Electronic Communications in Probability

Fractional Brownian motion with zero Hurst parameter: a rough volatility viewpoint

Eyal Neuman and Mathieu Rosenbaum

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Rough volatility models are becoming increasingly popular in quantitative finance. In this framework, one considers that the behavior of the log-volatility process of a financial asset is close to that of a fractional Brownian motion with Hurst parameter around 0.1. Motivated by this, we wish to define a natural and relevant limit for the fractional Brownian motion when $H$ goes to zero. We show that once properly normalized, the fractional Brownian motion converges to a Gaussian random distribution which is very close to a log-correlated random field.

Article information

Electron. Commun. Probab., Volume 23 (2018), paper no. 61, 12 pp.

Received: 1 November 2017
Accepted: 26 July 2018
First available in Project Euclid: 12 September 2018

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60G22: Fractional processes, including fractional Brownian motion 60G15: Gaussian processes 60G57: Random measures
Secondary: 60G18: Self-similar processes 28A80: Fractals [See also 37Fxx]

fractional Brownian motion log-correlated random field rough volatility multifractal processes

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Neuman, Eyal; Rosenbaum, Mathieu. Fractional Brownian motion with zero Hurst parameter: a rough volatility viewpoint. Electron. Commun. Probab. 23 (2018), paper no. 61, 12 pp. doi:10.1214/18-ECP158. https://projecteuclid.org/euclid.ecp/1536718014

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