This paper deals with the European option-valuation problem under a prediction model for the underlying asset prices with an external impact. We suggest an alternative model for risky asset prices in which the parameters are dependent on a non-Markovian process. This generalizes the regime-switching models with continuous-time Markov-chain processes. A notable problem of this model is that the process embedded in the parameter of the stock price is non-Markovian; in addition, the market is incomplete. The change from the historical probability to a risk-neutral one is investigated, and the set of equivalent martingale measures is determined. In addition, an infinitesimal generator is obtained, which allows numerical simulations of the non-Markovian and stock-price processes to be conducted. Several illustrations are provided.