Financial Crisis and the Efficient Market Hypothesis: What Have We Learned? (2009)
  366. The informative role of trading volume in an expanding spot and futures market (2016)
  367. Do markets overreact: International evidence (1999)
  368. Financial Modelling by Multiscale Fractional Brownian Motion (2005)
  369. Local estimation of the Hurst index of multifractional Brownian motion by Increment Ratio Statistic method (2013)
  370. Modelling NASDAQ Series by Sparse Multifractional Brownian Motion (2012)
  371. Identification of the Hurst Index of a Step Fractional Brownian Motion (2000)
  372. Overfitting of Hurst estimators for multifractional Brownian motion: A fitting test advocating simple models (2018)
  373. Price Volatility, Trading Volume, and Market Depth: Evidence from Futures Markets (1993)
  374. Self-Similarity Parameter Estimation for k-dimensional Processes (2013)
  375. Modeling stock prices by multifractional Brownian motion: an improved estimation of the pointwise regularity (2013)
  376. A Dynamical Assessment of Stock Market (In)Efficiency (2013)
  377. Multifractional processes in finance (2014)
  378. Efficient Markets and Behavioral Finance: a Comprehensive multifractional Model (2015)
  379. Asset Price Modeling: From Fractional to Multifractional Processes (2015)
  380. Market statistics and technical analysis, the role of volume (1994)
  381. Volume, Volatility, and Public News Announcements (2018)
  382. Theory of Financial Risk and Derivative Pricing. From Statistical Physics to Risk Management (2000)
  383. A Demon Of Our Own Design (2007)
  384. Information, volatility, volume and maturity: an investigation of stock index futures (1990)
  385. Market microstructure and asset pricing: On the compensation for illiquidity in stock returns (1996)
  386. Scaling analysis of stock markets (2014)
  387. Measuring multiscaling in financial time-series (2016)
  388. Multifractality in Asset Returns: Theory and Evidence (2002)
  389. The Hurst exponent over time: testing the assertion that emerging markets are becoming more efficient (2004)
  390. Hurst exponent and prediction based on weak-form efficient market hypothesis of stock markets (2008)
  391. The impact of 2008 and 2010 financial crises on the Hurst exponents of international stock markets: implications for efficiency and contagion (2014)
  392. Trading Volume and Serial Correlation in Stock Returns* (1993)
  393. Simulation of Multifractional Brownian Motion (1998)
  394. Exploiting stochastic dominance to generate abnormal stock returns (2014)
  395. Bayesian Estimation of the Multifractality Parameter for Image Texture Using a Whittle Approximation (2015)
  396. Simulation of multifractional Brownian motion (1998)
  397. An examination of the complementary volume-volatility information theories (2008)
  398. Commonality in liquidity (2001)
  399. Market Liquidity and Trading Activity (2001)
  400. Theory-Based Illiquidity and Asset Pricing (2009)
  401. Causality in quantiles and dynamic stock return–volume relations (2009)
  402. The bivariate GARCH approach to investigating the relation between stock returns, trading volume, and return volatility (2012)
  403. Inf 'e rence statistique pour les mouvements browniens fractionnaires et multifractionnaires (2000)
  404. Identification of multifractional Brownian motion (2005)
  405. Hurst exponent estimation of locally self-similar Gaussian processes using sample quantiles (2008)
  406. Estimating the parameters of a fractional Brownian motion by discrete variations of its Sample Paths (2001)
  407. From Self-Similarity to Local Self-Similarity: the Estimation Problem (1999)
  408. Weighted Kolmogorov-Smirnov test: Accounting for the tails (2012) — link
  409. Scaling in Stock Market Data: Stable Laws and Beyond (1997)
  410. An Empirical Analysis of Stock and Bond Market Liquidity (2005)
  411. Multifractional Stochastic Volatility Models (2014)
  412. Structural causes of the global financial crisis: a critical assessment of the ‘new financial architecture’ (2009)
  413. Maximum of a Fractional Brownian Motion: Analytic Results from Perturbation Theory. (2015)
  414. Extreme-value statistics of fractional Brownian motion bridges (2016)
  415. Intraday trading volume and return volatility of the DJIA stocks: A note (2003)
  416. Stochastic Analysis of the Fractional Brownian Motion (1999)
  417. Liquidity Risk Premia in Corporate Bond Markets (2012)
  418. Scaling behaviors in differently developed markets (2003)
  419. Fractal Market Hypothesis and Markov Regime Switching Model: A Possible Synthesis and Integration (2018)
  420. A Composite Index for Measuring Stock Market Inefficiency (2022)
  421. Imprints of log-periodic self-similarity in the stock market (1999)
  422. The Stochastic Dependence of Security Price Changes and Transaction Volumes: Implications for the Mixture-of-Distributions Hypothesis (1976)
  423. Market efficiency, long-term returns, and behavioral finance (1998)
  424. Trading activity, realized volatility and jumps (2010)
  425. Measuring the self-similarity exponent in L'evy stable processes of financial time series (2013)
  426. The modified mixture of distributions model: a revisit (2006)
  427. Fractal stock markets: International evidence of dynamical (in)efficiency (2017)
  428. Stock Prices and Volume (1992)
  429. The High Volume Return Premium (2001)
  430. Spectral Analysis of New York Stock Market prices (1963)
  431. Liquidity and execution costs in equity markets (1988)
  432. Liquidity and execution costs in equity markets (1988)
  433. Testing for Linear and Nonlinear Granger Causality in the Stock Price-Volume Relation (1994)
  434. An example of infinite dimensional quasi-helix (2003)
  435. Volume and skewness in international equity markets (2008)
  436. Estimating serial correlation and self-similarity in financial time series—A diversification approach with applications to high frequency data (2015)
  437. The intraday relationship between volume and volatility in LIFFE futures markets (1999)
  438. Common factors in prices, order flows, and liquidity. (2001)
  439. Approach to an irregular time series on the basis of the fractal theory (1988)
  440. A Unified Theory of Underreaction, Momentum Trading, and Overreaction in Asset Markets (1999)
  441. Some Lessons from the Financial Market Crisis* (2009)
  442. Variations quadratiques et estimation de l’exposant de H "older local d’un processus gaussien (1997)
  443. An equilibrium model of asset trading with sequential information arrival (1981)
  444. Information dissemination and portfolio choice (1983)
  445. The relationship between trading activity and stock market volatility: Does the volume threshold matter? (2019)
  446. Trading intensity and the volume-volatility relationship on the Tunis Stock Exchange (2018)
  447. Some fractional and multifractional Gaussian processes: A brief introduction (2015)
  448. Arbitrage Asymmetry and the Idiosyncratic Volatility Puzzle (2015)
  449. Prospect Theory: An Analysis of Decision under Risk (1979)
  450. Investment implications of the Fractal Market Hypothesis (2019)
  451. Intraday relationships between volatility in S&P500 futures prices and volatility in the S&P500 index (1990)
  452. Estimating the fractal dimension of a locally selfsimilar G aussian process using increments (1997)
  453. Return-volume dynamics in futures markets (1998)
  454. Volume and Volatility Surrounding Quarterly Redesignation of the Lead S&P 500 Futures Contract (2001)
  455. Heteroskedasticity in Stock Return Data: Volume versus GARCH Effects (1990)
  456. Introduction to the multifractal analysis of images (1998)
  457. Fractal and multifractal analysis: A review (2009)
  458. Testing the mixture-of-distributions hypothesis using “realized” volatility (2003)
  459. An algorithm for the characterization of time-series based on local regularity (2007)
  460. An Empirical Study of Volatility and Trading Volume Dynamics Using High-Frequency Data (2010)
  461. Multifractal Models in Finance: Their Origin, Properties and Applications (forthcoming)
  462. Fractals and Scaling in Finance. Discontinuity, Concentration, Risk Selecta Volume E (1997)
  463. An Introduction to Econophysics. Correlations and Complexity in Finance (2004)
  464. A Simple Model of Capital Market Equilibrium with Incomplete Information (1987)
  465. Identifying periods of market inefficiency for return predictability (2017)
  466. Dynamical generalized Hurst exponent as a tool to monitor unstable periods in financial time series (2012)
  467. Wavelets and multifractal formalism for singular signals: Application to turbulence data (1991)
  468. Surprising information, the MDH, and the relationship between volatility and trading volume (2010)
  469. A general class of multifractional processes and stock price informativeness (2018) — link
  470. Fractal market analysis – Applying chaos theory to investment and analysis (1994)
  471. A new method for estimating the parameter of fractional Brownian motion (1994)
  472. Multifractional Brownian Motion: Definition and Preliminary Results (1995)
  473. CED model for asset returns and fractal market hypothesis (1999)
  474. An Analysis of U.S. Stock Price Behavior using Wavelets (1995)
  475. Continuous martingales and Brownian motion (1999)
  476. Multifractal Processes (2002)
  477. Parameter estimation for fractional O rnstein– U hlenbeck processes: non-ergodic case (2011)
  478. Necessary and sufficient conditions for Hölder continuity of Gaussian processes (2014)
  479. Why Does Stock Market Volatility Change Over Time? (1989)
  480. Behavioral Capital Asset Pricing Theory (1994)
  481. Behavioral Portfolio Theory (2000)
  482. THE BEHAVIORAL LIFE-CYCLE HYPOTHESIS (1988)
  483. An empirical analysis of the stock price-volume relationship (1988)
  484. Risk Finance and Asset Pricing (2010)
  485. The Price Variability-Volume Relationship on Speculative Markets (1983)
  486. On some generalization of fractional Brownian motions (2006)
  487. Fractal market hypothesis and two power-laws (2000)
  488. Quantitative MR Image Analysis for Brain Tumor (2018)
  489. New Estimators of the Hurst Index for Fractional Brownian Motion (2014)
  490. Texture Estimation for Abnormal Tissue Segmentation in Brain MRI (2016)
  491. Not all estimators are born equal: The empirical properties of some estimators of long memory (2013)
  492. Estimators for Long-Range Dependence: an empirical study (1995)
  493. Time and dynamic volume$-$volatility relation (2006)
  494. Scaling, self-similarity and multifractality in FX markets (2003)
  495. Trading volume, bid–ask spread, and price volatility in futures markets (2000)
  496. The Relationship between Volatility and Trading Volume in the Chinese Stock Market: A Volatility Decomposition Perspective (2012)
  497. An Investigation of Transactions Data for NYSE Stocks (1985)
  498. ARFBF model for non stationary random fields and application in HRTEM images (2015)
  499. Intraday futures patterns and volume–volatility relationships: the German evidence (2014)
  500. New dynamics between volume and volatility (2019)
  501. Volatility is rough (2018)
  502. Estimating variance from high, low and closing prices (1991)
  503. Decoupling the Short- and Long-Term Behavior of Stochastic Volatility (2021) — link
  504. Rough volatility: Evidence from option prices (2018) — link
  505. A GMM approach to estimate the roughness of stochastic volatility (2022)
  506. Multifractional Stochastic Volatility Models (2014)
  507. Regularity of multifractional moving average processes with random Hurst exponent (2021)
  508. Pricing under rough volatility (2016)
  509. Volatility is rough (2014)
  510. Selfsimilar Processes (2002)
  511. Lack of robustness in two tests of normality against autocorrelation in sample data (1992)
  512. Goodness-of-fit tests with dependent observations (2011)
  513. Generalized Ornstein–Uhlenbeck processes and associated self-similar processes (2003)
  514. On fractional Ornstein-Uhlenbeck processes (2011)
  515. Roughening Heston (2019)
  516. From rough to multifractal volatility: The log S-fBM model (2022)
  517. The I nterpretation and E stimation of E ffective S ample S ize (1984)
  518. The frequency distribution of the annual maximum (or minimum) values of meteorological elements (1955)
  519. Simulation of stationary Gaussian processes in [0, 1] d (1994)
  520. S pectral R epresentation of F ractional B rownian M otion in $n$ D imensions and its P roperties (1995)
  521. P ortfolio S election (1952)
  522. B emerkungen zum B egriff der D ifferential- Q uotentcn ge- hrochener O rdnung (1967)
  523. N onhomogeneous fractional integration and multifractional processes (2008)
  524. E stimating Q uadratic V ariation U sing R ealized V ariance (2002)
  525. N onlinear biases in the roughness of a F ractional S tochastic R egularity M odel (2023)
  526. N onlinearity of the volume–volatility correlation filtered through the pointwise H urst– H "o lder regularity (2023)
  527. Assessing Risk in Graphically Presented Financial Series (2016)
  528. Does the Stock Market Overreact? (1985)
  529. Fractal analysis of surface roughness by using spatial data (1999)
  530. Wavelets Tools for Science & Technology (2001)
  531. Fractal Geometry: Mathematical Fondations and Applications (1990)
  532. Stable non-Gaussian random processes (1994)
  533. Curves and Fractal dimensions (1995)
  534. Sur certains processus stochastiques homogènes (1939)
  535. C oherent M easures of R isk (1999)
  536. Packing dimension of the image of fractional Brownian motion (1997) — link
  537. Stochastic Calculus for Fractional Brownian Motion and Related Processes. In: Lecture Notes in Mathematics (2008)
  538. From F air P rice to F air V olatility: T owards an E fficiency- C onsistent D efinition of F inancial R isk. (2025)

2 of 546 cited works are in the corpus; resolved via arXiv source bundles, Semantic Scholar and OpenAlex. Corpus-wide: 2518 linked papers.