In this talk, we consider the pricing of option derivatives under a generalized mixed fractional Brownian motion (GMFBM) model. As the name suggests, the GMFBM model is a generalization of all the FBM (fractional Brownian motion) models used in the literature, and is proved to be able to effectively capture the long-range dependence of the stock returns. To develop the pricing mechanics, we first derive a sufficient condition for the market modeled under the GMFBM to be arbitrage free. Then, under the risk-neutral assumption, for European-style options, we consider the pricing of credit default swaps (CDS) by investigating the two legs of the cash flow involved. The price we obtained involves elementary functions only, and can be easily implemented for practical purpose. For American-style options, we introduce a robust numerical method for American puts under this new model. By using portfolio analysis and applying the Wick-It  formula, a partial differential equation (PDE) governing the price of American puts under the GMFBM is successfully derived. Based on this, the pricing of American puts is then formulated as a linear complementarity problem (LCP). The newly obtained LCP under the GMFBM model is difficult to be solved accurately because of the numerical instability resulting from the degeneration of the governing PDE as time approaches zero. To overcome this difficulty, a numerical approach based on the upwind scheme is adopted. It is shown that the coefficient matrix of the current method is an M-matrix, which ensures the stability in the maximum-norm sense. A sharp theoretic error estimate for the current method is also provided, which is further verified numerically.