Stochastic covariance and first passage time for multidimensional stochastic processes.

这个研究项目有两个主要目标。首先研究了具有随机协方差的多元随机模型及其对金融市场风险和奇异产品定价的定量影响。第二，涉及多个障碍的多变量产品的评估，有和没有随机相关性和波动性。它们都是结合先进的随机协方差模型和首次通过时间问题的长期追求的一部分。 捕捉不同资产的联合行为将带来对金融风险的更好评估，并更准确地理解金融机构在过去十年中发行的一些复杂衍生品。模型必须捕捉尽可能多的程式化事实，同时在参数数量和它们的潜力方面足够容易处理，以导致封闭形式的简单表达式。许多复杂的多变量产品在实践中由于使用了不适当的模型而被错误定价。这引起了研究界的许多声音，声称对包含多元衍生品的高度非线性投资组合的风险评估不佳是信贷部门持续危机的主要原因之一，其中对抵押债务债券（CDO）的不正确评估引发了关键机构的破产。 我将提出并研究几个新的多变量过程，不仅存在随机协方差，而且在波动率和波动率的相关性水平上存在高阶随机。我还考虑了复杂的金融衍生品的定价，其收益取决于几个基础资产的集体行为，假设上述模型所描述的动态。我们的具体目标是一篮子和价差期权、山脉衍生品和有障碍和无障碍债务抵押债券。定价表达式将通过基于CCF、绿色函数、镜像法和偏微分方程解析解的技术得到。我将使用这些属性和技术对这些模型进行校准和测试，以获得真实的市场数据。 我预计，通过更好地评估该项目的定价方法，该项目的成果将对金融部门产生重大影响，加拿大银行和其他金融机构也将从中受益。

There are two general goals of this research project. First the study of multivariate stochastic models with stochastic covariance and their quantitative impact in the risk of financial markets and the pricing of exotic products. Secondly, the evaluation of multivariate products involving multiple barriers with and without stochastic correlation and volatility. They are both part of a longer-term pursue for combining advanced stochastic covariance models and first passage time problems. Capturing the joint behavior of different assets would bring a better assessment of financial risks and a more accurate understanding of some of the complex derivatives that financial institutions have issued in the last decade. The models must capture as many stylized facts as possible while being tractable enough in terms of number of parameters and their potential to lead to closed form simple expressions. Many complex multivariate products have been mispriced in practice by the use of inadequate models. This has raised many voices in the research community claiming that the poor evaluation of risk in highly nonlinear portfolios containing multivariate derivatives is one of the main reasons of the ongoing crisis of the credit sector, where the incorrect evaluation of Collateral Debt Obligations (CDO's) provoked the bankruptcy of key institutions. I will propose and examine several novel multivariate processes in the presence not only of stochastic covariance but also higher order stochastic at the level of volatility of volatility and volatility of correlation. I also consider the pricing of complex financial derivatives whose payoffs depend on the collective behavior of several underlying assets assuming the dynamic described by the models above. Basket and Spread Options, Mountain Range Derivatives and Collateralized Debt Obligations with and without barriers are our specific targets. Pricing expressions will be obtained by techniques based on the CCF, the Green function, the method of images and the analytical solutions of PDEs. I will use these properties and techniques for calibration and testing of these models to real market data. I expect the results derived from the project will have a significant impact in the financial sector through a better assessment of its pricing methodologies, with the corresponding benefit to Canadian banks and other financial institutions.