Measures of financial risk and derivative pricing

我的研究目标是为金融应用开发数学模型和理论，并将结果应用于金融市场。这些主题包括衍生证券的定价和对冲、风险度量和风险管理。金融风险的度量和管理是数学金融学中的重要课题。我的工作是研究与“连贯的风险度量”相关的问题。众所周知，一个连贯的风险度量是由一组概率度量描述的，也就是说，当且仅当集合中概率度量下的每个期望值都是非负的时，未来值是可接受的。当存在动态交易的可能性或现有信息的演变时，就需要进行多阶段的风险衡量。这个话题已经在报纸上(与Artzner，Delbaen，Eber和Heath)讨论过了。风险度量在金融理论中被广泛应用，我还研究复杂衍生证券的定价和套期保值。自从布莱克-斯科尔斯公式提供了定价和对冲衍生品的理论方法以来，数学理论使得无数衍生品的创造和定价成为可能。在现实中，一个市场是“不完整的”。在这种不完全的市场中，定价和对冲变得更加复杂。拟议中的研究将为金融机构(如银行)提供理论背景，并帮助它们设计专门的衍生品，找到一个“公平”的价格，并管理它们的风险。

The objective of my research is to develop the mathematical models and theory for financial applications and to apply the results to financial markets. The topics include pricing and hedging derivative securities, measures of risk, and risk management.Measurement and management of financial risks are important topics in mathematical finance. I work on problems related to a "coherent risk measure". It is known that a coherent risk measure is described by a set of probability measures, in the sense that the future value is acceptable if and only if every expected value under probability measures in the set are nonnegative. When there is the possibility of dynamic trading or the evolution of available information, the need for multi-period risk measurement arises. This topic has been treated in the papers (with Artzner, Delbaen, Eber and Heath). Risk measures are widely applied to finance theory.I also work on pricing and hedging of complicated derivative securities. Since Black-Scholes formula provided a theoretical method of pricing and hedging derivatives, the mathematical theory has permitted the creation and pricing of numerous derivatives. In reality, a market is "incomplete". Pricing and hedging in such incomplete markets become more complex. The proposed research will provide financial institutions (such as banks) with the theoretical backgrounds and help them design specialized derivatives, find a "fair" price, and manage their risks.