Finite Element Method For Option Pricing

Option is one of the most important derivatives in financial market. There are many types of Option, but few of them have available analytical solutions. So we have to develop numerical method for pricing Options which do not have analytical solutions. In this dissertation, we are devoted to applying Finite Element Method (FEM) for pricing Options. We resolve four different types of Option: Single-asset Option, Two-asset Basket Option, Continuously Observed Asian Option and Discretely Observed Asian Option.When pricing Single-asset Option, we investigate the American-style Options with free boundary. We also discuss the risk management parameters (Greeks) of Single-asset Option, such as Delta, Theta, Vega and Rho, and compare the different property of Greeks of American Option to those of European Option.The Two-asset Basket Option is the first two-dimension option problem investigated in this dissertation. We bulid the finite element model for the Basket Option, and apply triangle and quadrangle elements in the numerical computation. Besides, we use local refinement meshes in computations and obtain better results.Pricing the Asian Option with FEM is the key part in this dissertation. Asian Option can be classified into many categories according to different criterions, such as European type and American type; Fixed strike type and Floating strike type; Continuously Observed type and Discretely Observed type. We apply FEM for pricing different types of the Asian Option, and present the mathematical expression of the free boundary which the American Asian Option has. We also apply local refinement mesh schemes in numerical testings, and lead to significant efficiency gains over uniform meshes with comparable number of elements.Finally, we test many numerical examples for option pricing, and the numerical results demonstrate the accuracy, convergence and efficiency.