Stock Price Returns and the Joseph Effect: A Fractional Version of the Black-Scholes Model

Abstract

In mathematical finance, semimartingales are traditionally viewed as the largest class of stochastic processes which are economically reasonable models for stock price movements. This is mainly because stochastic integrals play a crucial role in the modern theory of finance, and semimartingales represent the largest class of stochastic processes for which a general theory of stochastic integration exists. However, some empirical evidence from actual stock price data suggests stochastic models that are not covered by the semimartingale setting.In this paper, we discuss the empirical evidence that suggests that long-range dependence (also called the Joseph effect) be accounted for when modeling stock price movements. We present a fractional version of the Black—Scholes model that (i) is based on fractional Brownian motion, (ii) accounts for long-range dependence and is therefore inconsistent with the semimartingale framework (except in the case of ordinary Brownian motion) and (iii) yields the ordinary (independent) Black—Scholes model as a special case. Mathematical problems of practical importance to finance (e. g., completeness, equivalent martingale measures, arbitrage, financial gains) for this class of fractional Black—Scholes models are dealt with in an elementary fashion, namely using the (hyperfinite) fractional versions of the corresponding Cox—Ross—Rubinstein models.

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