Portfolio Optimization and Martingale Measures

The paper studies connections between risk aversion and martingale measures in a discrete-time incomplete financial market. An investor is considered whose attitude toward risk is specified in terms of the index b of constant proportional risk aversion. Then dynamic portfolios are admissible if the terminal wealth is positive. It is assumed that the return (risk) processes are bounded. Sufficient (and nearly necessary) conditions are given for the existence of an optimal dynamic portfolio which chooses portfolios from the interior of the set of admissible portfolios. This property leads to an equivalent martingale measure defined through the optimal dynamic portfolio and the index 0 < b ≤ 1. Moreover, the option pricing formula of Davis is given by this martingale measure. In the case of b = 1; that is, in the case of the log-utility, the optimal dynamic portfolio defines the numéraire portfolio.     

基于多属性模糊决策的行为投资组合优化模型

Simon认为经典决策理论的假定过分地严格,在实际中往往难以满足。运用上、下偏矩的方法来估计未来财富不低于渴求水平的概率,并基于多属性模糊决策方法构建两个心理帐户的行为投资组合优化模型。模型中融入了不同质投资者的真实情感、信念及认知状态,实证分析的结果表明该模型具有很强的实用性和包容性。     

基于偏度的多期证券投资组合模型研究

在考虑资产收益率分布中正的偏度水平前提下,以风险价值VaR为约束条件,并引入非线性交易费用、税收等市场摩擦因素,建立以累积偏度最大为目标函数的多期投资组合优化模型,用罚函数法和PSO算法结合求解此模型,并给出实证分析。考虑到在买卖资产风险时交易费用等对投资收益的影响,投资者应该在每一期都对其资产组合进行调整分析,确保在每一期的开始都建立起符合需要的最优资产组合,这对投资者的连续投资行为具有一定的指导意义。     

OPTIONED PORTFOLIO SELECTION: MODELS AND ANALYSIS

The mean‐variance model of Markowitz and many of its extensions have been playing an instrumental role in guiding the practice of portfolio selection. In this paper we study a mean‐variance formulation for the portfolio selection problem involving options. In particular, the portfolio in question contains a stock index and some European style options on the index. A refined mean‐variance methodology is adopted in our approach to formulate this problem as multistage stochastic optimization. It turns out that there are two different solution techniques, both lead to explicit solutions of the problem: one is based on stochastic programming and optimality conditions, and the other one is based on stochastic control and dynamic programming. We introduce both techniques, because their strengths are very different so as to suit different possible extensions and refinements of the basic model. Attention is paid to the structure of the optimal payoff function, which is shown to possess rich properties. Further refinements of the model, such as the request that the payoff should be monotonic with respect to the index, are discussed. Throughout the paper, various numerical examples are used to illustrate the underlying concepts.     

基矛CVaR的投资组合优化模型研究

风险价值（VaR）是近年来国际金融机构所倡导的测度和控制金融风险的国际主流技术，但是它在投资组合损益服从非正态分布的情形时，不满足一致性风险度量，出现尾部损失测量的非充分性。为了使具有一致性的条件风险值度量（CVaR）克服VaR的不足，构建基于CVaR约束的投资组合优化模型，该模型虑及了投资组合资产的交易成本、交易限制、资金约束和投资者的风险承受度，为制定合理的最优投资组合提供了一种新的思路。     

基于CVaR的投资组合优化模型研究

风险价值(VaR)是近年来国际金融机构所倡导的测度和控制金融风险的国际主流技术,但是它在投资组合损益服从非正态分布的情形时,不满足一致性风险度量,出现尾部损失测量的非充分性。为了使具有一致性的条件风险值度量(CVaR)克服VaR的不足,构建基于CVaR约束的投资组合优化模型,该模型虑及了投资组合资产的交易成本、交易限制、资金约束和投资者的风险承受度,为制定合理的最优投资组合提供了一种新的思路。     

Should Stochastic Volatility Matter to the Cost-Constrained Investor?

Significant strides have been made in the development of continuous-time portfolio optimization models since Merton (1969) . Two independent advances have been the incorporation of transaction costs and time-varying volatility into the investor's optimization problem. Transaction costs generally inhibit investors from trading too often. Time-varying volatility, on the other hand, encourages trading activity, as it can result in an evolving optimal allocation of resources. We examine the two-asset portfolio optimization problem when both elements are present. We show that a transaction cost framework can be extended to include a stochastic volatility process. We then specify a transaction cost model with stochastic volatility and show that when the risk premium is linear in variance, the optimal strategy for the investor is independent of the level of volatility in the risky asset. We call this the Variance Invariance Principle.     

基于粒子群算法的证券组合投资模型的研究

从证券组合模型的概念,给出证券组合多目标决策模型。并用模糊优选法将证券组合多目标优化转化为单目标优化。同时,重点描述了粒子群算法,并采用自适应变异的粒子群算法对证券组合投资模型进行求解,最后编程实现证券组合模型的最优解,试验结果表明此方法取得了较好的效果。     

SIMULATION-BASED PORTFOLIO OPTIMIZATION FOR LARGE PORTFOLIOS WITH TRANSACTION COSTS

We consider a portfolio optimization problem where the investor's objective is to maximize the long-term expected growth rate, in the presence of proportional transaction costs. This problem belongs to the class of stochastic control problems with singular controls , which are usually solved by computing solutions to related partial differential equations called the free-boundary Hamilton–Jacobi–Bellman (HJB) equations . The dimensionality of the HJB equals the number of stocks in the portfolio. The runtime of existing solution methods grow super-exponentially with dimension, making them unsuitable to compute optimal solutions to portfolio optimization problems with even four stocks. In this work we first present a boundary update procedure that converts the free boundary problem into a sequence of fixed boundary problems. Then by combining simulation with the boundary update procedure, we provide a computational scheme whose runtime, as shown by the numerical tests, scales polynomially in dimension. The results are compared and corroborated against existing methods that scale super-exponentially in dimension. The method presented herein enables the first ever computational solution to free-boundary problems in dimensions greater than three.