Solution of a Linear Stochastic Differential Equation for the Impact of Inflation and Long-Term Growth Trends on Bank Valuation

This study investigates the dynamics of Nigerian bank stock prices using an exponential-growth geometric Brownian motion framework. The model extends the classical geometric Brownian motion (GBM) by incorporating a growth-adjustment parameter $k$ in the drift term, allowing it to capture both deterministic long-term growth and stochastic fluctuations. Three Brownian-motion paths are first presented to illustrate the underlying randomness, followed by sample price paths demonstrating how the same growth rate can produce both upward and downward trajectories depending on the shock realizations. A sensitivity analysis is then conducted to assess the effect of $k$ on the simulated paths. The results show that higher values of $k$ amplify exponential growth or decay and increase dispersion in absolute valuation. This framework provides insight into how long-term growth expectations interact with short-term volatility in the Nigerian banking sector.