We propose a tractable benchmark of portfolio choice under transaction costs. Our analysis is based on the model of Dumas and Luciano(1991), which concentrates on long-run asymptotics of a power utility investor to gain in tractability. In a Black-Scholes model with constant proportional transaction costs, we find that the optimal trading policy, its welfare, and the resulting trading volume are all simple functions of the model parameters, the investor's risk aversion and, crucially, of a liquidity premium that depends on the level of transaction costs. This liquidity premium is given as the solution of a one-dimensional equation, from which an asymptotic expansion of arbitrary order can be computed for small transaction costs. This in turn leads to asymptotic formulas for all other quantities of interest.
A key idea in our results - and in their proof - is that a market with constant investment opportunities with transaction costs is equivalent to another market without transaction costs, but with stochastic investment opportunities. This "shadow price" has previously been determined for log-investors. Here we construct it for investors with power utilities.
(joint work with Stefan Gerhold, Paolo Guasoni, and Walter Schachermayer)