1. An Intuitive Introduction to Fractional and Rough Volatilities
- Author
-
Jorge Leon and Elisa Alòs
- Subjects
Statistics::Theory ,Skorohod integral ,General Mathematics ,Structure (category theory) ,rough volatility ,fractional Brownian motion ,Itô’s formula ,Implied volatility ,Type (model theory) ,Malliavin calculus ,01 natural sciences ,010104 statistics & probability ,Mathematics::Probability ,derivative operator in the Malliavin calculus sense ,0502 economics and business ,Computer Science (miscellaneous) ,Filtration (mathematics) ,QA1-939 ,Applied mathematics ,0101 mathematics ,Engineering (miscellaneous) ,stochastic volatility models ,future average volatility ,Mathematics ,Hull and White formula ,050208 finance ,Fractional Brownian motion ,05 social sciences ,skews and smiles ,Semimartingale ,Volatility (finance) ,implied volatility - 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.
- Published
- 2021