Risk and Decision Analysis

A first type of Multifractional Process with Random Exponent (MPRE) was constructed several years ago in [Publ. Mat.49 (2005), 459–486] by replacing in a wavelet series representation of Fractional Brownian Motion (FBM) the Hurst parameter by a random variable depending on the time variable. In the present article, we propose another approach for constructing another type of MPRE. It consists in substituting to the Hurst parameter, in a stochastic integral representation of the high-frequency part of FBM, a random variable depending on the integration variable. The MPRE obtained in this way offers, among other things, the advantages to have a representation through classical Itô integral and to be less difficult to simulate than the first type of MPRE, previously introduced in [Publ. Mat.49 (2005), 459–486]. Yet, the study of Hölder regularity of this new MPRE is a significantly more challenging problem than in the case of the previous one. Actually, it requires to develop a new methodology relying on an extensive use of the Haar basis.

The aim of this paper is to provide simple models with a time-varying Hurst index. Such models should be simple as much as possible and well fit the estimated Hurst index. After a recall on the fractional and multifractional Brownian motion and on the statistical estimation of the Hurst index, we propose a fitting test for a model with a time-varying Hurst index. Then, we give an approach to select a simple model. Our approach is illustrated by numerous numerical simulations and then applied to market finance data.

This paper is an empirical study investigating the effect of using a fractional Black–Scholes model for pricing Call options in comparison to the classical Black–Scholes model. We estimate the Hurst parameter by nine different methods available in software R, for a set of fifteen assets from four different industries and for various lengths of used historical data. We use implied volatility for calculation of option price with a one month time to maturity. As a main result, we provide a table of tendencies of suitability for using each of the Hurst exponent estimation methods depending on the length of underlying data set.

Stochastic volatility models have proven relevant for various purposes, e.g. explaining the term structure of implied volatility or correlations in absolute returns. When used in conjunction with jump models, they provide powerful tools that have been applied successfully in recent years. Volatility is however but one aspect of risk: in the frame of jump processes, the (local) intensity of jumps is another one, which is equally important. It thus seems natural to supplement stochastic volatility models with “stochastic jump intensity” ones, where the local intensity of jumps evolves according to a random dynamics. The aim of this preliminary work is merely to set up such models as extensions of the celebrated CGMY process, leaving their exploration both for risk management and for option pricing to further studies.