Drawdown and drawup for fractional Brownian motion with trend

TL;DR

This paper studies the extreme behaviors of drawdown and drawup in fractional Brownian motion with trend, deriving asymptotic probabilities that depend on the Hurst index, relevant for financial modeling of stock prices.

Contribution

It provides new asymptotic results for the tail probabilities of maximum drawdown and drawup in fractional Brownian motion with trend, highlighting the influence of the Hurst index.

Findings

Asymptotic tail probabilities are derived for maximum drawdown and drawup.

Extremes of drawdown exhibit new asymptotic scenarios based on the Hurst index.

Results are applicable to modeling stock prices with fractional Brownian motion.

Abstract

In this paper, we consider the drawdown and drawup of the fractional Brownian motion with trend, which corresponds to the logarithm of geometric fractional Brownian motion representing the stock price in financial market. We derive the asymptotics of tail probabilities of the maximum drawdown and maximum drawup as the threshold goes to infinity, respectively. It turns out that the extremes of drawdown leads to new scenarios of asymptotics depending on Hurst index of fractional Brownian motion.