Fast algorithm for nonparametric arbitrage-free SPD estimation

State price density (SPD) contains important information concerning market expectations. An estimator of the SPD based on observed European option prices, taking into account the time of the trade, has been previously considered. Financial markets produce huge amounts of data and, due to time constraints, it is not always possible to calculate the estimator using all available data. Using a model for the covariance structure of the observed option prices, the algorithm identifies observations with little importance to the estimator. Dropping these observations increases the speed of computation and allows frequenter updating of the estimator. The algorithms efficiently use indices that combine information contained in the data. Fast algorithms are proposed and their properties are investigated using both simulated and real data sets.