Past exam of the mathematics course of the University of Cambridge / 2013 / iii / Paper 39 / 3 / c

For each terminal event $A$, market completeness supplies a portfolio $H_A$ replicating its indicator function. The two pricing identities giveThe constant payoff is also replicable, so these positive pricing variables are integrable under the stated finite pricing expectations. Equivalently, the preceding finite-atom result makes them finite-valued modulo null sets. Taking $A=\{X>Y\}$, the equality says that the nonnegative variable $(X-Y)1_A$ has expectation zero; thus $X\le Y$ almost surely. Reversing the roles givesThis is uniqueness of a one-period pricing density in a complete market. It uses replication of every event, not only matching a few marginal asset expectations.