Multidimensional Portfolio Optimization With Proportional Transaction Costs

Abstract

We provide a computational study of the problem of optimally allocating wealth among multiple stocks and a bank account, to maximize the infinite horizon discounted utility of consumption. We consider the situation where the transfer of wealth from one asset to another involves transaction costs that are proportional to the amount of wealth transferred. Our model allows for correlation between the price processes, which in turn gives rise to interesting hedging strategies. This results in a stochastic control problem with both drift-rate and singular controls, which can be recast as a free boundary problem in partial differential equations. Adapting the finite element method and using an iterative procedure that converts the free boundary problem into a sequence of fixed boundary problems, we provide an efficient numerical method for solving this problem. We present computational results that describe the impact of volatility, risk aversion of the investor, level of transaction costs, and correlation among the risky assets on the structure of the optimal policy. Finally we suggest and quantify some heuristic approximations.

Primal-dual methods for the computation of trading regions under proportional transaction costs

“…Based on Theorem 2.1 we may further analyze the regularity of the value function and the properties of the trading regions. From the results in Shreve and Soner [1994] and the numerical analysis in Akian et al [1996], Dai and Zhong [2010], Muthuraman and Kumar [2006] we expect the value function Φ = Φ(t, x 0 , x) to be continuously differentiable in t and in x 0 , and twice continuously differentiable…”

“…We mention that a different numerical approach for the stationary problem with consumption as in Davis and Norman [1990], Shreve and Soner [1994], Akian et al [1996] was proposed in Muthuraman [2007], Muthuraman and Kumar [2006]. There the authors employ a monotonically decreasing update of the no-trading region, which is motivated by a policy improvement procedure.…”

Dynamic Portfolio Optimization with Transaction Costs and State-Dependent Drift

“…The opposite is true here as higher risk aversion leads to less investment in the stock, see also Muthuraman and Kumar (2006). Cvitanić et al (2006) also find that the certainty equivalents that they examine achieve the highest values for the lowest risk aversion in different setups.…”

Synergy effect of cooperative investment

“…In models with transaction costs, a portfolio does not have exact equivalent in terms of a single numeraire, therefore the portfolio value (or gain) is often expressed as an n-component vector, where n is the number of assets, see [31]. In particular, a natural outcome of any trading strategy in a dollar-euro currency exchange market is a 2-dimensional random vector (X, Y), where X and Y denotes the gain in dollars and euros, correspondingly.…”