Estimation of time-dependent Hurst exponents with variational smoothing and application to forecasting foreign exchange rates

Abstract

Hurst exponents depict the long memory of a time series. For human-dependent phenomena, as in finance, this feature may vary in the time. It justifies modelling dynamics by multifractional Brownian motions, which are consistent with time-varying Hurst exponents. We improve the existing literature on estimating time-dependent Hurst exponents by proposing a smooth estimate obtained by variational calculus. This method is very general and not restricted to the sole Hurst framework. It is globally more accurate and easier than other existing non-parametric estimation techniques. Besides, in the field of Hurst exponents, it makes it possible to make forecasts based on the estimated multifractional Brownian motion. The application to high-frequency foreign exchange markets (GBP, CHF, SEK, USD, CAD, AUD, JPY, CNY and SGD, all against EUR) shows significantly good forecasts. When the Hurst exponent is higher than 0.5, what depicts a long-memory feature, the accuracy is higher.

Hurst Exponents and Delampertized Fractional Brownian Motions

“…For the fBm, among a rich litterature about estimators of Hurst exponents [26,1,24,31], we know a simple estimator of H based on absolute moments [29,10,15]:…”

“…Many studies of their predictability have already been published. Among them, we can cite techniques based on fractional dynamics and Hurst exponents [15,13], on multiscale decomposition [2,18], or on chaos theory [12,17,13]. In the latter case, one of the challenges consists in determining the Lyapunov horizon, under which forecasts are supposed reliable.…”

“…In particular, two features are of importance: the stationarity of the process and the presence or not of long memory. For its part, the fractional nature of FX rates series is now well established [35,5,15,36] and makes it possible to generalize dynamics based on the standard Brownian motion.…”

“…anti-persistence) of the increments. Therefore, in fields as diverse as finance [11,27,15,20] or medicine [33,34], researchers hunt for values of Hurst exponents above 1/2, in order to leverage the persistence of the series and make predictions. Or, on the contrary, values of Hurst exponents below 1/2 are expected to make us benefit from the mean reversion of the series, as in pair trading [32].…”

Non-parametric news impact curve: a variational approach

“…For a complete review on these techniques, we refer the reader to [41,7,42,2]. The literature in economics and finance mentions various applications of the calculus of variations, for example for optimal income allocation [45], hedge fund asset allocation [10], foreign exchange rate predictions [20] or studies of options price sensitivities with stochastic variational calculus (Malliavin calculus) [33]. A goal of the present paper is to show that signal-theory techniques are suitable in volatility modelling and can be use with success in financial issues.…”