Past exam of the mathematics course of the University of Cambridge / 2004 / iii / Paper 35 / 3 / d

The usual replication argument needs two hypotheses implicit in the pricing conclusion: the strategy must replicate the payoff, $V_N=X$, and its discounted gains must be integrable. Bounded predictable risky holdings suffice for the latter. Under these conditions, the equivalent martingale measure $\mathbb{Q}$ and part (c) giveThus discounted wealth is a martingale, and repeated conditioning givesFor a normalized initial bank account, $B_0=1$, deterministic terminal bank value, and trivial initial information, this reduces to the intended risk-neutral pricing formulaIf $B_N$ is random it must remain inside the expectation. If a claim is not replicable, the existence of an equivalent martingale measure alone does not make its arbitrage-free price unique.

The discounted wealth martingale integrability condition cannot simply be dropped. For a concrete counterexample, set $B_n=1$, $S_0=3/2$, $S_1=1+U$, and $S_2=1+U+\epsilon /2$, where $U$ is uniform on $(0,1)$ and $\epsilon$ an independent fair sign. These positive bounded prices form a martingale in the natural filtration. Start with zero holdings; at time one buy $1/U$ shares financed by borrowing $S_1/U$. This is previsible and self-financing, but its terminal wealth is $\epsilon /(2U)$, whose absolute expectation is infinite. It is therefore not a martingale. The stated pricing proof applies to the standard integrable replication class, rather than all unconstrained previsible strategies.