Fractal asset returns, arbitrage and option pricing

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].