9706985 Junping Wang Five important aspects on the valuation of options will be addressed. First, we propose a new mathematical formulation for the free boundary value problem. Second, we investigate the uniqueness and existence of the solution. Third, we use finite element methods to compute the option price based on our proposed formulas. Fourth, we provide iterative schemes to effectively solve the system of nonlinear algebraic equations arising from the finite element method. Fifth, we will develop a code package that is computationally efficient and robust. The proposed new mathematics features a weak variational approach to the time value of the option by using a Hilbert space method. The weak form for the valuation of options opens a door to the use of finite element methods together with grid local refinement in the approximation of the option pricing function and the free boundary by efficient numerical techniques such as domain decomposition and multigrid methods. In particular, this approach provides a very promising future for the computation of financial products involving multi-assets and securities, as the computational domain will be of multi-dimension in those applications. Financial derivatives are a major and fast growing area in modern financial markets. For example, according to the Swaps Monitor, the size of swaps alone, a particular kind of derivatives, was approximately $9 trillion in 1993,almost the size of the annual gross national income of the United States. The valuation of these derivatives is practically useful, important, and mathematically challenging. Advanced techniques in mathematics have been playing an important role in the understanding and valuation of various derivative securities ever since the first trading of these financial products. With increasing complexity of new and exotic financial products, the demand for new and efficient techniques in mathematics and computation becomes greater type derivatives, but also has far-reaching im pact on derivative valuations in general. The methods can be extended to currency options, interest rate options and exotic options such as Asian options and lookback options which have added difficulties due to the path-dependence of the payoff function. In particular, the methods can yield the value of various bonds based on the new and empirically relevant interest rate diffusions, resulting in the value of swaps and various mortgage-backed securities.
