Good Portfolio Strategies under Transaction Costs: A Renewal Theoretic Approach

Abstract

This paper looks at portfolio theory in the presence of costs of transactions. A fundamental result is given in Morton and Pliska (1995) ([15]) where renewal theoretic arguments and the theory of optimal stopping are used to derive optimal strategies for maximizing the asymptotic growth rate under purely fixed costs which are proportional to the portfolio value. Our paper is also devoted to maximizing the asymptotic growth rate but here we consider a combination of fixed and proportional costs. Motivated by various structural results in the work on optimal portfolio theory we introduce a class of natural trading strategies which can be described by four parameters, two for the stopping boundaries and two for the new risky fractions (fraction of the wealth invested in the stock). In this class the problem can be simplified by renewal theoretic arguments to treating one period between two trading times, where we then have to start the new risky fraction process according to the invariant distribution. This yields an explicit form for the asymptotic growth rate that can be maximized in these four parameters. The computation of best strategies in this class thus is simple, and we provide various examples. Preliminary considerations based on the fundamental results of Bielecki and Pliska (2000)([5]) and the results of this paper point out that in fact an allover optimal impulse control strategy can be found within this class.