The law of one price in quadratic hedging and mean-variance portfolio selection
Aleš Černý, Christoph Czichowsky
TL;DR
The paper addresses the law of one price (LOP) in a continuous-time, $L^2$-theoretic setting for quadratic hedging, showing that LOP is the minimal condition to avoid degeneracy in mean-variance hedging and to ensure well-defined pricing under trading. It introduces a novel LOP failure mechanism tied to trading from just before predictable stopping times, even for continuous price processes, and proves a Fundamental Theorem of Asset Pricing analogue: LOP is equivalent to the existence of a local $\mathscr{E}$-martingale state price density, enabling unique prices for square-integrable claims in an extended market. The work develops a conditional uniform boundedness principle for $L^0$-module functionals, and demonstrates that time-consistent pricing functionals can be represented via stochastic exponentials of a fixed local martingale. Crucially, it extends mean-variance hedging theory to general $S$ without requiring an equivalent local martingale measure, providing explicit conditional mean-variance frontiers and feedback hedges, and clarifying the role of LOP in market extensions and pricing uniqueness.
Abstract
The 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 selection framework without degeneracy. Crucially, the paper identifies a new mechanism through which LOP can fail in a continuous-time $L^2$ 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 claims in an extended market and subsequently plays an important role in quadratic hedging and mean-variance portfolio selection. Mathematically, we formulate a novel variant of the uniform boundedness principle for conditionally linear functionals on the $L^0$ 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.