Upper bound algorithms for pricing American-style options (also called American options) usually rely on the approximations of the optimal martingale. Tightness of the estimated upper bounds highly depends on the construction of the martingale. The nested simulation method constructs the martingale with the aid of an estimated lower bound process. It can be easily implemented. However, a limitation of the nested simulation method is its requirement of large computational effort. Several non-nested upper bound algorithms for pricing American-style options have been proposed in recent years. They are computationally attractive where nested simulations are not necessary. The computational cost is linear in the number of exercise dates. To incubate non-nested upper bound algorithms, researchers study the characteristics of the optimal martingale from various aspects, such as stability, martingale representation theorem, analytical representation and connection with delta hedging. For the ideas of analytical representation and connection with delta hedging, the construction of the martingale comes down to the estimation of delta process. It is well known that delta is one of the most important hedging parameters for American-style options. However, estimating hedging parameters at any date efficiently is not an easy task. In this thesis, based on the dynamic-programming representation of the value process, a least-squares method (LSM) is proposed to estimate the hedging parameters backwardly. With this method, estimation of the delta process can be done in an efficient way. Based on the delta estimates, two ideas are implemented to generate non-nested upper bounds. The first one is that the delta estimates can be included into the basis functions of the regression procedure when estimating the martingale. The other one is that the martingale can be constructed directly with no sub-simulation and no optimization. Through numerical experiments, it is found that the latter one is a better choice.