Asian Journal of Pure and Applied Mathematics

This study examines the effects of seasonal dividend behaviour and time-dependent initial conditions on bank stock-price dynamics using Geometric Brownian Motion (GBM). Two stochastic systems are considered: a seasonal model with a periodic initial condition and a benchmark model with a constant initial condition. The seasonal formulation is intended to represent recurring annual dividend, earnings, and macroeconomic cycles, whereas the benchmark formulation excludes deterministic seasonality and related regime effects. Both systems are expressed as stochastic differential equations and solved using Itô's formula to obtain closed-form solutions. The resulting models are illustrated through simulated paths for First Bank and Fidelity Bank under the parameter settings specified in the manuscript. The simulations indicate that the seasonal model retains a recurring cyclical structure, while Brownian shocks generate short-term dispersion around that pattern. Changes in seasonal amplitude affect the magnitude of the price oscillations without changing their timing. In the benchmark model, drift influences the central direction of the simulated price paths, whereas volatility determines the extent of short-term dispersion. Taken together, the two formulations provide a comparative framework for examining how deterministic seasonal structure and stochastic noise interact in the modelled bank share prices. The benchmark model also provides a reference against which the effects of the periodic initial condition can be assessed.