Pricing financial and insurance products in the multivariate setting

In finance and insurance there is often the need to construct multivariate distributions to take into account more than one source of risk, where such risks cannot be assumed to be independent. In the course of this thesis we are going to explore three models, namely the copula models, the trivariate reduction scheme and mixtures as candidate models for capturing the dependence between multiple sources of risk. This thesis contains results of three different projects. The first one is in financial mathematics, more precisely on the pricing of financial derivatives (multi-asset options) which depend on multiple underlying assets, where we construct the dependence between such assets using copula models and the trivariate reduction scheme. The second and the third projects are in actuarial mathematics, more specifically on the pricing of the premia that need to be paid by policyholders in the automobile insurance when more than one type of claim is considered. We do the pricing including all the information available about the characteristics of the policyholders and their cars (i.e. a priori ratemaking) and about the numbers of claims per type in which the policyholders have been involved (i.e. a posteriori ratemaking). In both projects we model the dependence between the multiple types of claims using mixture distributions/regression models: we consider the different types of claims to be modelled in terms of their own distribution/regression model but with a common heterogeneity factor which follows a mixing distribution/regression model that is responsible for the dependence between the multiple types of claims. In the second project we present a new model (i.e. the bivariate Negative Binomial-Inverse Gaussian regression model) and in the third one we present a new family of models (i.e. the bivariate mixed Poisson regression models with varying dispersion), both as suitable alternatives to the classically used bivariate mixed Poisson regression models