CAREER: Financial Engineering, Incomplete Markets and Investment Under Uncertainty

该学院早期职业发展（CAREER）计划奖为金融工程的研究和教育提供资金。重点是开发适合不完全金融市场的期权定价和估值方法。一旦放松对完美市场的不切实际的假设，金融市场就会出现不完全性。经典的布莱克和斯科尔斯估值不再适用于这种情况。基于经济学的效用无差别定价方法是近年来出现的一种处理不完全市场的方法，本文旨在将该方法推广到具有无限成熟度的美式期权。所使用的技术包括随机，偏微分方程和鞅。当标的资产的不可交易性导致不完全性时，这类期权在模拟公司面临的投资（和放弃）决策时非常有用。效用无差别定价涉及选择一个合适的效用函数来描述风险偏好。研究的第二个主要目标是发展时间一致性效用函数的概念。无限期限美式期权的考虑导致了中间现金流的估值问题和由于期限的复杂性。朴素的极限接受导致退化控制问题，因此必须注意开发新的方法。这项研究将为投资组合管理带来新的方法，并与动态风险度量相联系。这项研究的结果将在企业融资和真实的期权等不同领域产生影响。从这项研究中产生的改进的公司投资决策可以使公司的股东受益，从而使社会受益。 此外，随机最优控制和鞅的技术将是数学金融研究人员感兴趣的。金融业将受益于效用无差异定价技术的发展，因为它们在为衍生品定价时需要每天科普市场不完全性的实际问题。

This Faculty Early Career Development (CAREER) Program award provides funding for research and education in financial engineering. The focus is on the development of option pricing and valuation methodologies suitable for incomplete financial markets. Incompleteness arises in financial markets as soon as the unrealistic assumptions of a perfect market are relaxed. The classic Black and Scholes valuation is no longer applicable in such situations. The approach of utility indifference pricing, with roots in economics, has emerged recently to deal with incomplete markets, and the research aims to extend this methodology to American options with an infinite maturity. The techniques used include stochastics, partial differential equations and martingales. When incompleteness arises from non-tradability of the underlying asset, such options are useful in modeling the investment (and abandonment) decisions faced by firms. Utility indifference pricing involves choice of a suitable utility function to describe risk preferences. The second main goal of the research is to develop the concepts of time consistent utility functions. Consideration of infinite horizon American options leads to issues of valuing intermediate cashflows and complications due to the horizon. Naive limit taking results in degenerate control problems, so care must be taken to develop new approaches. This research will lead to new methods for portfolio management and links to dynamic risk measures.The results of this research will have impact in diverse areas such as corporate finance and real options. Improved corporate investment decisions arising from this research could benefit the shareholders of firms, and therefore society. Additionally, the techniques of stochastic optimal control and martingales will be of interest to researchers in mathematical finance. The finance industry will benefit from developments in utility indifference pricing techniques, as they need to cope daily with the practicalities of the incompleteness of markets when pricing derivatives.