Essays on variance risk

My PhD thesis consists of three papers which study the nature, structure, dynamics and price of variance risks. As tool I make use of multivariate affine jump-diffusion models with matrix-valued state spaces. The first chapter proposes a new three-factor model for index option pricing. A core feature of the model are unspanned skewness and term structure effects, i.e., it is possible that the structure of the volatility surface changes without a change in the volatility level. The model reduces pricing errors compared to benchmark two-factor models by up to 22%. Using a decomposition of the latent state, I show that this superior performance is directly linked to a third volatility factor which is unrelated to the volatility level. The second chapter studies the price of the smile, which is defined as the premia for individual option risk factors. These risk factors are directly linked to the variance risk premium (VRP). I find that option risk premia are spanned by mid-run and long-run volatility factors, while the large high-frequency factor does not enter the price of the smile. I find the VRP to be unambiguously negative and decompose it into three components: diffusive risk, jump risk and jump intensity risk. The distinct term structure patterns of these components explain why the term structure of the VRP is downward sloping in normal times and upward sloping during market distress. In predictive regressions, I find an economically relevant predictive power over returns to volatility positions and S&P 500 index returns. The last chapter introduces several numerical methods necessary for estimating matrix-valued affine option pricing models, including the Matrix Rotation Count algorithm and a fast evaluation scheme for the Likelihood function.