连续时间框架下资产混合策略研究
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摘要
个人的投资与消费、机构资产管理等都是非常重要的。面对现实生活中的大量不确定性因素以及多期投资问题,Markowitz的投资组合理论以及在其基础上发展起来的资本资产定价理论和套利定价理论则显得无能为力。本文在考虑现实生活中存在的各种影响投资和消费活动的约束条件下,采用随机最优控制技术,分别对以下几种投资组合管理问题进行了研究并对最优配置方案进行了求解,进而揭示其在现实应用中的意义,主要研究的问题包括:连续、多期投资组合/消费问题、摩擦市场的投资消费问题、随机利率和通货膨胀问题、随机环境下养老金配置问题以及有稀有事件发生的跳跃过程问题。全文分五个方面对连续时间框架下的资产混合策略进行研究:
1、假定证券预期收益和方差为外生状态变量的随机函数,以方差为风险测度,采用动态随机规划技术对投资消费财富最大化及风险最小化同时进行考虑的交易策略进行研究,并建立多目标模型,对模型进行求解并解的经济含义进行了分析。
2、对两资产、HARA效用函数下,有限期间和无限期间具有成比例交易费用的最优投资消费策略的价值函数的解析形式进行了研究。利用价值函数的凹性和近似性,通过变量代换的方式,将HJB方程从偏微分方程(PDE)转化常微分方程(ODE),对价值函数进行求解,给出交易区解析解的形式和非交易区所要满足的HJB方程。
3、基金经理的投资建议和两基金分离定理之间存在较大差异,这就是著名的Canner难题。以银行帐户、两种债券和股票作为可以交易的资产,采用了Hull&White两因素利率期限结构模型,对以通货膨胀为背景风险和长、短期利率为主要影响因素的资产配置问题进行了研究并得出最优的资产配置方案。通过对该方案的经济含义进行理论分析和实例分析,对投资建议和两基金分离定理之间的差异给出了合理的解释。
4、对一般框架下,以投资期末预期财富最大化为目标的缴费确定性养老金计划进行了研究。在一个具有随机投资机会集的经济环境中,基金管理者将面临着金融市场风险和以缴费者的随机收入为代表的背景风险。本文对这一问题构建了数学模型,然后合理的猜测价值函数以及其中的参量函数的具体形式,最终得出了该问题的解,给出最优配置方案。最后,对最优配置方案中较为复杂的第三项的具体求解方法进行了演示。
5、一般维纳过程可以模拟出连续时间和连续状态下的金融资产价格运动模型,但是,金融价格变动并非都是连续的。为了全面描绘资产价格的真实运动情况,引入了泊松过程来模拟随机过程的跳跃部分。泊松过程主要是用来构造了金融市场上“小概率事件”。事实上,股票价格的连续变动和突然变动两部分基本上描述了股票价格波动的主要特征。首先建立模型的框架,对金融市场结构进行介绍,给出了资产配置模型的动态预算方程,讨论了资产价格存在泊松过程的情况时的最优配置问题,以及有风险因素存在泊松过程的最优配置问题两个问题,分别给出了最优配置方案。由于随机偏微分方程求解的困难,分别引入了猜测价值函数形式以及Taylor级数展开的方法,对价值函数进行求解。最后,获得最优配置的近似解的解析表达式。
Investments and consumptions of an individual and assets managements of an institute are very important activities. The portfolios theory of Markowitz becomes helpless while faces the large amount uncertainties in the reality. In our thesis, we take account of the most common constrains on investments, consumptions and assets allocations. By adopting singular stochastic control technique, we solve problems on continuous-time, intertemporal Investments and consumptions under friction market, assets allocation with stochastic interest rate and inflation and pensions under stochastic environments. The analytic forms are achieved in most of the above problems. The assets mix strategies under continuous-time framework in this thesis are scheduled by the following:
1. It supposes that asset prices evolve through time according to Ito diffusion processes, the expected yield and variance are impacted by exogenous states variables. We studies dynamic strategies of maximizing the utility of Consumption and Investment and minimizing risk. The multi-objective models are established and the solutions are given.
2. We consider the optimal Consumption and Investment Strategy for a hyperbolic absolute risk aversion (HARA) investor who faces proportional transaction costs and maximizes expected utility of finite/ infinite horizon with only two-asset. Using the concavity and the homothetic property of the value function, the HJB can be reduced from a PDE to an ODE. The analytic forms of the value functions in the transaction regions are achieved and HJB equation. We also introduce the option pricing technique under transaction cost.
3. The advices of popular investment advisors are apparently inconsistent with the Separation Theorem; it is the so-called Canner Puzzle. A bank account, nominal bonds, and stocks can be traded. We provide the optimal asset allocation strategy with interest rate risk and inflation risk, which can be seen as a background risk. Uncertainty about future interest rates is represented by the Hull&White two-factor model .The solutions are given and the economic significations are analyzed by theoretic and illustrative calculations. The rational explications of the diversities between the popular investment advices and the Separation Theorem are given.
4. We study defined-contribution plans by maximizing the terminal expected wealth under a general framework. In an economics environment with stochastic investment opportunities, a fund manager is faced with market risks and backgroundrisks, which is represented with the salary risk. We first construct the mathematical model, then guess the value function and its parameters forms, and finally, we achieve the optimal strategy and the solution. We also make presentations on this strategy.
5. A Generalized Wiener Process can simulate the price process of a financial asset. But actually, the asset’s price changes non-continuous. We introduce a Poisson Process to simulate the“jump”stochastic process model. We first construct the model, and then introduce the financial market and the dynamics budget equation. We discuss the two problems that the asset price evolves with Poisson Process and the risk factor evolves with Poisson Process. The optimal allocation strategies are given for these two problems.
1、假定证券预期收益和方差为外生状态变量的随机函数,以方差为风险测度,采用动态随机规划技术对投资消费财富最大化及风险最小化同时进行考虑的交易策略进行研究,并建立多目标模型,对模型进行求解并解的经济含义进行了分析。
2、对两资产、HARA效用函数下,有限期间和无限期间具有成比例交易费用的最优投资消费策略的价值函数的解析形式进行了研究。利用价值函数的凹性和近似性,通过变量代换的方式,将HJB方程从偏微分方程(PDE)转化常微分方程(ODE),对价值函数进行求解,给出交易区解析解的形式和非交易区所要满足的HJB方程。
3、基金经理的投资建议和两基金分离定理之间存在较大差异,这就是著名的Canner难题。以银行帐户、两种债券和股票作为可以交易的资产,采用了Hull&White两因素利率期限结构模型,对以通货膨胀为背景风险和长、短期利率为主要影响因素的资产配置问题进行了研究并得出最优的资产配置方案。通过对该方案的经济含义进行理论分析和实例分析,对投资建议和两基金分离定理之间的差异给出了合理的解释。
4、对一般框架下,以投资期末预期财富最大化为目标的缴费确定性养老金计划进行了研究。在一个具有随机投资机会集的经济环境中,基金管理者将面临着金融市场风险和以缴费者的随机收入为代表的背景风险。本文对这一问题构建了数学模型,然后合理的猜测价值函数以及其中的参量函数的具体形式,最终得出了该问题的解,给出最优配置方案。最后,对最优配置方案中较为复杂的第三项的具体求解方法进行了演示。
5、一般维纳过程可以模拟出连续时间和连续状态下的金融资产价格运动模型,但是,金融价格变动并非都是连续的。为了全面描绘资产价格的真实运动情况,引入了泊松过程来模拟随机过程的跳跃部分。泊松过程主要是用来构造了金融市场上“小概率事件”。事实上,股票价格的连续变动和突然变动两部分基本上描述了股票价格波动的主要特征。首先建立模型的框架,对金融市场结构进行介绍,给出了资产配置模型的动态预算方程,讨论了资产价格存在泊松过程的情况时的最优配置问题,以及有风险因素存在泊松过程的最优配置问题两个问题,分别给出了最优配置方案。由于随机偏微分方程求解的困难,分别引入了猜测价值函数形式以及Taylor级数展开的方法,对价值函数进行求解。最后,获得最优配置的近似解的解析表达式。
Investments and consumptions of an individual and assets managements of an institute are very important activities. The portfolios theory of Markowitz becomes helpless while faces the large amount uncertainties in the reality. In our thesis, we take account of the most common constrains on investments, consumptions and assets allocations. By adopting singular stochastic control technique, we solve problems on continuous-time, intertemporal Investments and consumptions under friction market, assets allocation with stochastic interest rate and inflation and pensions under stochastic environments. The analytic forms are achieved in most of the above problems. The assets mix strategies under continuous-time framework in this thesis are scheduled by the following:
1. It supposes that asset prices evolve through time according to Ito diffusion processes, the expected yield and variance are impacted by exogenous states variables. We studies dynamic strategies of maximizing the utility of Consumption and Investment and minimizing risk. The multi-objective models are established and the solutions are given.
2. We consider the optimal Consumption and Investment Strategy for a hyperbolic absolute risk aversion (HARA) investor who faces proportional transaction costs and maximizes expected utility of finite/ infinite horizon with only two-asset. Using the concavity and the homothetic property of the value function, the HJB can be reduced from a PDE to an ODE. The analytic forms of the value functions in the transaction regions are achieved and HJB equation. We also introduce the option pricing technique under transaction cost.
3. The advices of popular investment advisors are apparently inconsistent with the Separation Theorem; it is the so-called Canner Puzzle. A bank account, nominal bonds, and stocks can be traded. We provide the optimal asset allocation strategy with interest rate risk and inflation risk, which can be seen as a background risk. Uncertainty about future interest rates is represented by the Hull&White two-factor model .The solutions are given and the economic significations are analyzed by theoretic and illustrative calculations. The rational explications of the diversities between the popular investment advices and the Separation Theorem are given.
4. We study defined-contribution plans by maximizing the terminal expected wealth under a general framework. In an economics environment with stochastic investment opportunities, a fund manager is faced with market risks and backgroundrisks, which is represented with the salary risk. We first construct the mathematical model, then guess the value function and its parameters forms, and finally, we achieve the optimal strategy and the solution. We also make presentations on this strategy.
5. A Generalized Wiener Process can simulate the price process of a financial asset. But actually, the asset’s price changes non-continuous. We introduce a Poisson Process to simulate the“jump”stochastic process model. We first construct the model, and then introduce the financial market and the dynamics budget equation. We discuss the two problems that the asset price evolves with Poisson Process and the risk factor evolves with Poisson Process. The optimal allocation strategies are given for these two problems.
引文
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