Three essays on modeling the dependence between financial assets

This thesis addresses two aspects of the dependence between financial assets. The first part is about the dependence between random vectors. The first chapter consists in a comparison of several algorithms that compute the optimal transport map for the quadratic cost between two (possibly continuous) probabilities over R^n. These algorithms compute couplings, called maximum correlation couplings, which have a property of extreme dependence that naturally appears in the definition of multivariate risk measures. The second chapter defines a notion of extreme dependence between random vectors based on the covariogram; the extreme couplings are characterized as maximum correlation couplings, up to a linear transform of one of the multivariate margins. A numerical method to compute these couplings is provided, and applications to the stress testing of multivariate dependence for portfolio allocation and the pricing of European options on several underlyings are detailed. The last part describes the spatial dependence between two Markovian diffusions, coupled with a state dependent correlation function. An integrated Kolmogorov forward PDE is established that relates the family of spatial copulas of the diffusion and the correlation function. Then the problem of attainable spatial dependence between two Brownian motions is addressed, and we show that some classical copulas are not admissible to describe the stationary dependence between Brownian motions.