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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 playa 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) BlackScholes 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. §1 INTRODUCTION In mathematical finance, semimartingales are typically viewed as a class of stochastic processes that is large enough to model a great variety of 'realistic' stock price behaviors. In addition, since stochastic integrals play a crucial role in the modern theory of finance and semimartingales are, in some sense (see Protter [63]), the largest class of stochastic processes for which a general stochastic integration theory exists, the semimartingale assumption has become a generally accepted modeling framework in the mathematical finance literature. In contrast 1991 Mathematics Subject Classification. 60F17, 60H05, 62M07, 90A09.

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Nonstandard probability theory and stochastic analysis, as developed by Loeb, Anderson, and Keisler, has the attractive feature that it allows one to exploit combinatorial aspects of a well-understood discrete theory in a continuous setting. We illustrate this with an example taken from financial economics: a nonstandard construction of the well-known Black-Scholes option pricing model allows us to view the resulting object at the same time as both (the hyperfinite version of) the binomial Cox-Ross-Rubinstein model (that is, a hyperfinite geometric random walk) and the continuous model introduced by Black and Scholes (a geometric Brownian motion). Nonstandard methods provide a means of moving freely back and forth between the discrete and continuous points of view. This enables us to give an elementary derivation of the Black-Scholes option pricing formula from the corresponding formula for the binomial model. We also devise an intuitive but rigorous method for constructing self-financing hedge portfolios for various contingent claims, again using the explicit constructions available in the hyperfinite binomial model, to give the portfolio appropriate to the Black-Scholes model. Thus, nonstandard analysis provides a rigorous basis for the economists' intuitive notion that the Black-Scholes model contains a built-in version of the Cox-Ross-Rubinstein model. Copyright 1991 Blackwell Publishers.