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Options With Underlying Asset Driven By A Fractional Brownian Motion: Crossing Barriers Estimates

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  • GIULIA ROTUNDO

    (Department of Business, Technological and Quantitative Studies, Faculty of Economics, University of Tuscia, via del Paradiso 47, 01100 Viterbo, Italy;
    Department of Economic and Financial Institutions, University of Macerata, Via Crescimbeni, 20-62100 - Macerata, Italy)

  • ROY CERQUETI

    (Department of Economic and Financial Institutions, University of Macerata, Via Crescimbeni, 20-62100 - Macerata, Italy)

Abstract

This paper aims at supplying a decision support system tool to investors having options written on an underlying asset driven by a fractional Brownian motion (fBm). The results presented here rely on the theory of nonlinear transformations of fBm and provide the calculus of the probability estimate that the underlying asset crosses nonlinear barriers. Recent results stating a Black and Scholes-like pricing formula for fBm monitor the expected behaviour of options on the basis of the dynamics of the underlying asset. We rely on the results drawn for plain vanilla options, leaving their extension to barrier options for future work. The theory of speculative bubbles due to endogenous causes provides a useful suggestion for the detection of periods in which these results should be used. The application of the above results is shown through the NASDAQ case study.

Suggested Citation

  • Giulia Rotundo & Roy Cerqueti, 2010. "Options With Underlying Asset Driven By A Fractional Brownian Motion: Crossing Barriers Estimates," New Mathematics and Natural Computation (NMNC), World Scientific Publishing Co. Pte. Ltd., vol. 6(01), pages 109-118.
    • Handle: RePEc:wsi:nmncxx:v:06:y:2010:i:01:n:s1793005710001633
    DOI: 10.1142/S1793005710001633
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    References listed on IDEAS

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    1. Dittmann, Ingolf & Granger, Clive W. J., 2002. "Properties of nonlinear transformations of fractionally integrated processes," Journal of Econometrics, Elsevier, vol. 110(2), pages 113-133, October.
    2. Tommi Sottinen, 2001. "Fractional Brownian motion, random walks and binary market models," Finance and Stochastics, Springer, vol. 5(3), pages 343-355.
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