随机市场中均值-方差模型最优投资策略的时间不相容性及其修正

由于方差算子在动态规划意义下不可分,导致随机市场中多期均值一方差模型的最优投资策略不满足时间相容性,即Bellman最优性原理．为此,首先提出了随机市场中比Bellman最优性原理更弱的时间相容性,并证明在投资区间的任意中间时刻,当投资者的财富不超过某一给定的财富阈值时,最优投资策略满足弱时间相容性;当投资者的财富超过该阈值时,最优投资策略将不再是弱时间相容的,且导致投资者变为非理性,即他会同时极小化终期财富的均值和方差．在这种情形下,通过放松自融资约束,对最优投资策略进行了修正,使得其满足：修正策略可使投资者回归理性;相对于终期财富,修正策略可以获得与最优投资策略相同的均值和方差．在策略修正过程中,投资者可以从市场中获得一个严格正的现金流．这些结果表明修正策略要优于原最优投资策略,拓展了现有关于确定市场下多期均值．方差模型的求解以及策略时间相容性的结论．     

具有交易费用和马氏调制的均值-方差组合选择

研究了均值-方差准则下,最优投资组合选择问题.投资者为了增加财富它可以在金融市场上投资.金融市场由一个无风险资产和n个带跳的风险资产组成,并假设金融市场具有马氏调制,买卖风险资产时,考虑交易费用.目标是,在终值财富的均值等于d的限制下,使终值财富的方差最小,即均值-方差组合选择问题.应用随机控制的理论解决该问题,获得了最优的投资策略和有效边界.     

Better than optimal mean–variance portfolio policy in multi-period asset–liability management problem

When the wealth is larger than some threshold in multi-period mean–variance asset–liability management, the pre-committed policy is no longer mean–variance efficient policy for the remaining investment horizon. To revise the policy, by relaxing self-financing constraint and allowing to withdraw some wealth, we derive a new dominating policy, which is better than the pre-committed policy. The revised policy can achieve the same mean–variance pairs attained by the pre-committed policy, and yields a nonnegative free cash flow stream over the investment horizon.     

Classical mean-variance model revisited: pseudo efficiency

Investigating the inverse problem of the classical Markowitz mean-variance formulation: Given a mean-variance pair, find initial investment levels and their corresponding portfolio policies such that the given mean-variance pair can be realized, we reveal that any mean-variance pair inside the reachable region can be achieved by multiple portfolio policies associated with different initial investment levels. Therefore, in the mean-variance world for a market of all risky assets, the common belief of monotonicity: ‘The larger you invest, the larger expected future wealth you can expect for a given risk (variance) level’ does not hold, which stimulates us to extend the classical two-objective mean-variance framework to an expanded three-objective framework: to maximize the mean and minimize the variance of the final wealth as well as to minimize the initial investment level. As a result, we eliminate from the policy candidate list the set of pseudo efficient policies that are efficient in the original mean-variance space, but inefficient in this newly introduced three-dimensional objective space.     

跳扩散模型下的非零和随机微分投资组合博弈

现实经济中,当股票价格受到一些重大信息影响而发生突发性的跳跃时,用跳扩散过程来描述股票价格的趋势更符合实际情况。基于这一观察,本文研究跳扩散模型下包含两个投资者的非零和投资组合博弈问题。假设金融市场中包含一种无风险资产和一种风险资产,其中风险资产的价格动态用跳扩散模型来描述。将该非零和博弈问题构造成两个效用最大化问题,每个投资者的目标是最大化终端时刻自身财富与其竞争对手财富差的均值-方差效用。运用随机控制理论,得到了均衡投资策略以及相应值函数的解析表达。最后通过数值仿真算例分析了模型相关参数变动对均衡投资策略的影响。仿真结果显示:当股价发生不连续跳跃,投资者在构造投资策略时考虑跳跃风险可以显著增加其效用水平;同时,随着博弈竞争的加剧,投资者为了在竞争中取得更好的表现,往往会采取更加激进的投资策略,增加对风险资产的投资。     

含有信用风险的跳扩散市场的最优组合选择

假定投资者将其财富分配在这样两种风险资产中,一种是股票,价格服从跳跃扩散过程;一种是有信用风险的债券,其价格服从复合泊松过程.在均值-方差准则下通过最优控制原理来研究投资者的最优投资策略选择问题,得到了最优投资策略及有效边界,最后通过数值例子分析了违约强度、债券预期收益率以及目标财富对最优投资策略的影响.     

跳扩散市场投资组合研究

研究了连续时间动态均值-方差投资组合选择问题.假设风险资产价格服从跳跃-扩散过程且具有卖空约束.投资者的目标是在给定期望终止时刻财富条件下,最小化终止时刻财富的方差.通过求解模型相应的Hamilton-Jacobi-Bellmen方程,得到了最优投资策略及有效前沿的显示解.结果显示,风险资产的卖空约束及价格过程的跳跃因素对最优投资策略及有效前沿的是不可忽略的.     

Vasicek利率模型下带有零息票债券的投资-消费模型

应用随机最优控制理论研究Vasicek利率模型下的投资-消费问题,其中假设无风险利率是服从Vasicek利率模型的随机过程,且与股票价格过程存在一般相关性.假设金融市场由一种无风险资产、一种风险资产和一种零息票债券所构成,投资者的目标是最大化中期消费与终端财富的期望贴现效用.应用变量替换方法得到了幂效用下最优投资-消费策略的显示表达式,并分析了最优投资-消费策略对市场参数的灵敏度.     

均值-方差准则下具有负债的随机微分博弈

研究了均值-方差准则下,具有负债的随机微分博弈.研究目标是:在终值财富的均值等于k的限制下,在市场出现最坏的情况下找到最优的投资策略使终值财富的方差最小.即:基于均值-方差随机微分博弈的投资组合选择问题.使用线性-二次控制的理论解决了该问题,获得了最优的投资策略、最优市场策略和有效边界的显示解.并通过对所得结果进行进一步分析,在经济上给出了进一步的解释.通过本文的研究,可以指导金融公司在面临负债和金融市场情况恶劣时,选择恰当的投资策略使自身获得一定的财富而面临的风险最小.     

Optimal time-consistent investment and reinsurance policies for mean-variance insurers

This paper investigates the optimal time-consistent policies of an investment-reinsurance problem and an investment-only problem under the mean-variance criterion for an insurer whose surplus process is approximated by a Brownian motion with drift. The financial market considered by the insurer consists of one risk-free asset and multiple risky assets whose price processes follow geometric Brownian motions. A general verification theorem is developed, and explicit closed-form expressions of the optimal polices and the optimal value functions are derived for the two problems. Economic implications and numerical sensitivity analysis are presented for our results. Our main findings are: (i) the optimal time-consistent policies of both problems are independent of their corresponding wealth processes; (ii) the two problems have the same optimal investment policies; (iii) the parameters of the risky assets (the insurance market) have no impact on the optimal reinsurance (investment) policy; (iv) the premium return rate of the insurer does not affect the optimal policies but affects the optimal value functions; (v) reinsurance can increase the mean-variance utility.     

Optimal Asset Allocation for Retirement Saving: Deterministic Vs. Time Consistent Adaptive Strategies

We consider optimal asset allocation for an investor saving for retirement. The portfolio contains a bond index and a stock index. We use multi-period criteria and explore two types of strategies: deterministic strategies are based only on the time remaining until the anticipated retirement date, while adaptive strategies also consider the investor’s accumulated wealth. The vast majority of financial products designed for retirement saving use deterministic strategies (e.g., target date funds). In the deterministic case, we determine an optimal open loop control using mean-variance criteria. In the adaptive case, we use time consistent mean-variance and quadratic shortfall objectives. Tests based on both a synthetic market where the stock index is modelled by a jump-diffusion process and also on bootstrap resampling of long-term historical data show that the optimal adaptive strategies significantly outperform the optimal deterministic strategy. This suggests that investors are not being well served by the strategies currently dominating the marketplace.     

Portfolio selection in stochastic markets with HARA utility functions

In this paper, we consider the optimal portfolio selection problem where the investor maximizes the expected utility of the terminal wealth. The utility function belongs to the HARA family which includes exponential, logarithmic, and power utility functions. The main feature of the model is that returns of the risky assets and the utility function all depend on an external process that represents the stochastic market. The states of the market describe the prevailing economic, financial, social, political and other conditions that affect the deterministic and probabilistic parameters of the model. We suppose that the random changes in the market states are depicted by a Markov chain. Dynamic programming is used to obtain an explicit characterization of the optimal policy. In particular, it is shown that optimal portfolios satisfy the separation property and the composition of the risky portfolio does not depend on the wealth of the investor. We also provide an explicit construction of the optimal wealth process and use it to determine various quantities of interest. The return-risk frontiers of the terminal wealth are shown to have linear forms. Special cases are discussed together with numerical illustrations.     

Continuous time mean-variance portfolio optimization with piecewise state-dependent risk aversion

In this paper, a continuous time mean-variance portfolio optimization problem is considered within a game theoretic framework, where the risk aversion function is assumed to depend on the current wealth level and the discounted (preset) investment target. We derive the explicit time consistent investment policy, and find that if the current wealth level is less (larger) than the discounted investment target, the future wealth level along the time consistent investment policy is always less (larger) than the discounted investment target.     

Dynamic mean-variance and optimal reinsurance problems under the no-bankruptcy constraint for an insurer

In this paper, we consider the optimal investment and optimal reinsurance problems for an insurer under the criterion of mean-variance with bankruptcy prohibition, i.e., the wealth process of the insurer is not allowed to be below zero at any time. The risk process is a diffusion model and the insurer can invest in a risk-free asset and multiple risky assets. In view of the standard martingale approach in tackling continuous-time portfolio choice models, we consider two subproblems. After solving the two subproblems respectively, we can obtain the solution to the mean-variance optimal problem. We also consider the optimal problem when bankruptcy is allowed. In this situation, we obtain the efficient strategy and efficient frontier using the stochastic linear-quadratic control theory. Then we compare the results in the two cases and give a numerical example to illustrate our results.     

Heston随机波动率市场中带VaR约束的最优投资策略

本文研究了Heston随机波动率市场下, 基于VaR约束下的动态最优投资组合问题。
假设Heston随机波动率市场由一个无风险资产和一个风险资产构成,投资者的目标为最大化其终端的期望效用。与此同时, 投资者将动态地评估其待选的投资组合的VaR风险,并将其控制在一个可接受的范围之内。本文在合理的假设下,使用动态规划的方法,来求解该问题的最优投资策略。在特定的参数范围内,利用数值方法计算出近似的最优投资策略和相应值函数, 并对结果进行了分析。     

Portfolio selection in stochastic markets with exponential utility functions

We consider the optimal portfolio selection problem in a multiple period setting where the investor maximizes the expected utility of the terminal wealth in a stochastic market. The utility function has an exponential structure and the market states change according to a Markov chain. The states of the market describe the prevailing economic, financial, social and other conditions that affect the deterministic and probabilistic parameters of the model. This includes the distributions of the random asset returns as well as the utility function. The problem is solved using the dynamic programming approach to obtain the optimal solution and an explicit characterization of the optimal policy. We also discuss the stochastic structure of the wealth process under the optimal policy and determine various quantities of interest including its Fourier transform. The exponential return-risk frontier of the terminal wealth is shown to have a linear form. Special cases of multivariate normal and exponential returns are disussed together with a numerical illustration.     

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??Under inflation influence, this paper investigate a stochastic differential game with reinsurance and investment. Insurance company chose a strategy to minimizing the variance of the final wealth, and the financial markets as a game ``virtual hand' chosen a probability measure represents the economic ``environment' to maximize the variance of the final wealth. Through this double game between the insurance companies and the financial markets, get optimal portfolio strategies. When investing, we consider inflation, the method of dealing with inflation is: Firstly, the inflation is converted to the risky assets, and then constructs the wealth process. Through change the original based on the mean-variance criteria stochastic differential game into unrestricted cases, then application linear-quadratic control theory obtain optimal reinsurance strategy and investment strategy and optimal market strategy as well as the closed form expression of efficient frontier are obtained; finally get reinsurance strategy and optimal investment strategy and optimal market strategy as well as the closed form expression of efficient frontier for the original stochastic differential game.     

基于通货膨胀风险和Heston随机波动率下的最优资产配置

本文考虑了基于均值-方差准则下的连续时间投资组合选择问题。为了对冲市场中的利率风险和通货膨胀风险,假定市场上存在可供交易的零息名义债券和零息通货膨胀指数债券。另外,投资者还可以投资一个价格具有Heston随机波动率的风险资产。首先建立了基于均值-方差框架下的最优投资组合问题,然后将原问题进行转换,利用随机动态规划方法和对偶Lagrangian原理,获得了均值-方差准则下的有效投资策略以及有效前沿的解析表达形式,最后对相关参数的敏感性进行了分析。     

通货膨胀时期的投资策略分析

目前,各个国家面临着不同程度的通贷膨胀.在此情况下,如何规避通货膨胀所带来的财富稀释是现阶段所有投资人追求的目标.本文从投资产品的价格出发给出了两阶段均值-方差投资问题的最优解.分析了交易费用、风险资产的期望回报率以及波动率对投资策略的影响.最后,我们依据实际算例分析为投资者提供指导,并且给出了政策建议.     

开环策略下多阶段均值-方差投资组合优化研究

首先研究开环策略下不同财富动态过程的多阶段均值-方差投资组合优化模型,讨论它们的实际意义和计算方法,其中投资比例财富动态过程模型为高度非线性非凸数学规划.进一步研究投资比例财富动态过程模型实际计算问题,并且通过构造辅助模型,给出投资比例两阶段模型的全局解求解方法并通过数值算例和仿真说明该方法的有效性和准确性.最后通过数值算例比较不同财富动态过程在开环策略下和闭环策略下前沿面的关系,结果表明在闭环策略下三种财富过程等价,但是在开环策略下资产财富模型的前沿面最高、资产调整模型的前沿面次之、投资比例多阶段模型的前沿面最低.