An Intuitive Introduction to Fractional and Rough Volatilities
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 cl...
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| Format: | Article |
| Language: | English |
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MDPI AG
2021-04-01
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| Series: | Mathematics |
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| Online Access: | https://www.mdpi.com/2227-7390/9/9/994 |
| _version_ | 1797535983299723264 |
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| author | Elisa Alòs Jorge A. León |
| author_facet | Elisa Alòs Jorge A. León |
| author_sort | Elisa Alòs |
| collection | DOAJ |
| description | 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. |
| first_indexed | 2024-03-10T11:53:07Z |
| format | Article |
| id | doaj.art-79d2c21f75eb4f64b2d9de545a13e402 |
| institution | Directory Open Access Journal |
| issn | 2227-7390 |
| language | English |
| last_indexed | 2024-03-10T11:53:07Z |
| publishDate | 2021-04-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Mathematics |
| spelling | doaj.art-79d2c21f75eb4f64b2d9de545a13e4022023-11-21T17:31:34ZengMDPI AGMathematics2227-73902021-04-019999410.3390/math9090994An Intuitive Introduction to Fractional and Rough VolatilitiesElisa Alòs0Jorge A. León1Department d’Economia i Empresa, Universitat Pompeu Fabra, and Barcelona GSE c/Ramon Trias Fargas, 25-27, 08005 Barcelona, SpainDepartamento de Control Automático, CINVESTAV-IPN, Apartado Postal 14-740, 07000 México City, MexicoHere, 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.https://www.mdpi.com/2227-7390/9/9/994derivative operator in the Malliavin calculus sensefractional Brownian motionfuture average volatilityHull and White formulaItô’s formulaSkorohod integral |
| spellingShingle | Elisa Alòs Jorge A. León An Intuitive Introduction to Fractional and Rough Volatilities Mathematics derivative operator in the Malliavin calculus sense fractional Brownian motion future average volatility Hull and White formula Itô’s formula Skorohod integral |
| title | An Intuitive Introduction to Fractional and Rough Volatilities |
| title_full | An Intuitive Introduction to Fractional and Rough Volatilities |
| title_fullStr | An Intuitive Introduction to Fractional and Rough Volatilities |
| title_full_unstemmed | An Intuitive Introduction to Fractional and Rough Volatilities |
| title_short | An Intuitive Introduction to Fractional and Rough Volatilities |
| title_sort | intuitive introduction to fractional and rough volatilities |
| topic | derivative operator in the Malliavin calculus sense fractional Brownian motion future average volatility Hull and White formula Itô’s formula Skorohod integral |
| url | https://www.mdpi.com/2227-7390/9/9/994 |
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