Multi‐period mean variance portfolio selection under incomplete information

This paper solves an optimal portfolio selection problem in the discrete‐time setting where the states of the financial market cannot be completely observed, which breaks the common assumption that the states of the financial market are fully observable. The dynamics of the unobservable market state is formulated by a hidden Markov chain, and the return of the risky asset is modulated by the unobservable market state. Based on the observed information up to the decision moment, an investor wants to find the optimal multi‐period investment strategy to maximize the mean‐variance utility of the terminal wealth. By adopting a sufficient statistic, the portfolio optimization problem with incompletely observable information is converted into the one with completely observable information. The optimal investment strategy is derived by using the dynamic programming approach and the embedding technique, and the efficient frontier is also presented. Compared with the case when the market state can be completely observed, we find that the unobservable market state does decrease the investment value on the risky asset in average. Finally, numerical results illustrate the impact of the unobservable market state on the efficient frontier, the optimal investment strategy and the Sharpe ratio. Copyright © 2016 John Wiley & Sons, Ltd.     

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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.     

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 portfolio‐consumption choice under stochastic inflation with nominal and indexed bonds

This research solves the intertemporal portfolio choice problems with and without interim consumption under stochastic inflation. We assume a one‐factor nominal interest rate and a one‐factor expected inflation rate, implying a two‐factor real interest rate in the economy. In contrast to other related research which adopts the one‐factor real interest rate model, the inflation‐indexed bond is not a redundant asset class even in a complete market. The infinitely risk‐averse investor would prefer to invest all her wealth in inflation‐indexed bonds maturing at the investment horizon. We also show that, with the two‐factor real interest rate model, the consumption‐wealth ratio is not determined by the real interest rate alone. The investor's consumption–wealth ratio is also affected by the nominal interest rate and expected inflation rate levels. The capital market is calibrated to U.S. stocks, bonds, and inflation data. The optimal weights show that aggressive investors hold more nominal bonds in order to earn the inflation risk premiums, while conservative investors concentrate on indexed bonds to hedge against the inflation risk. Copyright © 2011 John Wiley & Sons, Ltd.     

An examination of HMM‐based investment strategies for asset allocation

We develop and analyse investment strategies relying on hidden Markov model approaches. In particular, we use filtering techniques to aid an investor in his decision to allocate all of his investment fund to either growth or value stocks at a given time. As this allows the investor to switch between growth and value stocks, we call this first strategy a switching investment strategy. This switching strategy is compared with the strategies of purely investing in growth or value stocks by tracking the quarterly terminal wealth of a hypothetical portfolio for each strategy. Using the data sets on Russell 3000 growth index and Russell 3000 value index compiled by Russell Investment Services for the period 1995–2008, we find that the overall risk‐adjusted performance of the switching strategy is better than that of solely investing in either one of the indices. We also consider a second strategy referred to as a mixed investment strategy which enables the investor to allocate an optimal proportion of his investment between growth and value stocks given a level of risk aversion. Numerical demonstrations are provided using the same data sets on Russell 3000 growth and value indices. The switching investment strategy yields the best or second best Sharpe ratio as compared with those obtained from the pure index strategies and mixed strategy in 14 intervals. The performance of the mixed investment strategy under the HMM setting is also compared with that of the classical mean–variance approach. To make the comparison valid, we choose the same level of risk aversion for each set‐up. Our findings show that the mixed investment strategy within the HMM framework gives higher Sharpe ratios in 5 intervals of the time series than that given by the standard mean–variance approach. The calculated weights through time from the strategy incorporating the HMM set‐up are more stable. A simulation analysis further shows a higher performance stability of the HMM strategies compared with the pure strategies and the mean–variance strategy. Copyright © 2009 John Wiley & Sons, Ltd.     

Worst-case portfolio optimization with proportional transaction costs

We study optimal asset allocation in a crash-threatened financial market with proportional transaction costs. The market is assumed to be either in a normal state, in which the risky asset follows a geometric Brownian motion, or in a crash state, in which the price of the risky asset can suddenly drop by a certain relative amount. We only assume the maximum number and the maximum relative size of the crashes to be given and do not make any assumptions about their distributions. For every investment strategy, we identify the worst-case scenario in the sense that the expected utility of terminal wealth is minimized. The objective is then to determine the investment strategy which yields the highest expected utility in its worst-case scenario. We solve the problem for utility functions with constant relative risk aversion using a stochastic control approach. We characterize the value function as the unique viscosity solution of a second-order nonlinear partial differential equation. The optimal strategies are characterized by time-dependent free boundaries which we compute numerically. The numerical examples suggest that it is not optimal to invest any wealth in the risky asset close to the investment horizon, while a long position in the risky asset is optimal if the remaining investment period is sufficiently large.     

允许投资组合条件下概率准则

概率准则具有一定的现实意义，其投资决策是以期望贴现资产为导向的．本文讨论了完备标准动态金融市场中在允许投资组合条件下的概率准则问题，得到了准则函数，贴现资产过程以及最优允许投资组合过程的解析表达式．期望贴现资产越大，准则函数越小。     

Calculating risk neutral probabilities and optimal portfolio policies in a dynamic investment model with downside risk control

This paper presents a method for solving multiperiod investment models with downside risk control characterized by the portfolio’s worst outcome. The stochastic programming problem is decomposed into two subproblems: a nonlinear optimization model identifying the optimal terminal wealth distribution and a stochastic linear programming model replicating the identified optimal portfolio wealth. The replicating portfolio coincides with the optimal solution to the investor’s problem if the market is frictionless. The multiperiod stochastic linear programming model tests for the absence of arbitrage opportunities and its dual feasible solutions generate all risk neutral probability measures. When there are constraints such as liquidity or position requirements, the method yields approximate portfolio policies by minimizing the initial cost of the replication portfolio. A numerical example illustrates the difference between the replicating result and the optimal unconstrained portfolio.     

Optimal reinsurance–investment problem in a constant elasticity of variance stock market for jump‐diffusion risk model

In this paper, we consider the jump‐diffusion risk model with proportional reinsurance and stock price process following the constant elasticity of variance model. Compared with the geometric Brownian motion model, the advantage of the constant elasticity of variance model is that the volatility has correlation with the risky asset price, and thus, it can explain the empirical bias exhibited by the Black and Scholes model, such as volatility smile. Here, we study the optimal investment–reinsurance problem of maximizing the expected exponential utility of terminal wealth. By using techniques of stochastic control theory, we are able to derive the explicit expressions for the optimal strategy and value function. Numerical examples are presented to show the impact of model parameters on the optimal strategies. Copyright © 2011 John Wiley & Sons, Ltd.     

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.     

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.     

概率准则在投资模型中的应用研究

本文在Black-Scholes金融市场设置下，基于概率准则，研究连续时间金融市场最优动态资产组合的选择问题，导出了最优解的显式表达式，对结论给出了金融学解释，所得结果可以方便地应用于投资决策与管理实践中。     

An extension of the Clark–Haussmann formula and applications

ABSTRACT

This work considers a financial market stochastic model where the uncertainty is driven by a multidimensional Brownian motion. The market price of the risk process makes the transition between real world probability measure and risk neutral probability measure. Traditionally, the martingale representation formulas under the risk neutral probability measure require the market price of risk process to be bounded. However, in several financial models the boundedness assumption of the market price of risk fails; for example a financial market model with the market price of risk following an Ornstein–Uhlenbeck process. This work extends the Clark–Haussmann representation formula to underlying stochastic processes which fail to satisfy the standard requirements. Our methodology is classical, and it uses a sequence of mollifiers. Our result can be applied to hedging and optimal investment in financial markets with unbounded market price of risk. In particular, the mean variance optimization problem can be addressed within our framework.     

含不动产项目的保险公司再保险-投资策略

站在保险公司管理者的角度, 考虑存在不动产项目投资机会时保险公司的再保险--投资策略问题. 假定保险公司可以投资于不动产项目、风险证券和无风险证券, 并通过比例再保险控制风险, 目标是最小化保险公司破产概率并求得相应最佳策略, 包括: 不动产项目投资时机、 再保险比例以及投资于风险证券的金额. 运用混合随机控制-最优停时方法, 得到最优值函数及最佳策略的显式解. 结果表明, 当且仅当其盈余资金多于某一水平(称为投资阈值)时保险公司投资于不动产项目. 进一步的数值算例分析表明: (a)~不动产项目投资的阈值主要受项目收益率影响而与投资金额无明显关系, 收益率越高则投资阈值越低; (b)~市场环境较好(牛市)时项目的投资阈值降低; 反之, 当市场环境较差(熊市)时投资阈值提高.     

Optimal investment of DC pension plan under short-selling constraints and portfolio insurance

In this paper we investigate an optimal investment problem under short-selling and portfolio insurance constraints faced by a defined contribution pension fund manager who is loss averse. The financial market consists of a cash bond, an indexed bond and a stock. The manager aims to maximize the expected S-shaped utility of the terminal wealth exceeding a minimum guarantee. We apply the dual control method to solve the problem and derive the representations of the optimal wealth process and trading strategies in terms of the dual controlled process and the dual value function. We also perform some numerical tests and show how the S-shaped utility, the short-selling constraints and the portfolio insurance impact the optimal terminal wealth.     

A stochastic delay financial model

We compute the logarithmic utility of an insider when the financial market is modelled by a stochastic delay equation. Although the market does not allow free lunches and is complete, the insider can draw more from his wealth than the regular trader. We also offer an alternative to the anticipating delayed Black-Scholes formula, by proving stability of European call option prices when the delay coefficients approach the nondelayed ones.

     

Precommitment and equilibrium investment strategies for defined contribution pension plans under a jump–diffusion model

In this paper, we study an optimal investment problem under the mean–variance criterion for defined contribution pension plans during the accumulation phase. To protect the rights of a plan member who dies before retirement, a clause on the return of premiums for the plan member is adopted. We assume that the manager of the pension plan is allowed to invest the premiums in a financial market, which consists of one risk-free asset and one risky asset whose price process is modeled by a jump–diffusion process. The precommitment strategy and the corresponding value function are obtained using the stochastic dynamic programming approach. Under the framework of game theory and the assumption that the manager’s risk aversion coefficient depends on the current wealth, the equilibrium strategy and the corresponding equilibrium value function are also derived. Our results show that with the same level of variance in the terminal wealth, the expected optimal terminal wealth under the precommitment strategy is greater than that under the equilibrium strategy with a constant risk aversion coefficient; the equilibrium strategy with a constant risk aversion coefficient is revealed to be different from that with a state-dependent risk aversion coefficient; and our results can also be degenerated to the results of He and Liang (2013b) and Björk et al. (2014). Finally, some numerical simulations are provided to illustrate our derived results.     

Robust Optimal Portfolio Choice Under Markovian Regime-switching Model

We investigate an optimal portfolio selection problem in a continuous-time Markov-modulated financial market when an economic agent faces model uncertainty and seeks a robust optimal portfolio strategy. The key market parameters are assumed to be modulated by a continuous-time, finite-state Markov chain whose states are interpreted as different states of an economy. The goal of the agent is to maximize the minimal expected utility of terminal wealth over a family of probability measures in a finite time horizon. The problem is then formulated as a Markovian regime-switching version of a two-player, zero-sum stochastic differential game between the agent and the market. We solve the problem by the Hamilton-Jacobi-Bellman approach.      

Optimal investment and reinsurance for an insurer under Markov-modulated financial market

This study examines optimal investment and reinsurance policies for an insurer with the classical surplus process. It assumes that the financial market is driven by a drifted Brownian motion with coefficients modulated by an external Markov process specified by the solution to a stochastic differential equation. The goal of the insurer is to maximize the expected terminal utility. This paper derives the Hamilton–Jacobi–Bellman (HJB) equation associated with the control problem using a dynamic programming method. When the insurer admits an exponential utility function, we prove that there exists a unique and smooth solution to the HJB equation. We derive the explicit optimal investment policy by solving the HJB equation. We can also find that the optimal reinsurance policy optimizes a deterministic function. We also obtain the upper bound for ruin probability in finite time for the insurer when the insurer adopts optimal policies.     

Asset allocation under loss aversion and minimum performance constraint in a DC pension plan with inflation risk

In this paper we investigate an optimal investment strategy for a defined-contribution (DC) pension plan member who is loss averse, pays close attention to inflation and longevity risks and requires a minimum performance at retirement. The member aims to maximize the expected S-shaped utility from the terminal wealth exceeding the minimum performance by investing her wealth in a financial market consisting of an indexed bond, a stock and a risk-free asset. We derive the optimal investment strategy in closed-form using the martingale approach. Our theoretical and numerical results reveal that the wealth proportion invested in each risky asset has a V-shaped pattern in the reference point level, while it always increases in the rising lifespan; with a positive correlation between salary and inflation risks, the presence of salary decreases the member’s investment in risky assets; the minimum performance helps to hedge the longevity risk by increasing her investment in risky assets.