Pricing European Options under Logistic Stock-Price Dynamics with a Continuous Dividend Yield

Logistic stock-price dynamics have been proposed as a way of representing assets whose growth is bounded by competition, saturation or finite market capacity, but the derivations available in the literature retain a logistic drift while invoking a delta-hedging argument that removes it. This paper resolves that inconsistency and extends the framework to dividend-paying assets. It is first shown that, for a traded asset, no-arbitrage forces the logistic term out of the pricing equation; the logistic model is therefore developed as an incomplete-market model in which a state-dependent market price of risk fixes the pricing measure, and the pricing partial differential equation is obtained from the Feynman-Kac representation rather than from hedging. The resulting equation carries a continuous dividend yield and reduces to the Black-Scholes-Merton equation as the carrying capacity grows without bound. A closed-form solution of the price process is derived through the reciprocal transformation, which linearises the stochastic logistic equation and expresses the price as a geometric Brownian motion divided by an exponential functional. Prices are computed by a Crank-Nicolson scheme with Rannacher start-up; second-order convergence is demonstrated and the scheme is cross-validated against Monte Carlo simulation of the closed-form representation. Sensitivity analyses quantify the effect of carrying capacity, dividend yield and volatility, and a forward-price diagnostic measures the departure from put-call parity, for which a first-order analytic expression is derived and verified numerically.