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

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  • Ngai Hang Chan
  • Chi Tim Ng

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.

Suggested Citation

  • Ngai Hang Chan & Chi Tim Ng, 2009. "Stochastic integrals driven by fractional Brownian motion and arbitrage: a tale of two integrals," Quantitative Finance, Taylor & Francis Journals, vol. 9(5), pages 519-525.
    • Handle: RePEc:taf:quantf:v:9:y:2009:i:5:p:519-525
    DOI: 10.1080/14697680802626315
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