17 research outputs found
Existence of shadow prices in finite probability spaces
A shadow price is a process lying within the bid/ask prices of a market with proportional transaction costs, such that maximizing expected utility from consumption in the frictionless market with price process leads to the same maximal utility as in the original market with transaction costs. For finite probability spaces, this note provides an elementary proof for the existence of such a shadow pric
On the existence of shadow prices
For utility maximization problems under proportional transaction costs, it has been observed that the original market with transaction costs can sometimes be replaced by a frictionless {\em shadow market} that yields the same optimal strategy and utility. However, the question of whether or not this indeed holds in generality has remained elusive so far. In this paper we present a counterexample which shows that shadow prices may fail to exist. On the other hand, we prove that short selling constraints are a sufficient condition to warrant their existence, even in very general multi-currency market models with possibly discontinuous bid-ask-spreads
On the Existence of Shadow Prices
For utility maximization problems under proportional transaction costs, it
has been observed that the original market with transaction costs can sometimes
be replaced by a frictionless "shadow market" that yields the same optimal
strategy and utility. However, the question of whether or not this indeed holds
in generality has remained elusive so far. In this paper we present a
counterexample which shows that shadow prices may fail to exist. On the other
hand, we prove that short selling constraints are a sufficient condition to
warrant their existence, even in very general multi-currency market models with
possibly discontinuous bid-ask-spreads.Comment: 14 pages, 1 figure, to appear in "Finance and Stochastics
Shadow prices for continuous processes
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 counter-example
reveals that these hypotheses cannot be relaxed
Portfolio optimisation beyond semimartingales: shadow prices and fractional Brownian motion
While absence of arbitrage in frictionless financial markets requires price
processes to be semimartingales, non-semimartingales can be used to model
prices in an arbitrage-free way, if proportional transaction costs are taken
into account. In this paper, we show, for a class of price processes which are
not necessarily semimartingales, the existence of an optimal trading strategy
for utility maximisation under transaction costs by establishing the existence
of a so-called shadow price. This is a semimartingale price process, taking
values in the bid ask spread, such that frictionless trading for that price
process leads to the same optimal strategy and utility as the original problem
under transaction costs. Our results combine arguments from convex duality with
the stickiness condition introduced by P. Guasoni. They apply in particular to
exponential utility and geometric fractional Brownian motion. In this case, the
shadow price is an Ito process. As a consequence we obtain a rather surprising
result on the pathwise behaviour of fractional Brownian motion: the
trajectories may touch an Ito process in a one-sided manner without reflection.Comment: To appear in Annals of Applied Probability. We would like to thank
Junjian Yang for careful reading of the manuscript and pointing out a mistake
in an earlier versio