HURST EXPONENTS AND DELAMPERTIZED FRACTIONAL BROWNIAN MOTIONS

Matthieu Garcin ()
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Matthieu Garcin: Léonard de Vinci Pôle Universitaire, Research Center, 92916 Paris La Défense, France

International Journal of Theoretical and Applied Finance (IJTAF), 2019, vol. 22, issue 05, 1-26

Abstract: The inverse Lamperti transform of a fractional Brownian motion (fBm) is a stationary process. We determine the empirical Hurst exponent of such a composite process with the help of a regression of the log absolute moments of its increments, at various scales, on the corresponding log scales. This perceived Hurst exponent underestimates the Hurst exponent of the underlying fBm. We thus encounter some time series having a perceived Hurst exponent lower than 1/2, but an underlying Hurst exponent higher than 1/2. This paves the way for short- and medium-term forecasting. Indeed, in such series, mean reversion predominates at high scales, whereas persistence is overriding at lower scales. We propose a way to characterize the Hurst horizon, namely a limit scale between these opposite behaviors. We show that the delampertized fBm, which mixes persistence and mean reversion, is relevant for financial time series, in particular for high-frequency foreign exchange rates. In our sample, the empirical Hurst horizon is always above 1h and 23min.

Keywords: Foreign exchange rates; fractional Brownian motion; Hurst exponent; Lamperti transform; Ornstein–Uhlenbeck process; stationary process (search for similar items in EconPapers)
Date: 2019
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Citations: View citations in EconPapers (15)

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DOI: 10.1142/S0219024919500249

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