Drawdown and Drawup for Fractional Brownian Motion with Trend

Drawdown and Drawup for Fractional Brownian Motion with Trend 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 http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Journal of Theoretical Probability Springer Journals

Drawdown and Drawup for Fractional Brownian Motion with Trend

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Publisher
Springer Journals
Copyright
Copyright © 2018 by Springer Science+Business Media, LLC, part of Springer Nature
Subject
Mathematics; Probability Theory and Stochastic Processes; Statistics, general
ISSN
0894-9840
eISSN
1572-9230
D.O.I.
10.1007/s10959-018-0836-y
Publisher site
See Article on Publisher Site

Abstract

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

Journal of Theoretical ProbabilitySpringer Journals

Published: Jun 6, 2018

References

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