Quantification of the model risk in finance and related problems.

Summary The central objective of the thesis is to study various measures of model risk, expressed in monetary terms, that can be consistently applied to a heterogeneous collection of financial products. The first two chapters deal with this problem, first from a theoretical point of view, and then by conducting an empirical study focused on the natural gas market. The third chapter focuses on a theoretical study of the so-called basis risk. In the first chapter, we focused on the valuation of complex financial products, taking into account the model risk and the availability in the market of basic derivatives, also called vanilla. In particular, we have pursued the optimal transport approach (known in the literature) for the computation of price bounds and model-risk robust over- (under-) hedging strategies. In particular, we revive a construction of martingale probabilities under which the price of an exotic option reaches the so-called price bounds, focusing on the case of positive martingales. We also highlight significant symmetry properties in the study of this problem. In the second chapter, we approach the model risk problem from an empirical point of view, by studying the optimal management of a unit of natural gas and quantifying the effect of this risk on its optimal value. In this study, the valuation of the storage unit is based on the spot price, while its hedging is done with forward contracts. As mentioned before, the third chapter focuses on the basis risk, which arises when one wants to hedge a contingent asset based on an unprocessed asset (e.g. temperature) using a portfolio of processed assets in the market. A hedging criterion in this context is that of variance minimization, which is closely related to the so-called Föllmer-Schweizer decomposition. This decomposition can be deduced from the solution of a certain stochastic backward differential equation (SDE) directed by a possibly jumping martingale. When this martingale is a standard Brownian motion, the SRDEs are strongly associated with semilinear parabolic PDEs. In the general case we formulate a deterministic problem that extends the mentioned PDEs. We apply this approach to the important special case of the Föllmer-Schweizer decomposition, for which we give explicit expressions for the payoff decomposition of an option when the underlyings are exponential of additive processes.