Option pricing under the generalized mixed fractional brownian motion model

In this thesis, we consider the pricing of option derivatives under the so-called GMFBM (generalized mixed fractional Brownian motion) model, which is a generalization of all the FBM (fractional Brownian motion) models. Based on the martingale properties of FBM illustrated in [33], a new theorem illustrating that the GMFBM is not a semimartingale under certain conditions is successfully obtained. By using the portfolio analysis and applying the Wick-Itˆo formula, a PDE (partial differential equation) governing the price of option derivatives under the GMFBM model is then derived. A closed-form analytical solution of European options under the GMFBM model is also solved by using the Fourier transform. On the other hand, we consider the pricing of American-style options under this new model. We have derived the closed-form analytical solution for perpetual American options. For American options with finite-maturity, we formulate the pricing problem under the current model as a LCP (linear complementarity problem). Due to the numerical instability resulting from the degeneration of the governing PDE as time approaches zero, the LCP under the GMFBM model is more difficult to be solved accurately than the original B-S (Black-Scholes) model or the generalized B-S model discussed in [57]. Despite difficult, we have managed to design a stable numerical approach to solve for the option price accurately. It is shown that the coefficient matrix of the current method is an M-matrix, which ensures its stability in the maximum-norm sense. We also provide a sharp theoretic error estimate of the current method, which is further verified numerically. The results of various numerical experiments also suggest that this new approach is quite accurate, and could be easily extended to price other types of financial derivatives with an American-style exercise feature under the GMFBM model.