Given a particular market variable, which could be finite
dimensional (e.g., a price vector of a collection of stocks) or infinite
dimensional (e.g., a price trajectory of some security over some period of
time), we find the unique optimal European claim contingent on that variable in
the sense that, for a given price and risk tolerance level, this claim has the
highest expected return possible. The optimal contingent claims seem to be
attractive investment instruments and are proposed for trade in derivatives
markets.