Semimartingale properties of a generalized fractional Brownian motion and its mixtures with applications in asset pricing

TL;DR

This paper investigates the semimartingale properties of generalized fractional Brownian motion (GFBM), identifies parameter regions for semimartingale behavior, and explores applications in asset pricing, option pricing, and arbitrage theory.

Contribution

It characterizes the semimartingale regions of GFBM and its mixtures, introduces the associated equivalent Brownian measure, and applies these findings to financial modeling and arbitrage analysis.

Findings

GFBM is a semimartingale for specific parameter regions.

Mixed GFBM is a semimartingale for H in (1/2,1).

Provides an arbitrage-free semimartingale asset pricing model with long-range dependence.

Abstract

We study the semimartingale properties for the generalized fractional Brownian motion (GFBM) introduced by Pang and Taqqu (2019) and discuss the applications of the GFBM and its mixtures to financial asset pricing. The GFBM is self-similar and has non-stationary increments, whose Hurst index $H \in (0,1)$ is determined by two parameters. We identify the regions of these two parameter values where the GFBM is a semimartingale. We next study the mixed process made up of an independent BM and a GFBM and identify the range of parameters for it to be a semimartingale, which leads to $H \in (1/2,1)$ for the GFBM. We also derive the associated equivalent Brownian measure. This result is in great contrast with the mixed FBM with $H \in \{1/2\}\cup(3/4,1]$ proved by Cheridito (2001) and shows the significance of the additional parameter introduced in the GFBM. We then study the semimartingale asset pricing theory with the mixed GFBM, in presence of long range dependence, and applications in option pricing and portfolio optimization. Finally we discuss the implications of using GFBM on arbitrage theory, in particular, providing an example of semimartingale asset pricing model of long range dependence without arbitrage.