  Shadow prices simplify the derivation of optimal trading strategies in
markets with transaction costs by transferring optimization into a more
tractable, frictionless market. This paper establishes that a na\"ive shadow
price Ansatz for maximizing long term returns given average volatility yields a
strategy that is, for small bid-ask-spreads, asymptotically optimal at third
order. Considering the second-order impact of transaction costs, such a
strategy is essentially optimal. However, for risk aversion different from one,
we devise alternative strategies that outperform the shadow market at fourth
order. Finally, it is shown that the risk-neutral objective rules out the
existence of shadow prices.
