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On arbitrage and replication in the fractional Black–Scholes pricing model

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  • Sottinen Tommi
  • Valkeila Esko

Abstract

It has been proposed that the arbitrage possibility in the fractional Black-Scholes model depends on the definition of the stochastic integral. More precisely, if one uses the Wick–Itô–Skorohod integral one obtains an arbitrage-free model. However, this integral does not allow economical interpretation. On the other hand it is easy to give arbitrage examples in continuous time trading with self-financing strategies, if one uses the Riemann-Stieltjes integral. In this note we discuss the connection between two different notions of self-financing portfolios in the fractional Black–Scholes model by applying the known connection between these two integrals. In particular, we give an economical interpretation of the proposed arbitrage-free model in terms of Riemann–Stieltjes integrals.

Suggested Citation

  • Sottinen Tommi & Valkeila Esko, 2003. "On arbitrage and replication in the fractional Black–Scholes pricing model," Statistics & Risk Modeling, De Gruyter, vol. 21(2), pages 93-108, February.
    • Handle: RePEc:bpj:strimo:v:21:y:2003:i:2/2003:p:93-108:n:7
    DOI: 10.1524/stnd.21.2.93.19003
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    References listed on IDEAS

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    1. Salopek, D. M., 1998. "Tolerance to arbitrage," Stochastic Processes and their Applications, Elsevier, vol. 76(2), pages 217-230, August.
    2. Tommi Sottinen, 2001. "Fractional Brownian motion, random walks and binary market models," Finance and Stochastics, Springer, vol. 5(3), pages 343-355.
    3. L. C. G. Rogers, 1997. "Arbitrage with Fractional Brownian Motion," Mathematical Finance, Wiley Blackwell, vol. 7(1), pages 95-105, January.
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