Long memory options: LM evidence and simulations

This paper demonstrates the impact of the observed financial market persistence or long term memory on European option valuation by simple simulation. Many empirical researchers have observed the non-Fickian degrees of persistence in the financial markets different from the Fickian neutral independence (i.i.d.) of the returns innovations assumption of Black–Scholes’ geometric Brownian motion assumption. Moreover, Elliott and van der Hoek [Elliott, R.J., van der Hoek, J., 2003. A general fractional white noise theory and applications to finance. Math. Finance 13, 301–330] provide a theoretical framework for incorporating these findings into the Black–Scholes risk-neutral valuation framework. This paper provides the first graphical demonstration why and how such long term memory phenomena change European option values and provides thereby a basis for informed long term memory arbitrage. By using a simple mono-fractal fractional Brownian motion, it is easy to incorporate the various degrees of persistence into the Black–Scholes pricing formula. Long memory options are of considerable importance in corporate remuneration packages, since stock options are written on a company’s own shares for long expiration periods. It makes a significant difference in the valuation when an option is “blue” or when it is “red.” For a proper valuation of such stock options, the degrees of persistence of the companies’ share markets must be precisely measured and properly incorporated in the warrant valuation, otherwise substantial pricing errors may result.