J Theor Probab https://doi.org/10.1007/s10959-018-0836-y Drawdown and Drawup for Fractional Brownian Motion with Trend 1 1 Long Bai · Peng Liu Received: 20 February 2018 / Revised: 20 May 2018 © Springer Science+Business Media, LLC, part of Springer Nature 2018 Abstract We consider the drawdown and drawup of a fractional Brownian motion with trend, which corresponds to the logarithm of geometric fractional Brownian motion representing the stock price in a ﬁnancial market. We derive the asymptotics of tail probabilities of the maximum drawdown and maximum drawup, respectively, as the threshold goes to inﬁnity. It turns out that the extremes of drawdown lead to new scenarios of asymptotics depending on the Hurst index of fractional Brownian motion. Keywords Drawdown · Drawup · Fractional Brownian motion · Geometric fractional Brownian motion · Pickands constant · Piterbarg constant Mathematics Subject Classiﬁcation (2010) Primary 60G15 · Secondary 60G70 1 Introduction and Preliminaries Drawdown, deﬁned as the distance of the present value away from its historical running maximum, is an important indicator of downside risks in ﬁnancial risk management. For instance, the drawdown and the maximum drawdown have been customarily used as risk measures in ﬁnance where they measure the current drop of a stock
Journal of Theoretical Probability – Springer Journals
Published: Jun 6, 2018
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