Risk Analysis of Recurrent Neural Networks for Financial Time Series

Forecasting financial time series is a fundamental task for modern quantitative finance. Recurrent neural networks (RNNs) have been widely applied in this domain, yet their statistical behavior under the structural features of financial data remains poorly understood. We study the excess risk of RNNs for price movement prediction with k-line chart data. To obtain a financially interpretable risk characterization, we introduce a tractable data-generating model and decompose excess risk into approximation and estimation errors. The approximation analysis separates finite temporal representation, nonlinear function approximation, and normalization-induced information loss. Under binary cross-entropy loss, we prove that the finite-width term decays as $O((m-2p-2{)}^{-2})$. We further show that normalized observations inherit the $\beta$-mixing property of the underlying dynamics, yielding an estimation bound based on Rademacher complexity for dependent data. Combining the two error bounds, we reveal how financially interpretable factor into risk: the signal-to-noise ratio (SNR) proxy $B_s^2/{\sigma }^2$ governs approximation difficulty, while the persistence parameter \(\rho\) controls effective statistical information. Notably, a larger SNR proxy can increase approximation error and hence excess risk. We conduct controlled experiments on simulated data. The results exhibit trends consistent with the theoretical analysis, and empirically reveal the approximation–estimation tradeoff across model capacity and sample size.