Stochastic covariance and first passage time for multidimensional stochastic processes.

这项研究项目的总体目标有两个。首先，研究了具有随机协方差的多元随机模型及其在金融市场风险和奇异产品定价中的量化影响。第二，具有和不具有随机相关性和波动性的多障碍多元产品的评价。它们都是将高级随机协方差模型和首次通过时间问题结合起来的长期追求的一部分。 捕捉不同资产的联合行为将带来更好的金融风险评估，以及对金融机构在过去十年中发行的一些复杂衍生品的更准确理解。模型必须捕捉尽可能多的风格化事实，同时在参数数量和它们可能导致封闭形式的简单表达式方面足够容易处理。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。这在研究界引发了许多声音，他们声称，对包含多元衍生品的高度非线性投资组合的风险评估不力，是信贷部门持续危机的主要原因之一，其中对抵押品债务凭证(CDO)的错误评估引发了关键机构的破产。 在波动率的波动率和相关波动率的水平上，不仅存在随机协方差，而且还存在高阶随机性，我将提出并研究几个新的多变量过程。我还考虑了复杂金融衍生品的定价，其收益取决于几个标的资产的集体行为，假设上述模型描述的动态。篮子和价差期权、山脉衍生品和有无障碍的债务抵押债券是我们的具体目标。价格表达式将通过基于CCF、格林函数、图像法和偏微分方程组的解析解的技术来获得。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，该项目的结果将通过更好地评估其定价方法，对金融部门产生重大影响，并为加拿大银行和其他金融机构带来相应的好处。

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.