We propose several new models in ecophysics known as the Fractal Activity Time Geometric Brownian Motion (FATGBM) models with Student marginals. We summarize four models that construct stochastic processes of underlying prices with short-range and long-range dependencies. We derive solutions of option Greeks and compare with those in the Black-Scholes model. We analyse perfor-mance of delta hedging strategy using simulated time series data and verify that hedging errors are biased particularly for long-range dependence cases. We also apply underlying model calibration on S&P 500 index (SPX) and the U.S./Euro rate, and implement delta hedging on SPX options.