Prices of contracts with risky aspects are typically linked to specific uncertainties and probabilities of adverse scenarios. Insurance companies carry the risk of losses in exchange for a premium, which depends on the loss distribution. Another example where risk is exchanged for a fixed price is swap contracts. Electricity futures can be seen as swaps where the floating component are spot prices and the fixed component is a constant price for delivering electricity over a longer period. The primary goal of this thesis is the incorporation of model ambiguity for pricing these contracts. Moreover, we contemplate the complex structure of energy markets. For this reason, we also explore pricing a real option under model ambiguity. First of all, we study the theoretical properties of the distortion principle for insurance pricing. We find closed-form solutions for the optimal distortion premium under model ambiguity using Wasserstein distances. In various cases, we also find the distributions that reach the optimal prices. For the distortion principle, we can conclude that the price to pay for ambiguity only depends on the ambiguity radius and the distortion function, but not on the initial distribution. Additionally, we characterize the unboundedness of the robust distortion premium. Besides, we investigate the identification of distortion functions from observed prices. We propose a method to recover them from simulated prices in two cases: the average value-at-risk and power distortion principle. In the second part of this thesis, we bring together insurance pricing rules and electricity futures pricing rules. Due to the non-storability of electricity, many authors study different rules and empirical results to explain futures prices and the risk premia in this market. We extend the present literature and propose to explain the price formation of these contracts with three different quantities: the distortion premium, a correction factor and an ambiguity premium. [option]. The ambiguity premium is significant and increases with time-to-delivery for base futures. For these calculations, we specify a new regime-switching model for spot prices. [option1] These three factors capture a general mechanism of futures prices. We conclude the magnitude of futures increases with time-to-delivery. In addition, we recover a seasonal pattern of the risk premia and explain the changes in risk aversion depending on time-to-delivery. [option2] These three factors capture main characteristics of futures prices and the risk premia. Among them, we recover a seasonal pattern of the risk premia and explain the changes in risk aversion depending on time-to-delivery. The last part of this thesis studies an appropriate evaluation of a thermal power plant by incorporating model ambiguity. The different uncertainties that affect the expected profits of this real option are electricity prices, fuel prices and CO2 allowances. The power plant takes weekly decisions fixing the production for an entire week, while the uncertainties may affect the profit within weeks. Firstly, we discretize and quantize the uncertainties in a lattice process. To simulate different prices within weeks, we introduce an interpolation process called bridge process. Secondly, we propose a distance between lattice processes, which is tractable for solving dynamic problems backwards in time. This distance is a Wasserstein distance type with an underlying metric dependent on the state of the power plant. Our empirical results show that the larger the ambiguity radius is, the more conservative the production, and the less the achieved profit is. Although we solve a specific problem, our results can be applied to different multistage decision problems.
