New Mathematics and Innovative Numerical Methods for the Valuation of Options

用于期权估值的新数学和创新数值方法

• 批准号: 9706985
• 负责人: Junping Wang
• 金额: $ 7.5万
• 依托单位: Texas A&M Engineering Experiment Station
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1997
• 资助国家: 美国
• 起止时间: 1997-08-01 至 2001-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9706985&HistoricalAwards=false
• 关键词: New; Mathematics; Innovative; Numerical; Methods

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.

小行星9706985 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述。 其次，我们研究了解的唯一性和存在性。 第三，我们使用有限元方法来计算期权价格的基础上，我们提出的公式。 第四，我们提供了迭代方案，有效地解决系统的非线性代数方程所产生的有限元方法。 第五，我们将开发一个计算效率高且健壮的代码包。 建议的新数学的特点是弱变分方法的时间值的选择，通过使用希尔伯特空间方法。 弱形式的期权估值打开了一扇大门，使用有限元方法与网格局部细化在近似的期权定价函数和自由边界的有效的数值技术，如区域分解和多重网格方法。 特别是，这种方法提供了一个非常有前途的未来的计算金融产品，涉及多资产和证券，因为计算域将在这些应用中的多维。 金融衍生品是现代金融市场中一个重要且快速发展的领域。 例如，根据互换监测，1993年，仅互换这一特定种类的衍生工具的规模就约为9万亿美元，几乎相当于美国的年国民总收入。 这些衍生品的估值实际上是有用的，重要的，数学上具有挑战性。 自从这些金融产品首次交易以来，数学中的先进技术在理解和评估各种衍生证券方面一直发挥着重要作用。 随着新型金融产品的日益复杂，对新型高效的数学和计算技术的需求成为衍生产品的重要组成部分，同时也对衍生产品的估值产生了深远的影响。 该方法可以扩展到货币期权，利率期权和奇异期权，如亚式期权和回望期权，增加了困难，由于路径依赖的支付函数。特别是，这些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。