CAREER: Financial Engineering, Incomplete Markets and Investment Under Uncertainty

该学院早期职业发展(Career)计划奖为金融工程领域的研究和教育提供资金。重点是开发适用于不完全金融市场的期权定价和估值方法。一旦放松了对完美市场的不切实际的假设，金融市场就会出现不完全。经典的布莱克和斯科尔斯估值在这种情况下不再适用。效用无差别定价方法起源于经济学，是最近出现的一种处理不完全市场的方法，本研究旨在将这一方法推广到无限到期的美式期权。所使用的技巧包括随机学、偏微分方程式和鞅。当标的资产的不可交易性导致不完备性时，这种期权在模拟公司面临的投资(和放弃)决策时是有用的。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相容效用函数的概念。考虑无限远期美式期权导致了中间现金流的估值问题，以及由于远期而产生的复杂性。天真的取极限会导致控制问题的退化，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，研究结果将对公司财务和实物期权等多个领域产生影响。这项研究产生的更好的公司投资决策可能会使公司的股东受益，从而使社会受益。此外，随机最优控制和鞅的技术将引起数学金融学研究人员的兴趣。金融行业将受益于公用事业无差别定价技术的发展，因为它们在为衍生品定价时，需要每天应对市场不完备的实际情况。

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 The classic Black and Scholes valuation is no longer applicable in such situations.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.