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.
Highlights
It is well-known that the classical Black-Scholes model [1] describes the current market behavior when it is assumed that the volatility process σ is a constant
We consider in detail a model that combines the long and short memory properties of fractional Brownian motion (fBm) as an example of the approach introduced in this paper
(see (15) below), which can be obtained via the Itô’s formula and states that the price of the European option is given by a conditional expectation of the Black-Scholes option pricing formula where the constant volatility is changed by the future average volatility s t 7→
Summary
It is well-known that the classical Black-Scholes model [1] describes the current market behavior when it is assumed that the volatility process σ is a constant. A simple method to achieve this is to allow the volatility σ to be a process independent of the noise governing the stock prices (see Renault and Touzi [2], Stein and Stein [3], and Scott [4], amongst others). Under this model, some features, such as the smile, are analyzed using the Hull and White formula [5]. (see (15) below), which can be obtained via the Itô’s formula and states that the price of the European option is given by a conditional expectation of the Black-Scholes option pricing formula where the constant volatility is changed by the future average volatility s t 7→
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