Optimal portfolio policies under transaction costs

In a Black Scholes market - consisting of one stock whose prices evolve like a geometric Brownian motion and one risk free asset - an investor wants to maximize the asymptotic growth rate of his wealth (portfolio value). Without transaction costs the optimal policy would be given by the constant Merton fraction which is the fraction of the wealth to be invested in the stock. Facing transaction costs it is no longer adequate to keep this risky fraction constant.

We consider a combination of fixed (proportional to wealth) and proportional costs which punish the trading frequency as well as the magnitude of the transactions. Then an optimal trading strategy consists of a sequence of stopping times and the optimal transactions at those times. So we have to deal with impulse control strategies which can be described as solutions of quasi-variational inequalities.

Motivated by various structural results we first look at a restricted class of trading strategies which can be described by four parameters, two for the stopping boundaries and two for the new risky fractions. In this class the problem can be simplified by renewal arguments to one period between two trading times, where we have to weight the new risky fractions by their invariant distribution. This yields an explicit functional that has only to be maximized in these four parameters. So the computation of the best strategy in this class is very simple. Then we use the corresponding quasi-variational inequalities to prove that an optimal solution exists and that it can be found in this class. The approach also works for short selling and borrowing.

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