Time Changes of Markov Processes: Applications in Financial Mathematics

该项目涉及股票价格、汇率、利率、商品和能源价格等金融变量的随机建模，以及金融衍生品的定价，金融衍生品是作为缓解金融市场和信用风险的工具的金融工具。其目标是开发经验上现实的、易于分析的模型，描述金融变量的随机动态，以及实施这些模型的分析和计算方法。该项目集中在以下几个方面。(I)开发丰富的金融模型工具箱，包括国家依赖跳跃、随机波动和违约。这将基于马尔可夫过程的时间变化的方法学。所提出的模型结构是将马尔可夫过程与具有时间变化的可解析可处理的半群与可解析可处理的拉普拉斯变换配对，以设计具有期望性质的可解析可处理过程。(Ii)在这些模型中，基于拉普拉斯变换逆变换和对称马尔可夫过程的谱展开方法，开发有效的衍生证券定价工具。(3)发展了具有跳跃、随机波动率和违约的下一代统一信贷权益模型，该模型在具有杀戮的马尔可夫过程的时间变化的框架下，对与给定公司有关的所有财务义务，包括股票、债务、股票期权和信用衍生品，提供了统一的处理。在这个统一的框架下研究期权定价和公司债务估值。(Iv)开发一类新的利率、大宗商品和能源市场的均值回归跳跃模型，方法是使均值回归扩散受到时间变化的影响。这个项目的目的是为金融实践开发复杂的数学工具。本研究可望具有重要的实践意义。根据国际清算银行(2006年)的数据，全球衍生品市场的名义规模为343万亿美元。主要市场包括用于管理金融风险的利率、货币、股票、大宗商品、能源和信用衍生品。预计这项拟议的研究将在所有这些金融市场领域找到应用。本项目开发的模型在经验上切合实际，同时在分析和计算上易于处理，将有助于对用于管理市场和信用风险的金融工具进行一致、快速和准确的建模和定价。这将有助于提高金融市场的效率和稳定性。预计该项目还将对随机分析和应用概率产生更广泛的数学影响。本项目中开发的马尔可夫过程的分析易处理的时间变化将为随机分析的理论发展提供有用的实验室，并在使用马尔可夫过程对物理、生物、工程和经济现象进行建模的各种领域中找到应用。该项目将对教育和人力资源开发产生影响。这是西北大学在金融工程方面的长期发展努力的一部分，包括最近建立的博士学位。它将为学术界和工业界培养高素质的研究人员。

This project is concerned with stochastic modeling of financial variables, such as equity prices, foreign exchange rates, interest rates, and commodity and energy prices, and the pricing of financial derivatives, financial instruments that serve as tools to mitigate financial market and credit risk. The goal is to develop empirically realistic and analytically tractable models describing stochastic dynamics of financial variables and analytical and computational methods to implement these models. The project focuses in the following areas. (i) Develop a rich toolbox of financial models with state-dependent jumps, stochastic volatility, and default. This will be based on the methodology of time changes of Markov processes. The proposed model architecture is to pair Markov processes with analytically tractable semigroups with time changes with analytically tractable Laplace transforms to design analytically tractable processes with desired properties. (ii) Develop efficient pricing tools for derivative securities in these models based on Laplace transform inversion and, for symmetric Markov processes, on the spectral expansion method. (iii) Develop the next generation of unified credit-equity models with jumps, stochastic volatility, and default that provide a unified treatment for all financial obligations related to a given firm, including stock, debt, stock options, and credit derivatives in the framework of time changes of Markov processes with killing. Study options pricing and corporate debt valuation in this unified framework. (iv) Develop a novel class of models with mean-reverting jumps for interest rates, commodities, and energy markets by subjecting mean-reverting diffusions to time changes with jumps. The aim of this project is to develop sophisticated mathematical tools for financial practice. This research is expected to have significant practical impact. According to the Bank for International Settlements (2006), the size of the global derivatives markets is $343 trillion in notional amounts. Major market segments include interest rate, currency, equity, commodity, energy, and credit derivatives used to manage financial risks. The proposed research is expected to find applications in all of these financial market sectors. Empirically realistic and, at the same time, analytically and computationally tractable models developed in this project will facilitate consistent, fast, and accurate modeling and pricing of financial instruments used to manage market and credit risk. This will help improve efficiency and stability of financial markets. The project is also expected to have a broader mathematical impact on stochastic analysis and applied probability. Analytically tractable time changes of Markov processes developed in this project will provide a useful laboratory for theoretical developments in stochastic analysis, as well as find applications in a variety of areas that employ Markov processes for the modeling of physical, biological, engineering, and economic phenomena. The project will have an impact on education and human resources development. It is part of the long-term development effort at Northwestern University in financial engineering, including the recently established Ph.D. concentration. It will train highly qualified researchers for academia and industry.