New Mathematics and Innovative Numerical Methods for the Valuation of Options

9706985王俊平将讨论期权估值的五个重要方面。首先，我们提出了一个新的自由边值问题的数学表达式。其次，我们研究了解的唯一性和存在性。第三，基于本文提出的公式，采用有限元方法计算期权价格。第四，我们提供了迭代方案来有效地求解由有限元法产生的非线性代数方程组。第五，我们将开发一个计算效率高且健壮的代码包。提出的新数学方法的特点是利用希尔伯特空间方法对期权的时间值进行弱变分处理。期权估值的弱形式为使用有限元方法和网格局部细化方法来逼近期权定价函数和通过有效的数值技术（如域分解和多网格方法）逼近自由边界打开了大门。特别是，这种方法为涉及多资产和证券的金融产品的计算提供了一个非常有希望的未来，因为在这些应用中计算域将是多维的。金融衍生品是现代金融市场中一个重要且快速增长的领域。例如，根据掉期监测报告，1993年仅掉期这一特殊衍生品的规模就约为9万亿美元，几乎相当于美国每年的国民总收入。这些衍生品的估值实际上是有用的，重要的，并且在数学上具有挑战性。先进的数学技术在各种衍生证券的理解和估值中起着重要的作用，因为这些金融产品的第一次交易。随着新型金融产品的日益复杂，对新的、高效的数学和计算技术的需求越来越大，衍生品的估值也产生了深远的影响。该方法可以扩展到货币期权、利率期权以及亚洲期权和回溯期权等由于支付函数的路径依赖而增加了难度的奇异期权。特别是，该方法可以根据新的和经验相关的利率扩散产生各种债券的价值，从而产生掉期和各种抵押贷款支持证券的价值。

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.