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Fractal asset returns, arbitrage and option pricing

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  • Potgieter, Petrus H.

Abstract

In the discrete-time fractional random walk model a market with one risky asset affords an arbitrage opportunity as described by Cutland et al. [Cutland NJ, Kopp PE, Willinger W. Stock price returns and the Joseph effect: a fractional version of the Black–Scholes model. In: Russo Francesco, Bolthausen Erwin, Dozzi Marco, editors. Seminar on 6 stochastic analysis, random fields and applications, pp. 327–351. Seminar on stochastic analysis, random fields and applications. Ascona: Centro Stefano Franscini; 1993, Progress in probability 36. Birkhauser Verlag; 1995.] and Sottinen [Sottinen Tommi. Fractional Brownian motion, random walks and binary market models. Finance Stoch 2001;5(3):343–355]. We briefly discuss these results and compute a numerical example in a fractional binomial model as illustration and mention an option pricing model for assets the returns of which are driven by a fractional Brownian motion [Yaozhong Hu, Bernt Øksendal. Fractional white noise calculus and applications to finance. Infin Dimens Anal Quant Probability Rel Top 2003;6:1–32, ISSN 0219-0257; Fajardo J, Cajueiro DO. Volatility estimation and option pricing with fractional Brownian motion, October 2003. Available from: http://ideas.repec.org/p/ibm/finlab/flwp53.html].

Suggested Citation

  • Potgieter, Petrus H., 2009. "Fractal asset returns, arbitrage and option pricing," Chaos, Solitons & Fractals, Elsevier, vol. 42(3), pages 1792-1795.
    • Handle: RePEc:eee:chsofr:v:42:y:2009:i:3:p:1792-1795
    DOI: 10.1016/j.chaos.2009.03.095
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    References listed on IDEAS

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    1. Di Matteo, T. & Aste, T. & Dacorogna, M.M., 2003. "Scaling behaviors in differently developed markets," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 324(1), pages 183-188.
    2. Beretta, Alessandro & Roman, H. Eduardo & Raicich, Fabio & Crisciani, Fulvio, 2005. "Long-time correlations of sea-level and local atmospheric pressure fluctuations at Trieste," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 347(C), pages 695-703.
    3. King, Michael R., 2005. "Fractal analysis of eight glacial cycles from an Antarctic ice core," Chaos, Solitons & Fractals, Elsevier, vol. 25(1), pages 5-10.
    4. Tommi Sottinen, 2001. "Fractional Brownian motion, random walks and binary market models," Finance and Stochastics, Springer, vol. 5(3), pages 343-355.
    5. Cajueiro, Daniel O & Tabak, Benjamin M, 2004. "The Hurst exponent over time: testing the assertion that emerging markets are becoming more efficient," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 336(3), pages 521-537.
    6. Fajardo, J. & Cajueiro, D. O., 2003. "Volatility Estimation and Option Pricing with Fractional Brownian Motion," Finance Lab Working Papers flwp_53, Finance Lab, Insper Instituto de Ensino e Pesquisa.
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    2. Roy Cerqueti & Viviana Fanelli, 2021. "Long memory and crude oil’s price predictability," Annals of Operations Research, Springer, vol. 299(1), pages 895-906, April.

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