Market Origins of Rough Volatility

Definitions and mathematical statement

For a parameter $H\in(0,1)$, a normalized fractional Brownian motion $(B_t^H)_{t\geq 0}$ is a centered real Gaussian process with continuous paths and covariance \[ \mathbb{E}[B_t^H B_s^H] =\frac12\left(t^{2H}+s^{2H}-|t-s|^{2H}\right), \qquad s,t\geq0. \] Its increments satisfy \[ \mathbb{E}\left[(B_{t+h}^H-B_t^H)^2\right]=h^{2H}, \qquad t\geq0,\quad h>0. \] Here $H$ is the Hurst parameter. The regime $0<H<\tfrac12$ is conventionally called rough in volatility modeling; the supplied problem says only that $H$ is small and supplies neither a numerical value nor a precise range.

Research question. What mathematical or market-microstructural mechanisms explain the apparently universal roughness of financial volatility, commonly modeled using fractional Brownian motion with a small Hurst parameter $H$? In this question, apparent universality is an empirical motivation, not an established assumption or a mathematically quantified conclusion.

Unresolved formulation. The source does not specify the volatility observable, a class of market models, assumptions on market dynamics, the relevant time scales, a limiting procedure, or a criterion for agreement with fractional Brownian motion. In particular, it does not specify whether the fractional Brownian model describes volatility fluctuations, log-volatility, or another derived quantity. The Gaussian process defined above supplies a precise reference process, but does not itself identify a financial volatility process or a proposed mechanism. The source therefore poses an explanatory research question rather than a fully quantified conjecture. A precise theorem would require these choices and an explicit meaning of universality across markets; no such choices are imposed here.

Short English statement

Explain why financial volatility appears rough across markets and why models involving fractional Brownian motion with a small Hurst parameter fit this behavior. The source leaves the observable, mechanisms, scales, and meaning of universality unspecified, so this remains an explanatory research question requiring further mathematical formulation.

Sources

• Supplied local UnsolvedMath registry record: permanent ID $30003428$, title Market Origins of Rough Volatility, collection Oberwolfach Reports Open Problems, exact source problem code catalogue entry. No individual proposer is supplied; the proposed year is recorded as 2017.
• Supplied report citation: Mathematics of Quantitative Finance (2018), Oberwolfach Reports, DOI 10.4171/owr/2017/13, pp. 683--769. https://doi.org/10.4171/owr/2017/13. The supplied background describes Oberwolfach Reports as documenting mathematical workshops and their open-problem discussions. No exact problem page or named speaker is supplied. The citation's 2018 designation and the DOI's 2017 component are retained without resolving their relationship.
• Supplied literature reference, labeled "Benzaquen--Bouchaud--Mastromatteo, microstructural foundations": https://arxiv.org/abs/1609.05177. The label and link are retained as supplied; their correspondence has not been independently verified.
• Supplied literature reference, labeled "Gatheral--Jaisson--Rosenbaum, rough volatility evidence": https://arxiv.org/abs/1702.02777. The label and link are retained as supplied; their correspondence has not been independently verified.
• Literature assessment dated 2026-08-21: the supplied curation labels the problem open and states that microstructural and statistical models generate rough volatility and reproduce several stylized facts, while a universal origin of roughness across markets is not established. This is supplied dated curation derived from the OWR record, not an independently verified assessment; time-sensitive status and cited claims require independent verification.
• The registry supplies statement status corrected_verified, explained there as preservation of mathematical content while repairing notation, typography, and sentence structure. This is a supplied editorial designation, not independent mathematical or source verification of this draft.