Modified Fractional Brownian Motion and Option Pricing

Abstract

<p>The Black-Scholes model introduced by Black and Scholes (1973) and Merton (1973) has become synonymous with modern finance theory. It assumes that the dynamics of stock prices is well described by exponential Brownian motion, which is not consistent with empirical stock price returns, and then the dependence structure of stock price returns has been at the center of intense scrutiny for the last 30 or more years. This project studies modified fractional Brownian motions and shows that two different classes of modified fractional Brownian motions are equivalent to Brownian motion. We discuss option pricing under the hypothesis that the underlying asset price process satisfies a stochastic differential equation driven by a modified fractional Brownian motion. Parameter estimation and simulation methods are given. In particular, we investigate the ability of the self-similarity parameter H to explain the discrepancy between the Black-Scholes model and the reality of the market. The proposed method is applied to a real data set. The empirical results indicate that the model is better than the Black-Scholes model.</p>