We consider writing a derivative contract on some underlying asset in such a way that the derivative contract and underlying asset yield the same payo� distribution after one time period. Using the Hardy-Littlewood rearrangement inequality, we obtain an explicit solution for the cheapest measure preserving derivative contract in terms of the payo� distribution and pricing kernel of the underlying asset. We develop asymptotic theory for the behavior of an estimated optimal derivative contract formed from estimates of the pricing kernel and underlying measure. Our optimal derivative corresponds to a direct investment in the underlying asset if and only if the pricing kernel is monotone decreasing. When the pricing kernel is not monotone decreasing, an investment of one monetary unit in our optimal derivative yields a payo� distribution that �rst-order stochastically dominates an investment of one monetary unit in the underlying asset. Recent empirical research suggests that the pricing kernel corresponding to a major US market index is not monotone decreasing
