Growth Optimal Portfolios in Discrete-time Markets Under Transaction Costs

Details Growth Optimal Portfolios in Discrete-time Markets Under Transaction Costs Growth Optimal Portfolios in Discrete-time Markets Under Transaction Costs

2012-03-19

arxiv.org/abs/1203.4153v1

Computer Science

Systems and Control

Submitted to IEEE Transactions on Signal Processing, 30 pages

Scientific paper

We investigate portfolio selection problem from a signal processing perspective and study how an investor should distribute wealth over two assets in order to maximize the cumulative wealth. We construct portfolios that provide the optimal growth in i.i.d. discrete time two-asset markets under proportional transaction costs. As the market model, we consider arbitrary discrete distributions on the price relative vectors, which can also be used to approximate a wide class of continuous distributions. To achieve optimal growth, we use threshold portfolios, where we introduce a recursive update to calculate the expected wealth. We then demonstrate that under the threshold rebalancing framework, the achievable set of portfolios elegantly form an irreducible Markov chain under mild technical conditions. We evaluate the corresponding stationary distribution of this Markov chain, which provides a natural and efficient method to calculate the cumulative expected wealth. Subsequently, the corresponding parameters are optimized using a brute force approach yielding the growth optimal portfolio under proportional transaction costs in i.i.d. discrete-time two-asset markets. As a widely known financial problem, we also solve optimal portfolio selection in discrete-time markets constructed by sampling continuous-time Brownian markets. For the case that the underlying discrete distributions of the price relative vectors are unknown, we provide a maximum likelihood estimator that is also incorporated in the optimization framework.

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