Stochastic integrals driven by fractional Brownian motion and arbitrage: a tale of two integrals

Abstract

Recent research suggests that fractional Brownian motion can be used to model the long-range dependence structure of the stock market. Fractional Brownian motion is not a semi-martingale and arbitrage opportunities do exist, however. Hu and Øksendal [Infin. Dimens. Anal., Quant. Probab. Relat. Top., 2003, 6, 1–32] and Elliott and van der Hoek [Math. Finan., 2003, 13, 301–330] propose the use of the white noise calculus approach to circumvent this difficulty. Under such a setting, they argue that arbitrage does not exist in the fractional market. To unravel this discrepancy, we examine the definition of self-financing strategies used by these authors. By refining their definitions, a new notion of continuously rebalanced self-financing strategies, which is compatible with simple buy and hold strategies, is given. Under this definition, arbitrage opportunities do exist in fractional markets.

Keywords:

• Fractional Brownian motion
• Option pricing
• Arbitrage pricing
• Stochastic differential equations

Acknowledgements

We are indebted to the Editor and three referees for constructive comments that led to an improved version of the paper. This research is supported, in part, by HKSAR-RGC Earmarked Grants 400305 and 400306 and HKSAR-RGC General Research Fund 400408.