High-frequency trading with fractional Brownian motion

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.

Maxwell’s Demon and Information Theory in Market Efficiency: A Brillouin’s Perspective

“…Overall, this article sheds new lights on the informational content of fractional models. It confirms in a new manner previous results on the forecast of fBm in finance [39,38,32]. It also deepens the understanding of the stationary inverse Lamperti transform of the fBm [29].…”

“…for all s, t ∈ R. In finance, the propensity of the fBm to be forecast contradicts the EMH and is at the root of many statistical arbitrage strategies [39,38,32]. It is thus relevant to quantify the market information of such a dynamic.…”

The Optimal Limit Prices of Limit Orders under an Extended Geometric Brownian Motion with Bankruptcy Risk

“…A possible direction of future research is to extend model (1) with fractional Brownian motion. Recently, the fractional Brownian motion has fruitful applications in finance such as financial modelling, option pricing, optimal portfolio selection and high frequency trading, see Guasoni et al [13], Guasoni et al [14], Kříž and Szała [15] and Rostek and Schöbel [16] for more details. Another interesting issue of future research is to explore other real problems in finance market based on model (1) and test the efficiency of model (1) via analyzing the real financial data.…”

Analysis of optimal portfolios on finite and small-time horizons for a multi-dimensional correlated stochastic volatility model

“…With a relevant geometric formalism the authors show that the Bellman function is the unique viscosity solution of the corresponding HJB equation. On the other hand, in the high-frequency limit, the paper [6] finds an explicit formula for locally mean-variance optimal strategies. In the limit, it is shown that 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.…”