最优投资组合的极大极小模型的理论及算法

发布时间：2018-06-10 11:21

本文选题：极大极小方法 + 最优化　；参考：《湘潭大学》2012年硕士论文

【摘要】：金融学研究的核心问题之一是在不确定的环境下对资产进行有效地合理地配置.1952年,Markowitz[1～4]假定证券收益是随机变量,利用证券收益的方差度量投资风险,利用证券收益的均值来度量收益的好坏,针对投资组合问题提出了均值-方差模型.根据证券市场的实际情况,人们发现一些摩擦因素:交易费和税收等,他们对投资者的决策有直接影响,因此后来一大批学者把研究的重点放在了带摩擦因素的投资组合上. 本文研究了金融优化中不允许卖空下的无风险资产借贷下最优投资组合的极大极小模型和不允许卖空下的带交易费的无风险资产借贷下最优投资组合的极大极小模型.并分别分析了两种模型的数学特征,推导出有效的解析表达式,另外还给出了有效前沿表达式以及表达式的几何特征.在求解不允许卖空下的带交易费的无风险资产借贷下最优投资组合的极大极小模型时,由于交易费不可微,给求解表达式带来了不便,我们引进了次微分,根据次微分的定义,我们给出了交易费函数h(x)的次微分.然后利用最优化理论进行求解. 本文分为五章,第一章简单介绍了投资组合问题的研究背景和进展,对本文的主要工作做了介绍以及对金融优化所涉及的基本概念作了介绍,另外对本文所用到的符号进行说明.第二章,第三章和第四章,分析了不允许卖空下的无风险资产借贷下最优投资组合的极大极小模型,以及带不同类型交易费函数的不允许卖空下无风险资产借贷下最优投资组合的极大极小模型的数学特征,给出了有效投资组合的解析表达式以及表达式的几何特征.第五章是结论部分，，是对本文结果的总结以及对未来研究的展望。
[Abstract]:One of the core problems of financial research is to allocate assets effectively and reasonably under uncertain circumstances. In 1952, Markowitz assumed that securities returns were random variables, and used variance of securities returns to measure investment risk. The mean value of security returns is used to measure the returns, and a mean-variance model is proposed for portfolio problems. According to the actual situation in the securities market, people find some frictional factors, such as transaction fees and taxes, which have a direct impact on investors' decisions. As a result, a large number of scholars focused on the portfolio with frictional factors. This paper studies the minimax model of the optimal portfolio under the risk-free asset loan in financial optimization. A minimax model of the optimal portfolio with transaction fees allowed for loan of risk-free assets under short selling. The mathematical characteristics of the two models are analyzed, and the effective analytical expressions are derived. In addition, the effective frontier expressions and the geometric characteristics of the expressions are given. In solving the minimax model of the optimal portfolio under the risk-free asset loan with transaction cost, it is inconvenient to solve the expression because the transaction cost is not differentiable, so we introduce the subdifferential, which is based on the definition of subdifferential. We give the subdifferential of the transaction cost function HX). The first chapter briefly introduces the research background and progress of portfolio problem, introduces the main work of this paper and introduces the basic concepts involved in financial optimization. In addition, the symbols used in this paper are explained. In chapter two, chapter three and chapter four, we analyze the minimax model of optimal portfolio under risk-free asset borrowing without short selling. The mathematical characteristics of the minimax model of the optimal portfolio with different types of transaction cost functions are given. The analytic expression of the effective portfolio and the geometric characteristics of the expression are given. The fifth chapter is the conclusion part, is the summary of the results of this paper and prospects for future research.
【学位授予单位】：湘潭大学
【学位级别】：硕士
【学位授予年份】：2012
【分类号】：F224;F830.59

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