Forecasting with fractional Brownian motion: a financial perspective

Abstract

The fractional Brownian motion (fBm) extends the standard Brownian motion by introducing some dependence between non-overlapping increments. Consequently, if one considers for example that log-prices follow an fBm, one can exploit the non-Markovian nature of the fBm to forecast future states of the process and make statistical arbitrages. We provide new insights into forecasting an fBm, by proposing theoretical formulas for accuracy metrics relevant to a systematic trader, from the hit ratio to the expected gain and risk of a simple strategy. In addition, we answer some key questions about optimizing trading strategies in the fBm framework: Which lagged increments of the fBm, observed in discrete time, are to be considered? If the predicted increment is close to zero, up to which threshold is it more profitable not to invest? We also propose empirical applications on high-frequency FX rates, as well as on realized volatility series, exploring the rough volatility concept in a forecasting perspective.

Keywords:

• Fractional Brownian motion
• Hurst exponent
• Systematic trading
• Rough volatility
• Foreign-exchange rates

Acknowledgements

The author is grateful to Martino Grasselli, Chafic Merhy, and Konstantinos Tsianos for valuable discussions and comments.

Disclosure statement

No potential conflict of interest was reported by the author.

Notes

1 Precisely, in our empirical analysis, T = 504 business days.

2 Available at http://realized.oxford-man.ox.ac.uk/data/download.

3 The results with BIC, not displayed in a table, are similar.

4 This optimal threshold is not to be confused with an ex-post optimal threshold.