  This article clarifies the relationship between pricing kernel monotonicity
and the existence of opportunities for stochastic arbitrage in a complete and
frictionless market of derivative securities written on a market portfolio. The
relationship depends on whether the payoff distribution of the market portfolio
satisfies a technical condition called adequacy, meaning that it is atomless or
is comprised of finitely many equally probable atoms. Under adequacy, pricing
kernel nonmonotonicity is equivalent to the existence of a strong form of
stochastic arbitrage involving distributional replication of the market
portfolio at a lower price. If the adequacy condition is dropped then this
equivalence no longer holds, but pricing kernel nonmonotonicity remains
equivalent to the existence of a weaker form of stochastic arbitrage involving
second-order stochastic dominance of the market portfolio at a lower price. A
generalization of the optimal measure preserving derivative is obtained which
achieves distributional replication at the minimum cost of all second-order
stochastically dominant securities under adequacy.
