We first give a brief overview of American option pricing models and numerical methods. We treat American option models as a special class of obstacle problems. Finite element formulation is introduced together with error analysis of numerical solutions. Some interesting properties about sensitivity of the option price to the payoff function are proved. We also give a criterion for the convergence of numerical free boundaries (optimal exercise boundaries) under mesh refinement. Some future research plans will be discussed. Bio: Yongmin Zhang is a risk management consultant at Wells Fargo. Prior to the current position, he was a lead research analyst in Capital Market Research Group of Washington Mutual (now part of J. P. Morgan). His area is in fixed income and mortgage analysis. Before he joined this group, he was an assistant professor at State University of New York where he did research in turbulent flow and American options with more than thirty publications and taught numerous courses in applied mathematics and statistics. Prior to this appointment, he was a research scientist at SUNY Research Foundation. He was a co-principle investigator for various grants from US Department of Energy. He holds his Ph.D. in Applied Mathematics from University of Chicago.