Multifractional Stochastic Volatility Models

Abstract

The aim of this work is to advocate the use of multifractional Brownian motion (mBm) as a relevant model in financial mathematics. mBm is an extension of fractional Brownian motion where the Hurst parameter is allowed to vary in time. This enables the possibility to accommodate for varying local regularity, and to decouple it from long‐range dependence properties. While we believe that mBm is potentially useful in a variety of applications in finance, we focus here on a multifractional stochastic volatility Hull & White model that is an extension of the model studied in Comte and Renault. Using the stochastic calculus with respect to mBm developed in Lebovits and Lévy Véhel, we solve the corresponding stochastic differential equations. Since the solutions are of course not explicit, we take advantage of recently developed numerical techniques, namely functional quantization‐based cubature methods, to get accurate approximations. This allows us to test the behavior of our model (as well as the one in Comte and Renault) with respect to its parameters, and in particular its ability to explain some features of the implied volatility surface. An advantage of our model is that it is able both to fit smiles at different maturities, and to take volatility persistence into account in a more precise way than Comte and Renault.

Correction to Black--Scholes Formula Due to Fractional Stochastic Volatility

“…As a further generalization relative to a fractional Brownian motion based model the case of multi fractional Brownian motion based models is considered in [16]. This allows for a non-stationary local regularity or a time dependent Hurst exponent and then the implied volatility depends on weighted averages of the local Hurst exponent.…”

A Caputo fractional derivative of a function with respect to another function

“…For more than two centuries, this subject was relevant only in pure mathematics, and Euler, Fourier, Abel, Liouville, Riemann, Hadamard, among others, have studied these new fractional operators, by presenting new definitions and studying their most important properties. However, in the past decades, this subject has proven its applicability in many and different natural situations, such as viscoelasticity [11,26], anomalous diffusion [14,19], stochastic processes [9,29], signal and image processing [31], fractional models and control [24,32], etc. This is a very rich field, and for it we find several definitions for fractional integrals and for fractional derivatives [16,25].…”

Maximum likelihood estimation in the non-ergodic fractional Vasicek model

“…In recent years, many articles on various financial applications of the fractional Vasicek model (1) have appeared, see e.g. [8,9,12,13,30,40]). In order to use this model in practice, a theory of parameter estimation is necessary.…”