Several Numerical Methods For American Option Pricing

Option is one kind of financial derivatives people designed to avoid the market risk. Option pricing is the core question of theory research and practical application of financial derivatives. The option pricing is a frontier problem as well as a hot issue of financial engineering and financial mathematics at present. For American options can be exercised at any time up to the maturity dates, they are more flexibility in practice. However, no closed-form solutions exist for American options of their valuation in general case. Therefore it is important to study the numerical methods for American option pricing.In this paper, we propose several numerical methods for American option pricing. The main content of the thesis is presented as follows:In chapter1, we briefly introduce some basic option knowledge, the American option pricing model and the property of American option price. In addition, the research status of numerical solution for American option pricing problem is introduced too.In chapter 2, a compact finite difference method for American option pricing is proposed. By transformations we change the free boundary problem of partial differential equation into an ordinary differential equation initial value problem. And then we apply the compact difference scheme to the new problem. Moreover, stability of the method is proved and the algorithm is given. Some numerical experiments have been carried out. And a comparison, made among the binary tree algorithm, projective successive over-relaxation method and finite element method, shows that the new method is high efficient and convergent.In chapter 3, a memory gradient projection method for American option pricing is used. Firstly, we convert the linear complementarity problem into a variational inequality one, and then make variable transformation for the variational inequality problem. Secondly, we study the equivalent optimization problem with an inequality constraint and boundary conditions, whose necessary condition for the optimality is the variational inequality presentation of American options. To solve the obtained optimization problem, we use the memory gradient projection method. The detailed algorithm is suggested. Finally, we test the algorithm and compare it with the binary tree algorithm and projective successive over-relaxation method. Numerical experiments show that the new method is high efficient and convergent.In chapter 4, we provide a new method for American option pricing by drawing lessons from finite element method. First, we restrict the linear complementary problem to a limited area. And then a variable transformation is made. After that we convert the problem into a variational inequality problem. Second, we discretize the time and space area. At the same time, we prove the stability of this scheme. After that, we give a algorithm and solve the problem by it. At last, we carry out some numerical experiments to compare with binary tree algorithm and projective successive over-relaxation method. Results show that this algorithm is high efficient and convergent.