High-frequency trading with fractional Brownian motion
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High-frequency trading with fractional Brownian motion Paolo Guasoni1 · Yuliya Mishura2 · Miklós Rásonyi3
Received: 13 August 2019 / Accepted: 10 July 2020 © The Author(s) 2020
Abstract In the high-frequency limit, conditionally expected increments of fractional Brownian motion converge to a white noise, shedding their dependence on the path history and the forecasting horizon and making dynamic optimisation problems tractable. We find an explicit formula for locally mean–variance optimal strategies and their performance for an asset price that follows fractional Brownian motion. Without trading costs, risk-adjusted profits are linear in the trading horizon and rise asymmetrically as the Hurst exponent departs from Brownian motion, remaining finite as the exponent reaches zero while diverging as it approaches one. Trading costs penalise numerous portfolio updates from short-lived signals, leading to a finite trading frequency, which can be chosen so that the effect of trading costs is arbitrarily small, depending on the required speed of convergence to the high-frequency limit. Keywords Fractional Brownian motion · Transaction costs · High frequency · Trading · Mean–variance optimisation Guasoni is partially supported by SFI (16/IA/4443, 16/SPP/3347). Mishura is supported by the ToppForsk project Nr. 274410 of the Research Council of Norway with title STORM: Stochastics for Time-Space Risk Models. Rásonyi is supported by the NKFIH (National Research, Development and Innovation Office, Hungary) grant KH 126505 and by the “Lendület” grant LP 2015-6 of the Hungarian Academy of Sciences.
B M. Rásonyi
[email protected] P. Guasoni [email protected] Y. Mishura [email protected]
1
Dublin City University, School of Mathematical Sciences, Glasnevin, Dublin 9, Ireland
2
Taras Schevchenko National University of Kyiv, 64 Volodymyrska, 01033 Kyiv, Ukraine
3
Alfréd Rényi Institute of Mathematics, Budapest, Hungary
P. Guasoni et al.
Mathematics Subject Classification (2010) 91G10 · 91G80 JEL Classification G11 · G12
1 Introduction First proposed as a model of price dynamics by Mandelbrot [14], fractional Brownian motion (fBm) has since puzzled researchers and stirred controversy for its elusive properties, which have confounded both empirical and theoretical work. Long-range dependence in asset prices, the property that originally motivated the use of fBm to describe price dynamics, remains undecided; see Greene and Fielitz [7], Fama and French [6], Poterba and Summers [17], Lo [13], Jacobsen [12], Teverovsky et al. [22], Willinger et al. [23], Baillie [1]. Arbitrage, which has plagued the adoption of fBm in models of optimal investment (Rogers [19], Salopek [20], Dasgupta and Kallianpur [5], Cheridito [2]) disappears with frictions (Guasoni [8], Guasoni et al. [10], Czichowsky and Schachermayer [4], Czichowsky et al. [3]), leading to finite expected profits; see Guasoni et al. [9]. This paper finds locally mean–variance optimal trading strategies in fractional Brownian motion and characterises their co
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