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Reduced Basis Method and Space-Time Discretization for Electricity Markets
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Swing options are widely used in electricity markets for price and volume flexibility reasons they provide. Due to their multiple exercise property, the pricing of swing options requires sophisticated numerical methods. So far such method have mostly been based on Monte-Carlo methods. We want to price swing options efficiently, which leads to variational inequalities in the mathematical model. These variational inequalities also can be interpreted as obstacle problems, with which we will deal in the following. Fine discretizations that are needed for obstacle problems resolve in highdimensional problems and thus in long computing time. To reduce the dimensionality of such problems, we will use the Reduced Basis Method (RBM). The objective of the RBM is to efficiently reduce discretized parametrized partial differential equations. Problems are considered where not only a single solution is needed but solutions for a range of different parameter configurations for which we get computational speedup. In the context of variational inequalities, RBM have already been applied to the elliptic case. However, in the case of swing options - therefore time-dependent - we have to consider the class of parabolic variational inequalities. Recent work on parabolic variational inequalities combined with the reduced basis method has been done on American options. In the so far published work, the error estimator still depends on the sum of the error bounds in every single time step as known from the common theory of RBM for parabolic equations. Better error estimators for parabolic equations could be achieved by combining the reduced basis method with the space-time formulation. Using space-time formulations, we do not have a time-stepping scheme anymore, but take the time as an additional space on our variational formulation of the problem. In our work, we plan to combine the space-time discretization with the reduced basis method. We are confident to get good error estimators with this method.