Towards a Better Understanding of Fractional Brownian Motion and Its Application to Finance

Abstract

The aim of this work is to first build the underlying theory behind fractional Brownian motion and applying fractional Brownian motion to financial market. By incorporating the Hurst parameter into geometric Brownian motion in order to characterize the long memory among disjoint increments, geometric fractional Brownian motion model is constructed to model S &P 500 stock price index. The empirical results show that the fitting effect of fractional Brownian motion model is better than ordinary Brownian motion.

Company Valuation under Risk and Flexibility

“…The most frequently occurring values, as depicted in the charts, are 41403 and 39043, respectively. The observed forms of these two probability distribution plots exhibit characteristics reminiscent of the Poisson distribution and the Weibull distribution [11] . Notably, the graphical representation of the Weibull distribution shows a particularly striking resemblance.…”

“…Predict the equity value of Estee Lauder Companies from 2025 to infinity by deducting liabilities from assets. Following this, the real option method is employed as articulated in formula (11), where IV t denotes the intrinsic value of Estee Lauder Companies, namely, the value of equity. Subsequently, this value is discounted to the year 2021 using formula (12).…”