Topics in Mathematical Finance

在一个完整的金融市场中，每一个未定权益都可以被交易证券的受控投资组合完美地复制，因此存在一个定义明确的无套利价格。 在不完全市场中，或有债权通常是不可复制的。 在这种情况下，套利参数本身不足以确定唯一的价格，因此，更多的一般均衡为基础（或效用为基础）的方法必须使用。 本计画将研究以下与数理金融相关的课题：最适投资问题中价值函数二阶导数的存在性与最适投资策略关于初始财富的可微性;效用基础价格关于不可交易或有权益数的敏感性分析;“小”交易费用下最优投资问题的渐近分析;一个“大的”经济主体和金融市场的相互作用及其对衍生证券价格的影响。 一个特别注意的是在推导的“紧”（理想的，必要的和充分的）数学条件的结果成立。在过去的25年中，世界金融市场的一个重要发展是衍生产品市场的迅速扩大和扩展。 这之所以成为可能，主要是因为布莱克、斯科尔斯和默顿发起的无套利定价理论的创立。 最近发生的几起事件然而，最近几年的金融危机（例如，长期资本管理公司的濒临倒闭）表明，在实践中应谨慎使用无仲裁定价理论的结果，因为该理论没有考虑到金融市场的许多重要特征，如不完全性，流动性约束，交易成本等。在本项目中，我们计划开发技术，允许计算由于不同类型的市场不完善而对衍生品无仲裁价格的修正。

In a complete financial market every contingent claim can be perfectly replicated by a controlled portfolio of the traded securities and therefore admits a well-defined arbitrage free price. In an incomplete market, as a rule, contingent claims are not replicable. In this case, arbitrage arguments alone are not sufficient to determine unique prices and, hence, more general equilibrium based (or utility based) approach has to be used. This project will study the following related topics of Mathematical Finance: The existence of second derivatives of the value function in the problem of optimal investment and the differentiability of the optimal investment strategy with respect to initial wealth; sensitivity analysis of utility based prices with respect to the number of non-traded contingent claims; the asymptotic analysis of the problems of optimal investment under "small" transaction costs; the "equilibrium" derivation of the interaction between a "large" economic agent and a financial market and the impact of this interaction on the prices of derivative securities. A particular attention is on the derivation of "tight" (ideally, necessary and sufficient) mathematical conditions for the results to hold true.An important development in world financial markets over the last 25 years is the rapid broadening and expansion of derivatives markets. This became possible primarily because of the creation of the arbitrage-free pricing theory initiated by Black, Scholes and Merton. Several recent events (for example, the near-collapse of Long Term Capital Management) show, however, that the results of the arbitrage-free pricing theory should be used in practice rather cautiously as this theory does not take into account many important features of financial markets such as incompleteness, liquidity constraints, transaction costs and so on. In this project we plan to develop techniques that allow for the computation of the corrections to the arbitrage-free prices of derivatives due to different types of market imperfections.