Past exam of the mathematics course of the University of Cambridge / 2020 / ii / Paper 2 / 29K / v

Suppose the claim $H=(S_1{)}^2$, payable at time $1$, were replicable. Holdings chosen at time zero are constants, so the time-$1$ value of any self-financing portfolio has the affine formBut $S_1=S_0{Z}_1$ has a continuous positive density on $(0,\infty )$. No affine function of $S_1$ can equal the quadratic function $S_1^2$ almost surely. Therefore $H$ is not attainable and the market is not complete.

The same argument applies if $H$ is viewed as a time-$T$ claim: once $H$ is known at time $1$, its required discounted value at time $1$ is fixed, and the first-period portfolio value would still have to reproduce a nonaffine function of $S_1$.