Construction of Discrete Time Shadow Price

Abstract

In the paper expected utility from consumption over finite time horizon for discrete time markets with bid and ask prices and strictly concave utility function is considered. The notion of weak shadow price, i.e. an illiquid price, depending on the portfolio, under which the model without bid and ask price is equivalent to the model with bid and ask price is introduced. Existence and the form of weak shadow price is shown. Using weak shadow price usual (called in the paper strong) shadow price is then constructed.

Optimal Strategies for Utility from Terminal Wealth with General Bid and Ask Prices

“…For given (x, y) ∈ R 2 + we are looking forŝ ∈ [s, s] such that w(x, y, s, s) = w(x, y,ŝ). Such value if exists is called a shadow price (see [10] for more explanation).…”

“…The advantage of dynamic programming method is that it allows to work on approximation methods. This paper generalizes [10], where discrete time shadow price was constructed using discrete time system of Bellman equations and certain geometric properties of transaction zones. We extend the results of [10] showing that under quite general assumptions the existence of shadow price and construction of optimal strategies can be restricted to the study of one period static case.…”

“…This paper generalizes [10], where discrete time shadow price was constructed using discrete time system of Bellman equations and certain geometric properties of transaction zones. We extend the results of [10] showing that under quite general assumptions the existence of shadow price and construction of optimal strategies can be restricted to the study of one period static case. We furthermore show that construction of optimal strategies and shadow price can be extended to the case of various assets (multidimensional case).…”

“…, n should be nonnegative. As one can see in [10] such restriction is typical when asset prices satisfy so called full support condition. To simplify notations we also assume zero interest rates on the bank account.…”

“…Discrete time shadow price was studied using duality in [4]. In the paper [10] a direct method based on dynamic programming was proposed. The advantage of dynamic programming method is that it allows to work on approximation methods.…”