Stock price returns and the Joseph effect: a fractional version of the Black-Scholes model. (English) Zbl 0827.60021

Cutland, Nigel J.; Kopp, P. Ekkehard; Willinger, Walter

Bolthausen, Erwin (ed.) et al., Seminar on stochastic analysis, random fields and applications. Proceedings of a seminar held at the Centro Stefano Franscini, Ascona, Switzerland, June 7-12, 1993. Basel: Birkhäuser. Prog. Probab. 36, 327-351 (1995).

Summary: 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.
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.
For the entire collection see [Zbl 0822.00010].

MSC:

60F17	Functional limit theorems; invariance principles
91G20	Derivative securities (option pricing, hedging, etc.)
60G22	Fractional processes, including fractional Brownian motion
60H30	Applications of stochastic analysis (to PDEs, etc.)
60H05	Stochastic integrals
62M07	Non-Markovian processes: hypothesis testing

Keywords:

efficient markets; arbitrage opportunities; mathematical finance; long- range dependence; Black-Scholes model; fractional Brownian motion