Volatility Is Rough

Volatility Is Rough Jim Gatheral Thibault Jaisson Mathieu Rosenbaum

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Volatility Is Rough 2018 Quantitative Finance / arXiv 10.1080/14697688.2017.1393551 1410.3394 gatheral2018-rough-volatility gatheral2018-rough-volatility.txt, the project's staged copy b46d6deaa1606be58cddfcc3fc62b8f5ca13c5b0 arxiv-open quantitative-finance

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0 spans located by searching the stored text for the quote's exact words, allowing any whitespace between them existence Quotation withheld because this record is not cleared for redistribution. smoothness (roughness) of the volatility process Quotation withheld because this record is not cleared for redistribution. observational-study observational-study Quotation withheld because this record is not cleared for redistribution. mechanisms THE TERM'S TWO SENSES ARE SEPARATED BY THEIR HISTORY: 'long memory' initially meant the slow decay of the autocorrelation function - anything slower than exponential - and over time acquired the more precise meaning that the function is not integrable, and more precisely still that it decays as a power law with exponent less than one Quotation withheld because this record is not cleared for redistribution. mechanisms THE TECHNICAL SENSE IS DEFENDED AS THE ONLY ESTIMABLE ONE: the recent literature assumes long memory in the technical sense, and the authors add that meaningful results can probably ONLY be obtained under such a specification, since the asymptotic behaviour of the covariance function cannot be estimated without assuming a specific form for it Quotation withheld because this record is not cleared for redistribution. mechanisms THE QUESTION IS DECLARED UNANSWERABLE AS POSED, and the authors then supply a different instrument rather than a different answer: stated in terms of the asymptotic behaviour of the autocorrelation function the long memory question simply cannot be answered, but explicit expressions still allow the dependence structure to be analysed about this quote: Span re-sliced at drafting: the staged text splices footnote 1 AND a page number between 'This implies' and its own subordinate clause ('…see Section 2.1. 3 that when stated in term…'), so the quote begins after the splice. Extraction damage, not authorial. Damage class C. Quotation withheld because this record is not cleared for redistribution. mechanisms A WIDELY ACCEPTED STYLIZED FACT IS EXPLAINED AS AN ARTEFACT OF ITS OWN DETECTOR: the model has no long memory property, and yet the classical statistical procedures designed to detect volatility persistence TEND TO CONCLUDE THAT LONG MEMORY IS PRESENT in data the model generated Quotation withheld because this record is not cleared for redistribution. mechanisms the evidence for the stylized fact is dated rather than dismissed: the analyses supporting it use data that PREDATE high-frequency electronic trading, and the authors note the evidence has never satisfied the remaining doubters Quotation withheld because this record is not cleared for redistribution. mechanisms THE NEW NAME IS CHOSEN TO CARRY THE DEPARTURE: an existing model is adopted and renamed, with the new word picked to underline the single parameter inequality that separates it from its parent Quotation withheld because this record is not cleared for redistribution. mechanisms in the process used, ONE PARAMETER CONTROLS BOTH PROPERTIES AT ONCE - regularity and long memory stand in one-to-one correspondence through the Hurst parameter - so a claim about smoothness is inseparably a claim about memory Quotation withheld because this record is not cleared for redistribution. mechanisms A WIDESPREAD BELIEF IS MET WITH A COUNTEREXAMPLE RATHER THAN AN ARGUMENT: the model is offered as a counterexample to the widespread belief that the explosion of the volatility smile at short expiries implies the presence of jumps Quotation withheld because this record is not cleared for redistribution. mechanisms THE MODEL IS OFFERED AGAINST TWO SEPARATE REQUIREMENTS rather than one - consistency with the empirically observed properties of the volatility time series AND consistency with the shape of the volatility surface - and the claim is stated as POTENTIAL, not as achievement Quotation withheld because this record is not cleared for redistribution. mechanisms the smoothness estimate is reported as a RANGE ACROSS ASSETS rather than a single fitted number, and it is accompanied by two further empirical properties of the increments - approximate normality and a monofractal scaling of the moments - which together are what licenses the modelling choice about this quote: The span contains 'H¨older' for 'Hölder' - the diaeresis is emitted BEFORE its base letter by the PDF extraction. The quote is still an exact substring and anchors correctly; the damage is in what a reader of the tag sees, not in the provenance. None of the four shapes an internal tool looks for covers this, so it is annotated by hand. Quotation withheld because this record is not cleared for redistribution. the scaling result is established at usual observation time scales what this limits: the scaling result, over the range examined: AT LEAST AT USUAL TIME SCALES OF OBSERVATION Quotation withheld because this record is not cleared for redistribution. Smoothness estimates based on whole-trading-day realized variance are expected to be biased upwards because integration smooths the process. what this limits: the smoothness estimates specifically: WE EXPECT ESTIMATES OF THE SMOOTHNESS OF THE VOLATILITY PROCESS TO BE BIASED UPWARDS, INTEGRATION BEING A REG[ularising operation] - the direction of the bias is stated Quotation withheld because this record is not cleared for redistribution. Volatility is never observed directly, so the key quantity m(q, Δ) can only be computed from proxies. what this limits: every quantity computed in the paper: THE VOLATILITY PROCESS IS NOT DIRECTLY OBSERVABLE, AND AN EXACT COMPUTATION OF m(q, delta) IS NOT POSSIBLE IN PRACTICE Quotation withheld because this record is not cleared for redistribution. The paper only treats the volatility time series; whether the model is consistent with observed option prices is deferred. what this limits: the option-pricing implications: A MORE DETAILED ANALYSIS OF THE CONSISTENCY OF OUR MODEL WITH OPTION PRICES IS LEFT FOR A FUTURE ARTICLE Quotation withheld because this record is not cleared for redistribution. The property cannot be estimated without first assuming the functional form whose presence is at issue. what this limits: any estimate of long memory in volatility: the asymptotic behaviour of the covariance function cannot be estimated without assuming a specific form for it, so the specification is an input to the result rather than an output Quotation withheld because this record is not cleared for redistribution. As the term is now defined, the field's central question about volatility cannot be answered at all. what this limits: the debate over whether volatility has long memory: stated in terms of the asymptotic behaviour of the autocorrelation function, the paper says the question simply cannot be answered Quotation withheld because this record is not cleared for redistribution. Standard procedures report the property in data generated by a model that does not have it, which the authors offer as the reason the field accepted it as a stylized fact. what this limits: the widely accepted stylized fact that volatility has long memory: classical procedures for detecting volatility persistence conclude it is present in data simulated from a model that has no long memory property at all Quotation withheld because this record is not cleared for redistribution. fractional Brownian motion with a Hurst exponent below 1/2 used as the model of log-volatility, in place of the standard semimartingale volatility models of finance Quotation withheld because this record is not cleared for redistribution. a detailed test of the model against observed option prices - NOT performed, and deferred to a future article eliminates: the model fitting volatility time series while failing on the option prices it would be used to compute, which is the comparison the paper itself says its predecessors fail · outcome (stated by the tagger, no quotation): absent-declared-unreported Quotation withheld because this record is not cleared for redistribution. fractional Brownian motion fractional Brownian motion Quotation withheld because this record is not cleared for redistribution. realized volatility realized volatility Quotation withheld because this record is not cleared for redistribution. long memory long memory Quotation withheld because this record is not cleared for redistribution. Hurst parameter Hurst parameter Quotation withheld because this record is not cleared for redistribution. spot volatility spot volatility Quotation withheld because this record is not cleared for redistribution. volatility surface volatility surface Quotation withheld because this record is not cleared for redistribution. continuous semimartingales continuous semimartingales Quotation withheld because this record is not cleared for redistribution. volatility process volatility process Quotation withheld because this record is not cleared for redistribution. Black-Scholes framework Black-Scholes framework Quotation withheld because this record is not cleared for redistribution. deterministic function of time deterministic function of time Quotation withheld because this record is not cleared for redistribution. local volatility model local volatility model Quotation withheld because this record is not cleared for redistribution. time-inhomogeneous time-inhomogeneous Quotation withheld because this record is not cleared for redistribution. future volatility surfaces future volatility surfaces Quotation withheld because this record is not cleared for redistribution. exotic options exotic options Quotation withheld because this record is not cleared for redistribution. stochastic volatility models stochastic volatility models Quotation withheld because this record is not cleared for redistribution. continuous Brownian semi-martingale continuous Brownian semi-martingale Quotation withheld because this record is not cleared for redistribution. local volatility dynamics local volatility dynamics Quotation withheld because this record is not cleared for redistribution. stochastic volatility dynamics stochastic volatility dynamics Quotation withheld because this record is not cleared for redistribution. local-stochastic-volatility (LSV) models local-stochastic-volatility (LSV) models Quotation withheld because this record is not cleared for redistribution. reasonable dynamics reasonable dynamics Quotation withheld because this record is not cleared for redistribution. slow decay of the autocorrelation function slow decay of the autocorrelation function Quotation withheld because this record is not cleared for redistribution. not integrable not integrable Quotation withheld because this record is not cleared for redistribution. power-law with exponent less than 1 power-law with exponent less than 1 Quotation withheld because this record is not cleared for redistribution. asymptotic behavior of the covariance function asymptotic behavior of the covariance function Quotation withheld because this record is not cleared for redistribution. usual time scales of observation usual time scales of observation Quotation withheld because this record is not cleared for redistribution. dependence structure dependence structure about this quote: PAGE FURNITURE IS SPLICED INTO THE 'evidence' QUOTE - the text of footnote 1 ('1Actually H corresponds…') and a page number, between 'This implies' and 'that when stated'. The words on either side of it were never continuous on the page; read the quote with it deleted. Quotation withheld because this record is not cleared for redistribution. implied volatility implied volatility Quotation withheld because this record is not cleared for redistribution. volatility parameter volatility parameter Quotation withheld because this record is not cleared for redistribution. market price of that option market price of that option Quotation withheld because this record is not cleared for redistribution. strike price strike price Quotation withheld because this record is not cleared for redistribution. time to expiry time to expiry Quotation withheld because this record is not cleared for redistribution. power-law function of time to expiry power-law function of time to expiry Quotation withheld because this record is not cleared for redistribution. term structure of at-the-money (ATM) skew term structure of at-the-money (ATM) skew Quotation withheld because this record is not cleared for redistribution. sum of decaying exponentials sum of decaying exponentials Quotation withheld because this record is not cleared for redistribution. explosion of the volatility smile explosion of the volatility smile Quotation withheld because this record is not cleared for redistribution. presence of jumps presence of jumps Quotation withheld because this record is not cleared for redistribution. ATM volatility skew ATM volatility skew Quotation withheld because this record is not cleared for redistribution. smoothness parameter smoothness parameter Quotation withheld because this record is not cleared for redistribution. increments of the log-volatility increments of the log-volatility Quotation withheld because this record is not cleared for redistribution. approximately normally distributed approximately normally distributed Quotation withheld because this record is not cleared for redistribution. monofractal scaling property monofractal scaling property Quotation withheld because this record is not cleared for redistribution. fractional stochastic volatility (FSV) model fractional stochastic volatility (FSV) model Quotation withheld because this record is not cleared for redistribution. Rough FSV (RSFV) Rough FSV (RSFV) Quotation withheld because this record is not cleared for redistribution. Rough FSV Rough FSV Quotation withheld because this record is not cleared for redistribution. RFSV RFSV Quotation withheld because this record is not cleared for redistribution. FSV FSV Quotation withheld because this record is not cleared for redistribution. Rough FSV (RFSV) Rough FSV (RFSV) Quotation withheld because this record is not cleared for redistribution. RFSV model RFSV model Quotation withheld because this record is not cleared for redistribution. improved forecasts of realized volatility improved forecasts of realized volatility Quotation withheld because this record is not cleared for redistribution. volatility persistence volatility persistence Quotation withheld because this record is not cleared for redistribution. classical statistical procedures classical statistical procedures Quotation withheld because this record is not cleared for redistribution. RFSV volatility forecasts RFSV volatility forecasts Quotation withheld because this record is not cleared for redistribution. conventional AR and HAR volatility forecasts conventional AR and HAR volatility forecasts Quotation withheld because this record is not cleared for redistribution. market microstructure explanation market microstructure explanation Quotation withheld because this record is not cleared for redistribution. high degree of endogeneity of the market high degree of endogeneity of the market Quotation withheld because this record is not cleared for redistribution. algorithmic trading algorithmic trading Quotation withheld because this record is not cleared for redistribution. integrated variance integrated variance Quotation withheld because this record is not cleared for redistribution. model with uncertainty zones model with uncertainty zones Quotation withheld because this record is not cleared for redistribution. very short time intervals very short time intervals Quotation withheld because this record is not cleared for redistribution. daily realized variance estimates daily realized variance estimates Quotation withheld because this record is not cleared for redistribution. Oxford-Man Institute of Quantitative Finance Realized Library Oxford-Man Institute of Quantitative Finance Realized Library Quotation withheld because this record is not cleared for redistribution. integration integration Quotation withheld because this record is not cleared for redistribution. biased upwards biased upwards Quotation withheld because this record is not cleared for redistribution. smoothness parameter sq smoothness parameter sq Quotation withheld because this record is not cleared for redistribution. regressing log m(q, ∆) against log ∆ regressing log m(q, ∆) against log ∆ Quotation withheld because this record is not cleared for redistribution. fractional Ornstein-Uhlenbeck process fractional Ornstein-Uhlenbeck process Quotation withheld because this record is not cleared for redistribution. mathematical tractability mathematical tractability Quotation withheld because this record is not cleared for redistribution. reasonableness of the model at very large times reasonableness of the model at very large times Quotation withheld because this record is not cleared for redistribution. very long reversion time scale very long reversion time scale Quotation withheld because this record is not cleared for redistribution. choice H < 1/2 choice H < 1/2 Quotation withheld because this record is not cleared for redistribution. observed smoothness of the volatility process observed smoothness of the volatility process Quotation withheld because this record is not cleared for redistribution. term structure of volatility skew term structure of volatility skew Quotation withheld because this record is not cleared for redistribution. mean reversion by practitioners mean reversion by practitioners Quotation withheld because this record is not cleared for redistribution. fBM like-behavior of the log-volatility fBM like-behavior of the log-volatility Quotation withheld because this record is not cleared for redistribution. stationarity property stationarity property Quotation withheld because this record is not cleared for redistribution. one-hour window one-hour window Quotation withheld because this record is not cleared for redistribution. daily eight-hour windows daily eight-hour windows Quotation withheld because this record is not cleared for redistribution. regularizing effect of the integral operator regularizing effect of the integral operator Quotation withheld because this record is not cleared for redistribution. volatility of volatility volatility of volatility Quotation withheld because this record is not cleared for redistribution. persistent periods of high volatility persistent periods of high volatility Quotation withheld because this record is not cleared for redistribution. low volatility periods low volatility periods Quotation withheld because this record is not cleared for redistribution. fractal-type behavior fractal-type behavior Quotation withheld because this record is not cleared for redistribution. range of observation scales range of observation scales Quotation withheld because this record is not cleared for redistribution. volatility process over one day volatility process over one day Quotation withheld because this record is not cleared for redistribution. volatility process over a decade volatility process over a decade Quotation withheld because this record is not cleared for redistribution. straight lines with slope 1.86 straight lines with slope 1.86 Quotation withheld because this record is not cleared for redistribution. power law behavior of the autocorrelation function power law behavior of the autocorrelation function Quotation withheld because this record is not cleared for redistribution. wrongly convinced wrongly convinced Quotation withheld because this record is not cleared for redistribution. classical estimation procedures classical estimation procedures Quotation withheld because this record is not cleared for redistribution. spurious long memory of volatility spurious long memory of volatility Quotation withheld because this record is not cleared for redistribution. classical power law sense classical power law sense Quotation withheld because this record is not cleared for redistribution. RFSV forecast RFSV forecast Quotation withheld because this record is not cleared for redistribution. AR and HAR forecasts AR and HAR forecasts Quotation withheld because this record is not cleared for redistribution. longer horizons longer horizons Quotation withheld because this record is not cleared for redistribution. more parsimonious more parsimonious Quotation withheld because this record is not cleared for redistribution. forecast time horizon forecast time horizon Quotation withheld because this record is not cleared for redistribution. relevant regression window relevant regression window Quotation withheld because this record is not cleared for redistribution. forecasting horizon forecasting horizon Quotation withheld because this record is not cleared for redistribution. volatility over the last week volatility over the last week Quotation withheld because this record is not cleared for redistribution. volatility over the last month volatility over the last month Quotation withheld because this record is not cleared for redistribution. Hawkes processes Hawkes processes Quotation withheld because this record is not cleared for redistribution. extensions of Poisson processes extensions of Poisson processes Quotation withheld because this record is not cleared for redistribution. intensity at a given time intensity at a given time Quotation withheld because this record is not cleared for redistribution. past jumps past jumps Quotation withheld because this record is not cleared for redistribution. point process point process Quotation withheld because this record is not cleared for redistribution. kernel kernel Quotation withheld because this record is not cleared for redistribution. L1 norm L1 norm Quotation withheld because this record is not cleared for redistribution. power law tail power law tail Quotation withheld because this record is not cleared for redistribution. degree of endogeneity of the market degree of endogeneity of the market Quotation withheld because this record is not cleared for redistribution. one given order one given order Quotation withheld because this record is not cleared for redistribution. many other orders many other orders Quotation withheld because this record is not cleared for redistribution. electronic high frequency trading electronic high frequency trading Quotation withheld because this record is not cleared for redistribution. splitting of large orders splitting of large orders Quotation withheld because this record is not cleared for redistribution. metaorder metaorder Quotation withheld because this record is not cleared for redistribution. superposition effect superposition effect Quotation withheld because this record is not cleared for redistribution. irregular volatility irregular volatility Quotation withheld because this record is not cleared for redistribution. macroscopic scaling limit of Hawkes processes macroscopic scaling limit of Hawkes processes Quotation withheld because this record is not cleared for redistribution. integrated fractional process integrated fractional process Quotation withheld because this record is not cleared for redistribution. cumulated order flow cumulated order flow Quotation withheld because this record is not cleared for redistribution. Rough Fractional Stochastic Volatility (RFSV) model Rough Fractional Stochastic Volatility (RFSV) model Quotation withheld because this record is not cleared for redistribution. FSV model FSV model Quotation withheld because this record is not cleared for redistribution. one major difference one major difference Quotation withheld because this record is not cleared for redistribution. RFSV model → improved forecasts of realized volatility The RFSV model enables improved forecasts of realized volatility. Enables is absent from the eight governed types and is not direct physical causation. Quotation withheld because this record is not cleared for redistribution. classical statistical procedures → long memory Classical persistence diagnostics tend to conclude long memory in data generated by a model that has none. Misidentifies-as is absent from the edge vocabulary. Quotation withheld because this record is not cleared for redistribution. RFSV forecast → AR and HAR forecasts The RFSV forecast consistently outperforms AR and HAR, especially at longer horizons. Outperforms is absent from the eight types and contrast alone would discard the direction of performance. Quotation withheld because this record is not cleared for redistribution. relevant regression window → forecasting horizon The relevant history window is linear in the forecasting horizon. Proportional-to is absent from the governed edge types. Quotation withheld because this record is not cleared for redistribution. macroscopic scaling limit of Hawkes processes → integrated fractional process The near-critical long-tail Hawkes scaling limit can be seen as an integrated fractional process. Scaling-limit-is-represented-by is absent and a plain is-a would overstate an asymptotic result. Quotation withheld because this record is not cleared for redistribution. cumulated order flow → integrated variance The paper reports a linear relation between cumulated order flow and integrated variance. Correlates-with/linearly-related-to is absent from the eight types. Quotation withheld because this record is not cleared for redistribution. long memory - SENSE 1, THE ORIGINAL AND LOOSE ONE: slow decay of the autocorrelation function, meaning anything slower than exponential Quotation withheld because this record is not cleared for redistribution. long memory - SENSE 2, THE LATER TECHNICAL ONE the field drifted to: a non-integrable autocorrelation function, more precisely one decaying as a power law with exponent below 1, and the recent literature assumes this sense Quotation withheld because this record is not cleared for redistribution. volatility - AS A PROXY, ALWAYS: the process is not directly observable, so every quantity here is computed from an estimated daily stand-in taken at a fixed time of day Quotation withheld because this record is not cleared for redistribution. Local volatility - exact European-option fit with time-inhomogeneous and unrealistic future dynamics Quotation withheld because this record is not cleared for redistribution. Conventional stochastic volatility - more realistic dynamics without exact consistency with observed European option prices Quotation withheld because this record is not cleared for redistribution. Local-stochastic volatility - market hybrid targeting both exact fit and reasonable dynamics Quotation withheld because this record is not cleared for redistribution. Implied volatility - the Black-Scholes volatility input required to reproduce an option's market price Quotation withheld because this record is not cleared for redistribution. Volatility surface - implied volatility plotted across strike price and time to expiry Quotation withheld because this record is not cleared for redistribution. Short-maturity volatility-smile explosion - reproducible without jumps when volatility is fractionally rough Quotation withheld because this record is not cleared for redistribution. Rough FSV - fractional stochastic volatility with H below one half rather than the long-memory H above one half Quotation withheld because this record is not cleared for redistribution. Spot-volatility observation - intrinsically latent and therefore replaced by daily fixed-time proxies Quotation withheld because this record is not cleared for redistribution. Whole-day realized variance - a regularized spot proxy expected to bias smoothness upward Quotation withheld because this record is not cleared for redistribution. Stationary rough-volatility specification - fractional Ornstein-Uhlenbeck process with a very long reversion scale Quotation withheld because this record is not cleared for redistribution. Practitioner mean reversion - unusually high volatility tends down and unusually low volatility tends up Quotation withheld because this record is not cleared for redistribution. Window-length bias - longer integration raises estimated H and lowers estimated volatility-of-volatility Quotation withheld because this record is not cleared for redistribution. RFSV scale resemblance - a daily volatility path qualitatively resembles a decade-long path Quotation withheld because this record is not cleared for redistribution. Spurious memory diagnosis - standard estimators report power-law memory on simulations whose model has none Quotation withheld because this record is not cleared for redistribution. RFSV forecast parsimony - one H parameter rather than horizon-dependent AR/HAR coefficients about this quote: PAGE FURNITURE IS SPLICED INTO THE 'evidence' QUOTE - the text of footnotes 13 and 14 and a page number, between 'requires the parameter' and 'H to forecast'. The words on either side of it were never continuous on the page; read the quote with it deleted. Quotation withheld because this record is not cleared for redistribution. Forecast-history rule - past window approximately matches the future horizon Quotation withheld because this record is not cleared for redistribution. Hawkes endogeneity - one order triggers many more as algorithms respond to other participants' orders Quotation withheld because this record is not cleared for redistribution. Order splitting - many observed orders belong to a slowly executed metaorder Quotation withheld because this record is not cleared for redistribution. Microstructural roughness mechanism - near-critical endogenous feedback plus long-tailed order influence produces a fractional macroscopic limit Quotation withheld because this record is not cleared for redistribution. builds-on → fractional Brownian motion (from mathematics/physics) Quotation withheld because this record is not cleared for redistribution. builds-on → the fractional stochastic volatility (FSV) model of Comte and Renault [16] Quotation withheld because this record is not cleared for redistribution. extends → the FSV model of Comte and Renault [16], modified to H < 1/2 Quotation withheld because this record is not cleared for redistribution. contradicts → the accepted stylized fact of power-law long memory in volatility Quotation withheld because this record is not cleared for redistribution. replicates → the stylized fact of near-Gaussian log-volatility increments reported by Andersen et al. ([2], [3]) Quotation withheld because this record is not cleared for redistribution. builds-on → Fukasawa [25] on the ATM volatility skew generated by fBM-driven volatility Quotation withheld because this record is not cleared for redistribution. long memory —operationalised-as→ Hurst parameter long memory Hurst parameter strength of the relation: 1 Quotation withheld because this record is not cleared for redistribution. spot volatility —operationalised-as→ realized volatility spot volatility realized volatility strength of the relation: 1 Audit 2026-09-28: DOUBTFUL - read against its full span and kept, but the span writes 'daily spot variances' and 'daily realized variance estimates', not 'spot volatility' / 'realized volatility'. Quotation withheld because this record is not cleared for redistribution. stochastic volatility dynamics —contrasted-with→ local volatility dynamics stochastic volatility dynamics local volatility dynamics the relation itself is separately evidenced · strength of the relation: 1 · note: The stochastic dynamics are said to be more realistic than local-volatility dynamics, though still price-inconsistent. Quotation withheld because this record is not cleared for redistribution. power-law function of time to expiry —contrasted-with→ sum of decaying exponentials power-law function of time to expiry sum of decaying exponentials the relation itself is separately evidenced · strength of the relation: 1 · note: Observed skew is a power law, in contrast to the conventional model's constant/exponential shape. Quotation withheld because this record is not cleared for redistribution. Rough FSV —same-as→ RFSV Rough FSV RFSV the relation itself is separately evidenced · strength of the relation: 1 · note: The model name is immediately followed by its acronym in parentheses. Quotation withheld because this record is not cleared for redistribution. Rough FSV (RFSV) —distinct-from→ FSV Rough FSV (RFSV) FSV the relation itself is separately evidenced · strength of the relation: 1 · note: The paper explicitly contrasts the rough H below one-half regime with FSV. Quotation withheld because this record is not cleared for redistribution. choice H < 1/2 —causes→ observed smoothness of the volatility process choice H < 1/2 observed smoothness of the volatility process the relation itself is separately evidenced · strength of the relation: 1 · note: The choice enables the model to reproduce the observed smoothness. Quotation withheld because this record is not cleared for redistribution. choice H < 1/2 —causes→ term structure of volatility skew choice H < 1/2 term structure of volatility skew the relation itself is separately evidenced · strength of the relation: 1 · note: The same choice enables agreement with the observed skew term structure. Quotation withheld because this record is not cleared for redistribution. one given order —causes→ many other orders one given order many other orders the relation itself is separately evidenced · strength of the relation: 1 · note: The paper states that one order endogenously generates many others. Quotation withheld because this record is not cleared for redistribution. superposition effect —causes→ irregular volatility superposition effect irregular volatility the relation itself is separately evidenced · strength of the relation: 0.6 — the record marks this weak · note: The authors say they believe the combined phenomena induce irregular volatility. Audit 2026-09-28: DOUBTFUL - read against its full span and kept, but 'We believe' the two phenomena lead to the superposition effect; hedged. Quotation withheld because this record is not cleared for redistribution. Rough Fractional Stochastic Volatility (RFSV) model —distinct-from→ FSV model Rough Fractional Stochastic Volatility (RFSV) model FSV model the relation itself is separately evidenced · strength of the relation: 1 · note: The models are separated by the explicitly named major difference H below versus above one half. Quotation withheld because this record is not cleared for redistribution. H ≈ 0.1 — Hurst exponent of log-volatility (H<1/2 means rough) Quotation withheld because this record is not cleared for redistribution. H = 1/2 — the classical Brownian-motion boundary case Quotation withheld because this record is not cleared for redistribution. Log-volatility acts essentially like fractional Brownian motion with a Hurst exponent near 0.1 at any reasonable time scale, which leads to the Rough FSV (RFSV) model with H < 1/2. Quotation withheld because this record is not cleared for redistribution. Although volatility is NOT long-memory under RFSV, classical persistence tests spuriously detect long memory in RFSV-generated data — explaining why long memory became an accepted stylized fact. Quotation withheld because this record is not cleared for redistribution. Hurst parameter H Hurst parameter H Quotation withheld because this record is not cleared for redistribution. RFSV model RFSV model Quotation withheld because this record is not cleared for redistribution. EVIDENCE CORRECTED (RE-GROUNDED) · fractional Brownian motion fractional Brownian motion EVIDENCE CORRECTED: RE-GROUNDED. The old span welded 'model the log of volatility using' - Gatheral's OWN model, a fBM with H < 1/2 - to 'motion (fBM for…' 5,695 characters EARLIER, which belongs to Comte and Renault's H > 1/2 model. It conflated the two Hurst regimes, which is the single distinction this paper exists to draw. Quotation withheld because this record is not cleared for redistribution. mean mean about this quote: THE QUOTE STARTS INSIDE A FORMULA AND ENDS IN A TABLE: it opens on the tail of the flattened expectation symbol that its sentence defines; this copy places that sentence's last words (the period the mean is taken over) before the quote; and after the quote's last words the text runs into the labels of the forecasting results table. Quotation withheld because this record is not cleared for redistribution. regression regression Quotation withheld because this record is not cleared for redistribution. sample-size sample-size about this quote: QUOTE REPAIRED: the extractor split a hyphen-space and the quote was recorded faithfully with the damage in it, which made it unanchorable. The repaired string is an EXACT SUBSTRING of the staged text - that is why it was accepted, not because a rule produced it. The text as originally recorded is kept in evidence as extracted. Quotation withheld because this record is not cleared for redistribution. standard-error standard-error Quotation withheld because this record is not cleared for redistribution. variance variance about this quote: A FOOTNOTE CALL SITS INSIDE THIS QUOTE: the '6' between 'the S&P and' and 'NASDAQ indices' is the paper's footnote marker, not part of the sentence; read the quote with it deleted. 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The full text of this article is not redistributable, so it is not included here. The evidence sentences are withheld with it; the tag layer and its grounding status remain. The article itself is at https://doi.org/10.1080/14697688.2017.1393551.