Forecasting with fractional Brownian motion: a financial perspective

TL;DR

This paper explores forecasting methods for fractional Brownian motion in finance, providing theoretical accuracy formulas, strategy optimization insights, and empirical applications to FX rates and volatility.

Contribution

It introduces new theoretical formulas for forecasting accuracy and strategy optimization within the fractional Brownian motion framework in finance.

Findings

Theoretical formulas for hit ratio, expected gain, and risk of fBm-based strategies.

Guidelines on selecting lagged increments for forecasting.

Empirical validation on FX rates and volatility series.

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.