Building an Optimal Portfolio in Discrete Time in the Presence of Transaction Costs

Abstract

Portfolio theory covers different approaches to the construction of a portfolio offering maximum expected returns for a given level of risk tolerance where the goal is to find the optimal investment rule. Each investor has a certain utility for money which is reflected by the choice of a utility function. In this article, a risk averse power utility function is studied in discrete time for a large class of underlying probability distribution of the returns of the asset prices. Each investor chooses, at the beginning of an investment period, the feasible portfolio allocation which maximizes the expected value of the utility function for terminal wealth. Effects of both large and small proportional transaction costs on the choice of an optimal portfolio are taken into account. The transaction regions are approximated by using asymptotic methods when the proportional transaction costs are small and by using expansions about critical points for large transaction costs.     

Portfolio selection in discrete time with transaction costs and power utility function: a perturbation analysis

In this article, we study a multi-period portfolio selection model in which a generic class of probability distributions is assumed for the returns of the risky asset. An investor with a power utility function rebalances a portfolio comprising a risk-free and risky asset at the beginning of each time period in order to maximize expected utility of terminal wealth. Trading the risky asset incurs a cost that is proportional to the value of the transaction. At each time period, the optimal investment strategy involves buying or selling the risky asset to reach the boundaries of a certain no-transaction region. In the limit of small transaction costs, dynamic programming and perturbation analysis are applied to obtain explicit approximations to the optimal boundaries and optimal value function of the portfolio at each stage of a multi-period investment process of any length.     

Multi-period portfolio selection problem under uncertain environment with bankruptcy constraint

The complexity of financial markets leads to different types of indeterminate asset returns. For example, asset returns are considered as random variables, when the available data is enough. When the available data is too small or even no available data to estimate a probability distribution, we have to invite some domain experts to evaluate the belief degrees of asset returns. Then, asset returns can be described as uncertain variables. In this paper, we discuss a multi-period portfolio selection problem under uncertain environment, which maximizes the final wealth and minimizes the risk of investment. Unlike the common method to describe the multi-period portfolio selection problem as a bi-objective optimization model, we formulate this uncertain multi-period portfolio selection problem by a new method in three steps with two single objective optimization models. And, we consider the influence of transaction cost and bankruptcy of investor. Then, the proposed uncertain optimization models are transformed into the corresponding crisp optimization models and we use the genetic algorithm combined with penalty function method to solve them. Finally, a numerical example is given to show the effectiveness and practicability of proposed models and method.     

Growth Optimal Portfolio Selection Under Proportional Transaction Costs with Obligatory Diversification

A continuous time long run growth optimal or optimal logarithmic utility portfolio with proportional transaction costs consisting of a fixed proportional cost and a cost proportional to the volume of transaction is considered. The asset prices are modeled as exponent of diffusion with jumps whose parameters depend on a finite state Markov process of economic factors. An obligatory portfolio diversification is introduced, accordingly to which it is required to invest at least a fixed small portion of our wealth in each asset.     

Multi-objective possibilistic model for portfolio selection with transaction cost

In this paper, we introduce the possibilistic mean value and variance of continuous distribution, rather than probability distributions. We propose a multi-objective Portfolio based model and added another entropy objective function to generate a well diversified asset portfolio within optimal asset allocation. For quantifying any potential return and risk, portfolio liquidity is taken into account and a multi-objective non-linear programming model for portfolio rebalancing with transaction cost is proposed. The models are illustrated with numerical examples.     

A Further Study of the Choice Between Two Hedging Strategies–the Continuous Case

In our previous work, the choice between two popular hedging strategies was studied under the assumption that the hedge position of the underlying portfolio follows a discrete-time Markov chain with boundary conditions. This paper aims to investigate the same problem for the continuous case. We first assume that the underlying hedge position follows an arbitrary continuous-time Markov process; we give the general formulas for long-run cost per unit time under two cost structures: (1) a fixed transaction cost (2) a non-fixed transaction cost. Then we consider the case where the underlying hedge position follows a Brownian motion with drift; we show that (i) re-balancing the hedge position to the initial position is always more cost-efficient than re-balancing it to the boundary for a fixed transaction cost; (ii) when the cost function satisfies certain conditions, re-balancing the hedge position to the initial position is more cost-efficient than re-balancing it to the boundary for a non-fixed transaction cost.     

Portfolio adjusting optimization with added assets and transaction costs based on credibility measures

In response to changeful financial markets and investor’s capital, we discuss a portfolio adjusting problem with additional risk assets and a riskless asset based on credibility theory. We propose two credibilistic mean–variance portfolio adjusting models with general fuzzy returns, which take lending, borrowing, transaction cost, additional risk assets and capital into consideration in portfolio adjusting process. We present crisp forms of the models when the returns of risk assets are some deterministic fuzzy variables such as trapezoidal, triangular and interval types. We also employ a quadratic programming solution algorithm for obtaining optimal adjusting strategy. The comparisons of numeral results from different models illustrate the efficiency of the proposed models and the algorithm.     

Portfolio adjusting optimization with added assets and transaction costs based on credibility measures

In response to changeful financial markets and investor’s capital, we discuss a portfolio adjusting problem with additional risk assets and a riskless asset based on credibility theory. We propose two credibilistic mean–variance portfolio adjusting models with general fuzzy returns, which take lending, borrowing, transaction cost, additional risk assets and capital into consideration in portfolio adjusting process. We present crisp forms of the models when the returns of risk assets are some deterministic fuzzy variables such as trapezoidal, triangular and interval types. We also employ a quadratic programming solution algorithm for obtaining optimal adjusting strategy. The comparisons of numeral results from different models illustrate the efficiency of the proposed models and the algorithm.     

Utility maximization in markets with bid–ask spreads

A one-period financial market model with transaction costs is considered in this paper. Redefining the risky asset price process in a suitable way, we obtain an explicit solution to the utility maximization problem when the risk preferences of the investor are based on the exponential utility function and a liability can be included in her portfolio. The arbitrage-free interval price for a general liability, as well as its replication price, is characterized in terms of expectations with respect to equivalent martingale measures. The indifference price is derived and its asymptotic limit when the risk aversion is going to infinity is analysed.     

On an investment-consumption model with transaction costs: an asymptotic analysis

In this paper we examine the Akian, Menaldi and Sulem (1996) model for the optimal management of a portfolio, when there are transaction costs which are equal to a fixed percentage of the amount transacted. We analyse this model in the realistic limit of small transaction costs. Although the full problem is a free boundary diffusion problem in as many dimensions as there are assets in the portfolio, we find explicit solutions for the optimal trading policy in this limit. This makes the solution for a realistically large number of assets a practical possibility.     

QUADRATURE METHODS FOR HIGHLY OSCILLATORY SINGULAR INTEGRALS

We address the evaluation of highly oscillatory integrals,with power-law and logarithmic singularities.Such problems arise in numerical methods in engineering.Notably,the evaluation of oscillatory integrals dominates the run-time for wave-enriched boundary integral formulations for wave scattering,and many of these exhibit singularities.We show that the asymptotic behaviour of the integral depends on the integrand and its derivatives at the singular point of the integrand,the stationary points and the endpoints of the integral.A truncated asymptotic expansion achieves an error that decays faster for increasing frequency.Based on the asymptotic analysis,a Filon-type method is constructed to approximate the integral.Unlike an asymptotic expansion,the Filon method achieves high accuracy for both small and large frequency.Complex-valued quadrature involves interpolation at the zeros of polynomials orthogonal to a complex weight function.Numerical results indicate that the complex-valued Gaussian quadrature achieves the highest accuracy when the three methods are compared.However,while it achieves higher accuracy for the same number of function evaluations,it requires signi cant additional cost of computation of orthogonal polynomials and their zeros.     

基于排序预测的带交易费用在线投资组合策略

在线投资组合决策过程中频繁调整资产头寸会产生较多的交易费用。本文提出了一个综合考虑预期收益和交易费用的在线投资组合策略。通过预测资产的排序计算组合的预期收益,利用相对熵距离衡量交易费用,构造了一个极大化预期收益和极小化交易费用的优化模型,从而得到了一个在线投资组合更新策略。然后,从理论上证明了该策略具有BH泛证券性,即该策略与离线的最优购买并持有策略具有相同的渐近平均指数收益率。最后,采用中美股票市场实际数据,对该策略进行了数值分析。结果表明,该策略的表现优于已有的在线投资组合策略,且对模型的参数不敏感。     

Portfolio-optimization models for small investors

Since 2010, the client base of online-trading service providers has grown significantly. Such companies enable small investors to access the stock market at advantageous rates. Because small investors buy and sell stocks in moderate amounts, they should consider fixed transaction costs, integral transaction units, and dividends when selecting their portfolio. In this paper, we consider the small investor’s problem of investing capital in stocks in a way that maximizes the expected portfolio return and guarantees that the portfolio risk does not exceed a prescribed risk level. Portfolio-optimization models known from the literature are in general designed for institutional investors and do not consider the specific constraints of small investors. We therefore extend four well-known portfolio-optimization models to make them applicable for small investors. We consider one nonlinear model that uses variance as a risk measure and three linear models that use the mean absolute deviation from the portfolio return, the maximum loss, and the conditional value-at-risk as risk measures. We extend all models to consider piecewise-constant transaction costs, integral transaction units, and dividends. In an out-of-sample experiment based on Swiss stock-market data and the cost structure of the online-trading service provider Swissquote, we apply both the basic models and the extended models; the former represent the perspective of an institutional investor, and the latter the perspective of a small investor. The basic models compute portfolios that yield on average a slightly higher return than the portfolios computed with the extended models. However, all generated portfolios yield on average a higher return than the Swiss performance index. There are considerable differences between the four risk measures with respect to the mean realized portfolio return and the standard deviation of the realized portfolio return.     

THE BOUNDARY VALUE PROBLEMS FOR QUASILINEAR HIGHER ORDER ELLIPTIC EQUATIONS WITH A SMALL PARAMETER

In this paper we deal with the general boundary value problems for quasilinear higherorder elliptic equations with a small parameter before higher derivatives.By using themethod of multiple scales,we have proved that if the solution of degenerated boundary valueproblem exists,then under certain assumptions as the small parameter is sufficiently small,the solution of the origional boundary value problem exists as well and it is unique in a certainfunction space.Besides,the asymptotic expansion of the solution has been constructed.     

Asymptotic Initial-Value Method for Singularly-Perturbed Boundary-Value Problems for Second-Order Ordinary Differential Equations

A computational method is presented to solve a class of nonturning-point singularly-perturbed two-point boundary-value problems for second-order ordinary differential equations with a small parameter multiplying the highest derivative, subject to Dirichlet-type boundary conditions. In this method, first we construct a zeroth order asymptotic expansion for the solution of the given boundary-value problem. Then, this problem is integrated to get an equivalent initial-value problem for first-order ordinary differential equations. This initial-value problem is solved by either a classical method or a fitted operator method after approximating some of the terms in the differential equations by using the zeroth order asymptotic expansion. This method is effective and easy to implement. An error estimate is derived for the numerical solution. Examples are given to illustrate the method.     

Simultaneous pursuit of out-of-sample performance and sparsity in index tracking portfolios

Index tracking is a passive investment strategy in which a fund (e.g., an ETF: exchange traded fund) manager purchases a set of assets to mimic a market index. The tracking error, i.e., the difference between the performances of the index and the portfolio, may be minimized by buying all the assets contained in the index. However, this strategy results in a considerable transaction cost and, accordingly, decreases the return of the constructed portfolio. On the other hand, a portfolio with a small cardinality may result in poor out-of-sample performance. Of interest is, thus, constructing a portfolio with good out-of-sample performance, while keeping the number of assets invested in small (i.e., sparse). In this paper, we develop a tracking portfolio model that addresses the above conflicting requirements by using a combination of L0- and L2-norms. The L2-norm regularizes the overdetermined system to impose smoothness (and hence has better out-of-sample performance), and it shrinks the solution to an equally-weighted dense portfolio. On the other hand, the L0-norm imposes a cardinality constraint that achieves sparsity (and hence a lower transaction cost). We propose a heuristic method for estimating portfolio weights, which combines a greedy search with an analytical formula embedded in it. We demonstrate that the resulting sparse portfolio has good tracking and generalization performance on historic data of weekly and monthly returns on the Nikkei 225 index and its constituent companies.     

Dynamic portfolio selection with market impact costs

This paper concerns optimal dynamic portfolio choice with quadratic utility when there are market impact costs. The optimal policy is difficult to characterize, so we look instead for sub-optimal policies. Our proposed suboptimal policy solves a tractable dynamic portfolio choice problem where the cost of trading is captured in the objective instead of the price dynamics. A multiple time scale asymptotic expansion shows that our proposed policy has sensible structural properties, while numerical experiments show promising performance and robustness properties.     

三阶非线性微分方程三点边值问题的渐近解

本文通过引入伸长变量和使用边界层校正项的方法构造了一类三阶非线性微分方程三点边值问题的形式渐近解,然后利用高阶微分不等式理论,证明了此解的一致有效性.     

Asymptotic properties of solutions of parametric optimal control problems with varying index of the singular arcs

We consider a family of parametric linear-quadratic optimal control problems with terminal and control constraints. This family has the specific feature that the class of optimal controls is changed for an arbitrarily small change in the parameter. In the perturbed problem, the behavior of the corresponding trajectory on noncritical arcs of the optimal control is described by solutions of singularly perturbed boundary value problems. For the solutions of these boundary value problems, we obtain an asymptotic expansion in powers of the small parameter ?. The asymptotic formula starts from a term of the order of 1/? and contains boundary layers. This formula is used to justify the asymptotic expansion of the optimal control for a perturbed problem in the family. We suggest a simple method for constructing approximate solutions of the perturbed optimal control problems without integrating singularly perturbed systems. The results of a numerical experiment are presented.     

Asset liquidity and the valuation of derivative securities

We consider the valuation of European-style derivative securities under limited asset liquidity through the dynamic management of a portfolio of assets effected through continuous transaction. The valuation arises from the optimal realization of a performance index relative to the set of all feasible portfolio trajectories. An approximation procedure based upon the method-of-lines finite element method is developed and analyzed; numerical examples are presented in order to demonstrate the viability of the approach.