Stochastic covariance and first passage time for multidimensional stochastic processes.

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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.