Optimal investment with random endowments and transaction costs: duality theory and shadow prices

Abstract

This paper studies the utility maximization on the terminal wealth with random endowments and proportional transaction costs. To deal with unbounded random payoffs from some illiquid claims, we propose to work with the acceptable portfolios defined via the consistent price system (CPS) such that the liquidation value processes stay above some stochastic thresholds. In the market consisting of one riskless bond and one risky asset, we obtain a type of super-hedging result. Based on this characterization of the primal space, the existence and uniqueness of the optimal solution for the utility maximization problem are established using the duality approach. As an important application of the duality theorem, we provide some sufficient conditions for the existence of a shadow price process with random endowments in a generalized form similar to [5] as well as in the usual sense using acceptable portfolios.

Continuous-time duality for superreplication with transient price impact

“…The duality theory for these models is now developed in great detail (see [32,17,33,34,14,28]). This has been used to study utility maximization via its relation to shadow prices ( [35,24,18,19,9]) and has also been instrumental in the development of asymptotic approaches for small transaction costs ( [42] and the references therein).…”

On the market viability under proportional transaction costs

“…For example, we can allow the liquidation value processes to be bounded below by some stochastic process instead of a uniform constant. See the definition of -acceptable portfolios in market models with transaction costs in Bayraktar and Yu (2015) and Yu (2017).…”

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

“…This generalized shadow price process is defined via a "sandwiched" couple consisting of a predictable and an optional strong supermartingale, and pertains to all strategies which remain solvent under transaction costs. This notion is afterward extended by Bayraktar and Yu to the case with random endowment in [1]. Inspirited by Hugonnier and Kramkov [28], the authors of [1] considered a utility maximization problem with transaction costs and additionally with unbounded random endowment confined by maximal trading strategies and a uniform integrability condition.…”

“…This notion is afterward extended by Bayraktar and Yu to the case with random endowment in [1]. Inspirited by Hugonnier and Kramkov [28], the authors of [1] considered a utility maximization problem with transaction costs and additionally with unbounded random endowment confined by maximal trading strategies and a uniform integrability condition. They constructed generalized shadow prices, whenever the duality result holds and a sufficient condition on the dual optimizer is assumed.…”

“…In contrast to [13,1], if we consider the utility maximization problem with utility functions defined on the whole real line, the picture changes. In [14], Czichowsky and Schachermayer confirmed that the existence of strictly consistent price systems (see [39,8] for a definition) with "finite entropy" guarantees the existence of classical shadow prices.…”