Multivariate modelling of financial markets

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This proposal concerns the multivariate modeling of financial Markets beyond the classical Multivariate Geometric Brownian motion, by using innovative ideas, concepts and techniques that contribute to the theory of modern mathematical finance. Special attention is deserved to the modeling of the dependence structure between the different assets. The proposed models introduce jumps, stochastic volatility, stochastic covariance, fractional processes and sometimes combinations of theses features. For the proposed models, fundamental problems as pricing and sensitivity analysis are considered in this proposal, using both exact and approximate pricing approaches. For some of the proposed models, mainly for those that include a Fractional Brownian motion component and therefore may not be semimartingales, it is relevant to study the existence of arbitrage opportunities, and the hedging strategies associated to them. This approach emphasize on the fine structure of the trajectories instead of the probabilistic structure of the model. The statistical inference of the parameters of these multivariate models constitute also a very relevant problem that will be studied. The study of these models, and in particular the study of the dependence structure between the assets, provides a better understanding of the dynamics of financial markets, which is important from the risk management perspective.