Abstract
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.
Highlights
The fractional Brownian motion is a very popular modeling object in many fields such as hydrology, see for example [29], telecommunications and network traffic, see [23, 28] among others and finance, see the seminal paper [11]
Rough volatility models are becoming increasingly popular in quantitative finance
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
Summary
The fractional Brownian motion (fBm for short) is a very popular modeling object in many fields such as hydrology, see for example [29], telecommunications and network traffic, see [23, 28] among others and finance, see the seminal paper [11]. In this work, we wish to build a suitable sequence of normalized fBms and describe its limit as H goes to zero This will lead us to a possible definition of the fractional Brownian motion for H = 0. We show in this paper that the limit of XH as H goes to zero is “almost" a log-correlated Gaussian field, see Section 2 for an accurate result. The multifractal random walk model for the log-price of an asset in [1] satisfies such property It is defined as Yt = BM([0,t]), where B is a Brownian motion and t. We introduce our main theorem, that is an accurate statement about the convergence of the normalized fBm towards a LGF as H goes to zero.
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