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

Černý, Aleš and Czichowsky, Christoph ORCID: 0000-0002-3513-6843 (2024) The law of one price in quadratic hedging and mean–variance portfolio selection. Finance and Stochastics. ISSN 0949-2984 (In Press)

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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 L2 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 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 L0 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.

Item Type:	Article
Additional Information:	© 2024
Divisions:	Mathematics
Subjects:	Q Science > QA Mathematics H Social Sciences > HG Finance
JEL classification:	G - Financial Economics > G1 - General Financial Markets > G11 - Portfolio Choice; Investment Decisions G - Financial Economics > G1 - General Financial Markets > G12 - Asset Pricing; Trading volume; Bond Interest Rates C - Mathematical and Quantitative Methods > C6 - Mathematical Methods and Programming > C61 - Optimization Techniques; Programming Models; Dynamic Analysis
Date Deposited:	21 Oct 2024 09:18
Last Modified:	21 Nov 2024 10:21
URI:	http://eprints.lse.ac.uk/id/eprint/125805

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