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.
Highlights
First proposed as a model of price dynamics by Mandelbrot [14], fractional Brownian motion 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 fractional Brownian motion (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]
This paper finds locally mean–variance optimal trading strategies in fractional Brownian motion and characterises their convergence and performance in the highfrequency limit
Summary
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. Fractional drifts are critically dependent on the specific interval: as the interval length declines to zero, the conditionally expected returns converge in law, but not as random variables in any reasonable sense It is worthwhile comparing the findings in this paper to the recent results in Guasoni et al [9], as both articles study optimal trading strategies for fractional Brownian motion, though in very different settings. The present local mean– variance criterion is well posed even without frictions as the instantaneous Sharpe ratio remains bounded for any H ∈ (0, 1), arbitrage is feasible on any interval because arbitrage profits remain dispersed Both [9] and the present paper lead to finite maximal Sharpe ratios that are asymmetric in H , but their skews are reversed and arise for different reasons: while the asymptotically optimal strategies in [9] have higher Sharpe ratios near zero than near one, they are not necessarily optimal as the strategies maximise a risk-neutral objective, not the Sharpe ratio.
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