| In this thesis we study numerical pricing of American options on stocks and zero-coupon bonds. For American options on stocks, based on new exact formulations of American option problems on bounded regions, we establish error estimates for finite element approximations of American option prices under admissible regularity. To the best of our knowledge, there were no earlier known results using this approach. Numerical results are also presented to examine our theoretical results and to compare them with other approaches by means of several examples, which show that our methods provide very rapid and accurate option prices, early exercise prices, and hedge ratios. For American put options on zero-coupon bonds under the CIR model, we show the existence and uniqueness of the weak solution by formulating the corresponding free boundary problems as parabolic variational inequalities. Since the degenerate term in the highest order derivative is removed, this formulation leads to a type of finite element methods which are numerically stable in a stronger sense. Besides, we also study finite volume methods for the original free boundary problem. Numerical examples show that our methods provide very accurate approximations of option prices. Stability and convergence are also obtained for the two methods. In addition, we give an error checking method which is a practical gauge of whether or not a numerical method converges and has achieved a good accuracy. |
