The Law of One Price in Quadratic Hedging and Mean–Variance Portfolio Selection

34 Pages Posted: 31 Oct 2022 Last revised: 22 Jan 2024

See all articles by Aleš Černý

Aleš Černý

Bayes Business School, City, University of London

Christoph Czichowsky

London School of Economics & Political Science (LSE) - Department of Mathematics

Date Written: October 27, 2022

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 allocation 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 contingent claims in an extended market and subsequently plays an important role in mean-variance hedging. 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.

Keywords: Law of one price, E-density, efficient frontier, quadratic hedging, mean-variance portfolio selection

JEL Classification: G11, G12, C61

Suggested Citation

Černý, Aleš and Czichowsky, Christoph, The Law of One Price in Quadratic Hedging and Mean–Variance Portfolio Selection (October 27, 2022). Available at SSRN: https://ssrn.com/abstract=4260768 or http://dx.doi.org/10.2139/ssrn.4260768

Aleš Černý (Contact Author)

Bayes Business School, City, University of London ( email )

Northampton Square
London, EC1V 0HB
United Kingdom

Christoph Czichowsky

London School of Economics & Political Science (LSE) - Department of Mathematics ( email )

Houghton Street
GB-London WC2A 2AE
United Kingdom

HOME PAGE: http://https://www.lse.ac.uk/Mathematics/people/Christoph-Czichowsky

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