On the existence of shadow prices for optimal investment with random endowment

Abstract

In this paper, we consider a numéraire-based utility maximization problem under constant proportional transaction costs and random endowment. Assuming that the agent cannot short sell assets and is endowed with a strictly positive contingent claim, a primal optimizer of this utility maximization problem exists. Moreover, we observe that the original market with transaction costs can be replaced by a frictionless shadow market that yields the same optimality. On the other hand, we present an example to show that in some case when these constraints are relaxed, the existence of shadow prices is still warranted. MSC2010. 91B16; 91G10

Utility maximization problem with random endowment and transaction costs: when wealth may become negative

“…For example, Loewenstein [34] confirmed that shadow prices exist for continuous bid-ask price processes whenever short positions are ruled out. This result is recently generalized by Benedetti et al [3] with Kabanov's general multi-currency market models and by Gu et al [23] with positive random endowment. However, several counterexamples have been found to show that shadow prices in the classical sense may fail to exist without further assumptions, see [37,12,3,13,16].…”

On utility maximization without passing by the dual problem

“…When U is defined on (0,∞), a direct approach to the primal problem of utility maximization is well-known from [27], and it has already been exploited in markets with constraints (see [19]) or with frictions (see [16,15]). For U with domain R our method seems the first to avoid solving the dual problem.…”

Optimal Investment and Contingent Claim Valuation With Exponential Disutility Under Proportional Transaction Costs

“…Additionally, shadow prices may not be tractable, leading to the use of asymptotic expansions and/or restrictions in the magnitude of transaction costs. In the context of continuous-time models, see the earlier paper of Cvitanić and 2022-05-20-regretpaper-final-ITJAF Claim valuation with exponential disutility under transaction costs 3 Karatzas (1996), as well as more recent contributions by Kallsen and Muhle-Karbe (2010), Gerhold et al (2013), Gerhold et al (2014), Herczegh and Prokaj (2015), , Schachermayer (2016, 2017), Lin and Yang (2016) and Gu et al (2017).…”