An intuitive introduction to fractional and rough volatilities
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- dc.contributor.author Alòs, Elisa
- dc.contributor.author León, Jorge A.
- dc.date.accessioned 2023-06-15T06:02:38Z
- dc.date.available 2023-06-15T06:02:38Z
- dc.date.issued 2021
- dc.description.abstract Here, we review some results of fractional volatility models, where the volatility is driven by fractional Brownian motion (fBm). In these models, the future average volatility is not a process adapted to the underlying filtration, and fBm is not a semimartingale in general. So, we cannot use the classical Itô’s calculus to explain how the memory properties of fBm allow us to describe some empirical findings of the implied volatility surface through Hull and White type formulas. Thus, Malliavin calculus provides a natural approach to deal with the implied volatility without assuming any particular structure of the volatility. The aim of this paper is to provides the basic tools of Malliavin calculus for the study of fractional volatility models. That is, we explain how the long and short memory of fBm improves the description of the implied volatility. In particular, we consider in detail a model that combines the long and short memory properties of fBm as an example of the approach introduced in this paper. The theoretical results are tested with numerical experiments.
- dc.format.mimetype application/pdf
- dc.identifier.citation Alòs E, León JA. An intuitive introduction to fractional and rough volatilities. Mathematics. 2021;9(9):994. DOI: 10.3390/math9090994
- dc.identifier.doi http://dx.doi.org/10.3390/math9090994
- dc.identifier.issn 2227-7390
- dc.identifier.uri http://hdl.handle.net/10230/57177
- dc.language.iso eng
- dc.publisher MDPI
- dc.relation.ispartof Mathematics. 2021;9(9):994.
- dc.rights © 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).
- dc.rights.accessRights info:eu-repo/semantics/openAccess
- dc.rights.uri http://creativecommons.org/licenses/by/4.0/
- dc.subject.keyword derivative operator in the Malliavin calculus sense
- dc.subject.keyword fractional Brownian motion
- dc.subject.keyword future average volatility
- dc.subject.keyword Hull and White formula
- dc.subject.keyword Itô’s formula
- dc.subject.keyword Skorohod integral
- dc.subject.keyword stochastic volatility models
- dc.subject.keyword implied volatility
- dc.subject.keyword skews and smiles
- dc.subject.keyword rough volatility
- dc.title An intuitive introduction to fractional and rough volatilities
- dc.type info:eu-repo/semantics/article
- dc.type.version info:eu-repo/semantics/publishedVersion