Fractional Brownian markets with time-varying volatility and high-frequency data

TL;DR

This paper extends fractional Brownian motion models for stock prices by incorporating time-varying volatility, deriving option prices and asymptotic distributions of volatility estimators, enhancing modeling realism.

Contribution

It introduces a model with time-varying volatility in fractional Brownian markets and derives option prices and estimator distributions for this setting.

Findings

Derived option prices depending on volatility and Hurst parameter.

Established a central limit theorem for quadratic variation estimators.

Provided asymptotic distributions for volatility estimators.

Abstract

Diffusion processes driven by Fractional Brownian motion (FBM) have often been considered in modeling stock price dynamics in order to capture the long range dependence of stock price observed in reality. Option prices for such models had been obtained by Necula (2002) under constant drift and volatility. We obtain option prices under time varying volatility model. The expression depends on volatility and the Hurst parameter in a complicated manner. We derive a central limit theorem for the quadratic variation as an estimator for volatility for both the cases, constant as well as time varying volatility. That will help us to find estimators of the option prices and to find their asymptotic distributions.