Options with underlying asset driven by a fractional brownian motion: crossing barriers estimates

Electronic version of an article published as New Mathematics and Natural Computation Vol. 06, No. 01, pp. 109-118 (2010) DOI: 10.1142/S1793005710001633 ©World Scientific Publishing Company https://www.worldscientific.com/doi/abs/10.1142/S1793005710001633 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.