Shadow Prices for Continuous Processes

Abstract

In a financial market with a continuous price process and proportional transaction costs, we investigate the problem of utility maximization of terminal wealth. We give sufficient conditions for the existence of a shadow price process, i.e., a least favorable frictionless market leading to the same optimal strategy and utility as in the original market under transaction costs. The crucial ingredients are the continuity of the price process and the hypothesis of "no unbounded profit with bounded risk". A counterexample reveals that these hypotheses cannot be relaxed. MSC 2010 Subject Classification: 91G10, 93E20, 60G48JEL Classification Codes: G11, C61

Robust utility maximisation in markets with transaction costs

“…The next lemma states that the sets C θ (x) and D θ (y) are polar to each other. It follows directly from Proposition 2.9 of [12].…”

“…The condition U (0) = 0 is used only to simplify calculations. Condition (11) is mild and so is (12): as shown in Corollary 4.2(i) of [32], for every utility function U with reasonable asymptotic elasticity, its conjugate V satisfies (12). The studies [11], [23] assumed a smooth U which is strictly concave on its entire domain, we do not need either smoothness or strict concavity of U .…”

Optimal Investment with Random Endowments and Transaction Costs: Duality Theory and Shadow Prices

“…The acceptable portfolio process differs from the admissible portfolio process and the existence of equivalent local martingale measures forŜ are required to build the duality theory in the shadow price market without transaction costs. Therefore, unlike [7], even when the stock price process S is continuous, the existence of consistent local martingale system (Z 0 , Z 1 ) (see [1] for its definition and the equivalent characterization) is no longer the sufficient condition for the existence of a classic shadow price process. The existence of a CPS becomes crucial in our project with random endowments.…”

“…Comparing with Proposition 3.7 in [5] and Theorem 3.1 in [7], the existence of a classic shadow price process under random endowments becomes much more delicate and may fail in general even for continuous price processes. In our framework, it is even not enough to require that the dual optimizer (Y 0, * (y, r), Y 1, * (y, r)) satisfies the condition that Y 0, * (y, r) is a martingale and Y 1, * (y, r) is a local martingale.…”

“…Remark 4.1. Our definition of the classic shadow price processŜ is more restrictive than [5] and [7] and a classic shadow price processŜ satisfies NFLVR condition by its definition. The acceptable portfolio process differs from the admissible portfolio process and the existence of equivalent local martingale measures forŜ are required to build the duality theory in the shadow price market without transaction costs.…”