The paper focuses on the option price subdiffusive model under the unusual behavior of the market, when the price may not be changed for some time, which is a quite common situation in modern illiquid financial markets or during global crises. In the model, the risk- free bond motion and classical geometrical Brownian motion (GBM) are time-changed by an inverted inverse Gaussian(IG) subordinator. We explore the correlation structure of the subdiffusive GBM stock returns process, discuss option pricing techniques based on the martingale option pricing method and the fractal Dupire equation, and demonstrate how it applies in the case of the IG subordinator.