Two Classes Of Numerical Methods For Solving The American Option Pricing Problem

The American option pricing problem can be described mathematically by the well-known Black-Scholes equation with a free boundary. It is very di?cult to solve this problemsince the free boundary is unknown.Muthuraman proposed a numerical method to solve the Black-Scholes equation, calledthe moving boundary approach. However, there is a limit in this approach, which demandsthat the starting boundary must be less than or equal to the free boundary, otherwise theboundary renew strategy fails. In Chapter 3 in this thesis, an improved moving boundaryapproach is proposed which can overcome the disadvantage of the original approach andnumerical tests show the e?ciency of our improved approach.The penalty function method is another important numerical method for solving theBlack-Scholes equation. By adding a continuous penalty term with a small parameter ? intothe equation, we can transform the original free boundary problem into a nonlinear problemwith a fix boundary. Once the resulting nonlinear equation is solved, we can then obtainan approximate solution of the original option pricing problem by choosing a very smallparameter ?.However, there are two di?culties in the penalty function method. First, the value ofthe option must satisfy the positivity constraint, which brings a great di?culty to constructa numerical scheme satisfying such a constraint to solve the resulting nonlinear equationwith an extra penalty term being included, as pointed out by Nielsen. Secondly, the penaltyterm is a nonlinear function of the option value. If we deal with the penalty term withimplicit scheme, then a nonlinear equation must be solved which needs iteration. If we dealwith the penalty term with an explicit scheme, though the iteration process can be avoided,we have to pay the cost of losing accuracy since the the accuracy order is the first order onlywhen an explicit scheme is used.In Chapter 4 in this thesis, the American multi-asset option pricing problem is consid-ered. The Eulerian-Lagrangian splitting skill is firstly used to solve the resulting nonlinearproblem subject to a fixed boundary after the penalty function term has been added to the original equation. The main idea is that we divide the resulting nonlinear equation intotwo equations and then solve them separately. In the step of Euler, the equation whichincludes the penalty term can be solved analytically, thus the di?culty of the option valueto satisfy the positivity constraint is conquered e?ciently. Secondly the accuracy of compu-tational scheme is improved in the step of Lagrange, thus the whole accuracy can be greatlyimproved up. As two numerical examples, both the single- and multi-asset option pricingproblems are tested and numerical results show the e?ciency of the new method.