A Novel Black-Scholes Framework: Mean Reverting European Logistic Option Pricing with Jump Diffusion and Transaction Costs

Traditional Black-Scholes models assume frictionless markets with continuous price movements, limiting their applicability in emerging economies where transaction costs and sudden market shocks significantly influence asset pricing dynamics. These limitations became particularly evident during the COVID-19 pandemic when markets experienced unprecedented volatility and liquidity constraints. This study derives a novel Black-Scholes differential equation that incorporates mean reversion, jump diffusion processes, and transaction costs within a European logistic option pricing framework to address real-world market complexities. We extend geometric Brownian motion by integrating Vasicek mean reversion dynamics, Poisson jump processes for market discontinuities, and explicit transaction cost modeling. The enhanced stochastic differential equation is: dS(t) = (? – ?k – ?)(S? – ln S(t))S(t)(S* – S(t))dt + ?S(t)(S* – S(t))dZt + S(t)dq, where ? represents mean reversion speed, ? is jump intensity, ? denotes transaction costs, and dq captures Poisson jumps. Parameters are estimated using maximum likelihood methods with conditional density functions. Empirical validation using four major Nairobi Securities Exchange companies (2020-2022) demonstrates superior performance. Our model produces consistently lower, more realistic volatility estimates compared to five benchmark models. For Equity Bank, volatility estimates were 0.97 (2020) versus 1.36-1.68 from traditional models. ANOVA analysis confirms statistical significance of transaction cost effects (F = 1690.54, p < 2.8E-276). The model effectively captured crisis-period dynamics while maintaining stability through mean reversion mechanisms. The enhanced framework provides more accurate asset pricing for emerging markets by simultaneously accounting for bounded growth, price reversions, jump risks, and trading frictions. This offers substantial improvements for portfolio optimization, risk management, and derivative pricing in volatile market environments.