Stochastic covariance and first passage time for multidimensional stochastic processes.

这个研究项目有两个总体目标。首先研究了具有随机协方差的多变量随机模型及其在金融市场风险和外来产品定价中的定量影响。其次，对具有或不具有随机相关和波动率的多障碍产品进行评价。它们都是将高级随机协方差模型和首次通过时间问题相结合的长期追求的一部分。捕捉不同资产的共同行为将有助于更好地评估金融风险，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。模型必须捕获尽可能多的程式化事实，同时在参数数量和它们导致封闭形式简单表达式的潜力方面足够易于处理。许多复杂的多元产品在实践中由于使用不适当的模型而被错误定价。这引起了研究界的许多声音，声称对包含多变量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，其中抵押债务凭证（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元过程，不仅存在随机协方差，而且在波动率和相关波动率的波动率水平上也存在高阶随机。我还考虑了复杂金融衍生品的定价，这些衍生品的收益取决于几种基础资产的集体行为，假设上述模型描述的动态。一篮子期权和点差期权、山脉衍生品和有或无障碍的债务抵押债券是我们的具体目标。定价表达式将通过基于CCF、Green函数、图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，该项目的成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，并对加拿大银行和其他金融机构产生相应的好处。

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.