We propose several new models in finance 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 performance 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.
Nikolai Leonenko (NL) would like to thank for support and hospitality during the programme "Fractional Differential Equations" and the programmes "Uncertainly Quantification and Modelling of Material" and "Stochastic systems for anomalous diffusion" in Isaac Newton Institute for Mathematical Sciences, Cambridge. Also NL was partially supported under the ARC Discovery Grant DP220101680 (Australia), LMS grant 42997 (UK), grant FAPESP 22/09201-8 (Brazil) and Croatian Science Foundation (HRZZ) grant Scaling in Stochastic Models (IP-2022-10-8081). Nataliya Shchestyuk (NS) would like to thank for support provided by Knowledge Foundation.