Stochastic Modeling of Risk Aversion and its Implications for Derivative Pricing and Risk Management

在许多金融应用中，由于对冲约束或其他市场摩擦等原因，投资者不可避免地面临无法对冲的风险。衍生证券的价值自然取决于如何管理风险，以及投资者？在风险和回报之间的偏好。本课题将以员工股票期权（ESO）和信用衍生品为研究对象，系统地探讨将投资者的风险偏好纳入衍生品价值评估的问题。该项目的目标是设计有效和强大的估值模型，可以校准给定的市场数据，并应用于构建最佳的风险管理战略符合投资者？的风险偏好。该研究涉及开发适用的数学工具来建模，估计和分析这些衍生品的复杂风险结构，这对它们的价值有重大影响。此外，该项目还包括一些随机控制和最优停止问题，自然出现在这些估值问题的分析和数值研究。通过量化它们的风险和收益，本研究还将揭示这些衍生品的一些最优合约设计问题。员工股票期权在美国是薪酬的重要组成部分。出于对股东成本的担忧，美国财务会计准则委员会自2004年以来一直要求公司估计和报告股票期权的授予成本。经理股票期权定价的核心挑战在于期权行使时机的不确定性，这取决于员工的风险规避和期权的各种合同限制。另一方面，信贷衍生工具是一种金融工具，其回报取决于违约事件的发生，例如公司破产。信用衍生品的交易量在过去十年中急剧增长，但估值和风险管理技术却没有跟上。最近的信贷市场危机反映了主要金融机构对复杂的信贷衍生工具的风险管理不力。因此，本研究旨在建立一个明确考虑投资者风险偏好、合同特征和市场条件的估值框架，以便准确量化这些衍生工具的风险。这些估值问题不仅对个人或机构投资者很重要，对监管机构也很重要。

In many financial applications, investors are inevitably exposed to unhedgeable risks due to, for example, hedging constraints or other market frictions. The value of derivative securities naturally depends on how risks are managed, as well as the investor?s preferences between risk and return. This project will systematically investigate some recent problems that involve incorporating the investor's risk preferences into the valuation of derivatives, with a focus on employee stock options (ESOs) and credit derivatives. The project objective is to design efficient and robust valuation models that can be calibrated given market data and applied to construct the optimal risk management strategies consistent with the investor?s risk preferences. The research involves developing applicable mathematical tools to model, estimate, and analyze the complex risk structures of these derivatives, which have a significant bearing on their values. Moreover, the project includes the analytical and numerical studies of a number of stochastic control and optimal stopping problems that naturally arise in these valuation problems. By quantifying their risks and returns, the research will also shed light on some optimal contract design problems for these derivatives.Employee stock options are an important component of compensation in the US. Concerned about their cost to shareholders, the US Financial Accounting Standards Board since 2004 has required firms to estimate and report the granting cost of ESOs. The central challenge of ESO valuation lies in the uncertain timing of option exercises, which depends on the employee's risk aversion and various contractual restrictions of the options. On the other hand, credit derivatives are financial instruments whose payoffs are contingent on the occurrence of default event(s), such as the bankruptcy of a firm. The trading volume of credit derivatives has grown dramatically in the past decade, but the valuation and risk management technologies have not kept up. The recent crisis in the credit markets reflects ineffective risk management of complex credit derivatives by major financial institutions. Hence, this research aims to develop a valuation framework that explicitly accounts for the investor's risk preferences, contractual features and market conditions, so as to accurately quantify the risks of these derivatives. These valuation problems are important not only for individual or institutional investors, but also for regulators.