  The \emph{law of one price (LOP)} broadly asserts that identical financial
flows should command the same price. We show that, when properly formulated,
LOP is the minimal condition for a well-defined mean--variance portfolio
allocation framework without degeneracy. Crucially, the paper identifies a new
mechanism through which LOP can fail in a continuous-time $L^2(P)$ setting
without frictions, namely `trading from just before a predictable stopping
time', which surprisingly identifies LOP violations even for continuous price
processes. Closing this loophole allows to give a version of the ``Fundamental
Theorem of Asset Pricing'' appropriate in the quadratic context, establishing
the equivalence of the economic concept of LOP with the probabilistic property
of the existence of a local $\scr{E}$-martingale state price density. The
latter provides unique prices for all square-integrable contingent claims in an
extended market and subsequently plays an important role in mean-variance
portfolio selection and quadratic hedging.
