This thesis' goal is to model power forward contracts in the electricity marked with use of L\'evy processes. First we obtain some theory regarding these jump processes, which will enable us to built exponential additive models for the forwards. We obtain the L\'evy-Ito-decomposition and the L\'evy-Khinchin representation, which enable us to find the characteristic function of the marginal density to the process in progress, and also decide conditions to put on in order to obtain martingale property of the processes in progress. We will follow the tradition of modelling forwards with fixed maturity, and then view the forwards with delivery over a given period as an average over these fixed forwards.
For volatility specifications depending on delivery time, which there is statistical evidence for in the markets, we are not in general able to find the general analytical dynamics of the forward, so we use Monte Carlo simulations of the prices to obtain the structure. We find that much of the dynamics to the forwards with fixed maturity time is inherited to the flow forward, but the studies are complicated with time demanding simulations. Finally we will study the so called BRS-approximation, which is indeed much faster in simulations, and we find that it works very well for instance in option pricing.