Long Memory Options: Valuation

Abstract

This paper demonstrates the impact of the observed financial market persistence or long term memory on European option valuation. Many empirical researchers have observed non-Fickian degrees of persistence or long memory 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 (2003) have now also provided a theoretical framework for incorporating these findings into the Black-Scholes risk-neutral valuation framework. Risk-neutral valuation is equivalent to valuation by real world probabilities. 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-fractional Brownian motion, it is easy to incorporate the various degrees of persistence into the binomial and Black-Scholes pricing formulas. 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. Therefore, 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. A small error in the measurement of the persistence exponent value may lead to substantial pricing errors.

Keywords: Options, Long Memory, Persistence, Hurst Exponent, Executive Remuneration

JEL Classification: G14, G15, G33, C14