We present a lattice-based algorithm to price both European and American-style derivatives under the same fractional Brownian motion environment in which Hu and Oksendal (2003), Necula (2004), and Zhuang and Song (2023) obtain explicit-form formulas for European options. To manage the path-dependency induced by the asset value process, we apply a procedure recalling the forward shooting grid method. Hence, we discretize the asset evolution by constructing a grid made up of a limited number of representative buckets at each time step. For each bucket in the grid, we establish a binomial dynamics to identify the successors at the next observation epoch. Since the grid is composed by representative buckets, it may occur that successors do not appear in the grid. In these cases, we invoke interpolation techniques to proceed backward on the grid and compute the derivative price at inception. Numerical investigations show the accuracy and efficiency of the proposed model.
