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

TL;DR

This paper establishes the fundamental role of the law of one price in quadratic hedging and mean-variance portfolio selection, identifying conditions for its validity and linking it to local martingale measures in continuous-time markets.

Contribution

It introduces a new failure mechanism for the law of one price in continuous-time models and provides a version of the Fundamental Theorem of Asset Pricing tailored for quadratic hedging.

Findings

LOP is the minimal condition for well-defined mean-variance portfolios.

Identifies a new failure mode of LOP in continuous-time models.

Establishes the equivalence between LOP and the existence of a local $ ext{ extscr}$-martingale measure.

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.