A Fixed Point Algorithm For The Linear Complementarity Problem Arising From American Option Pricing

Option is one of the most important derivatives in financial market. There aremany types of Option, but few of them have available analytical solutions. So it isnecessary for us to develop numerical method for pricing Options which do not haveanalytical solutions. And the American option pricing is the most classical problemamong them. In this paper, we transform the original Black-Scholes-Merton modelinto linear complementarity problems (LCP) by discretization. LCP has beendeveloped for many years and has been a very important mathematical instrument. Infact, the methods for solving LCPs have a mature theoretical basis. But, to find a morefeasible and proper method, we need to take the specificity of the discretized modelinto consideration. In addition, when using the finite differential methods to theoriginal problem, we also have several differential schemes to choose, such as explicitscheme, implicit scheme and Crank-Nicolson scheme. With the knowledge above, weneed to design the corresponding algorithm, according to the special structure of theproblem presented in this paper.In this paper, after comprehensive reviewing the literatures on American optionpricing problem, we proposed a fixed point algorithm to solve the discretized LCPs.The main research work and results are as follows:1. Firstly, the source of option pricing problem, its development and thedevelopment of financial markets further demonstrates the necessity for theresearches on option pricing. It also discusses the development of optionpricing model, the existing models and the various numerical methods forthese models.2. Secondly, this paper describes the American option pricing problem and setup the mathematical model. Besides, the finite differential method especiallythe Crank-Nicolson scheme and their relative properties are introduced. Then,transform the original method into LCPs that we need.3. Thirdly, this paper proposes the new fixed point algorithm for the American.And according to the special structure of the problem, we prove thecoefficient matrix is a P-matrix. By combining this property, this papergives the convergence of the algorithm. 4. Finally, the new algorithm is implemented by MATLAB. After the analysis ofthe numerical results, it is easy to find that the proposed algorithm is moreeffective and stable than the PSOR method. What is more important, theaccuracy of the fixed point algorithm improves with the refinement ofdifference.