Finite Volume Method For American Option And American Option Under CEV Model

This paper develops an efficient finite volume method for pricing problem of American option, and then apply it to the American option under the CEV model, which is more complicated than standard American option.Since the American option problem is defined on an unbounded and irregular domain, we need some techniques to truncate the domain for computational purpose. In this paper, after changing the original problem into a linear problem, we will adopt front-fixing method and perfect match layer (PML) method for the irregular boundary and the unbounded boundary, respectively.For the unknown free boundary, we will use Newton iteration in each temperial step, where we solve the free boundary and option price simultaneously. For the constant constant elasticity of variance (CEV) model, the problem will be transform into a bounded problem when a>\, which is easily to deal with. Here, we only consider the situation when α<1, which can be solved by the similar arguments.After given the model and the algorithm of the two aforementioned problem, We will also prove the positive definiteness of this two problem in this paper. For the two algorithm, each step can be just seems as a process of solving a linear equations system. So if we can prove the coefficient matrix is an M-matrix, the problem is positive. Now we can prove our method is correct and reasonable. Through the numerical experiment, we can see our method is useful in real problem, and its result fits the consequence of other methods. Also it is much smoother and faster which is very satisfactory.Times are changing, more kinds of option will be created. The research of the American option pricing problem will never ends. We will continue to work hard in this field.