A Predictive Model of Currency Exchange Rates Based on Stochastic Fractional Power-Law Dynamics

Abstract

This work proposes a stochastic fractional power-law model for currency exchange rate forecasting. The model incorporates nonlocal temporal effects through the Caputo fractional derivative and nonlinear scaling dynamics via a power-law formulation, providing a flexible framework for representing complex temporal behavior in financial time series. Model parameters are estimated by fitting an explicit analytical calibration expression, constructed under the fractional chain-rule framework adopted in this study, to historical currency exchange-rate data using nonlinear optimization techniques. This expression is employed specifically as a tractable parameterization for model calibration and is not claimed as a general exact closed-form solution of the nonlinear problem involving the standard Caputo derivative. The forecasting stage is carried out through numerical simulations using a FORK-2-based stochastic discretization, with stochastic perturbations incorporated via a Wiener process. The proposed methodology is applied to daily EUR/MXN and EUR/CAD exchange rate series, and forecasts are generated through multiple Monte Carlo simulations over different prediction horizons. The results suggest that the fractional formulation can improve forecasting accuracy when longer training periods are employed. In addition, the nonlinear power-law structure increases the model’s flexibility and provides additional structural flexibility. Nevertheless, the integer-order formulation generally exhibits greater predictive stability, largely independent of the training period and the inclusion of the nonlinear power-law extension.