The law of one price in quadratic hedging and mean–variance portfolio selection

Abstract The law of one price (LOP) broadly asserts that identical financial flows should command the same price. We show that when properly formulated, the 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 the LOP can fail in a continuous-time $L^{2}$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>L</mml:mi> <mml:mn>2</mml:mn> </mml:msup> </mml:math> -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 us to give a version of the “fundamental theorem of asset pricing” appropriate in the quadratic context, establishing the equivalence of the economic concept of the LOP with the probabilistic property of the existence of a local 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. Mathematically, we formulate a novel variant of the uniform boundedness principle for conditionally linear functionals on the $L^{0}$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>L</mml:mi> <mml:mn>0</mml:mn> </mml:msup> </mml:math> -module of conditionally square-integrable random variables. We then study the representation of time-consistent families of such functionals in terms of stochastic exponentials of a fixed local martingale.