论文部分内容阅读
本文在连续时间不完备市场框架下,考虑了投资者终端时刻资产负债比率的期望效用最大化问题。假设金融市场由1个无风险资产与多个风险资产构成,其中风险资产的价格过程由几何布朗运动刻画;投资者在整个投资时间水平内面临一个由几何布朗运动刻画的外生负债。利用随机动态规划方法,给出了相应的HJB方程与验证定理,并得到了最优投资策略与最优值函数的解析表达式。进一步,通过敏感性分析与数值算例发现:(1)外生负债的预期增长率与当前时刻的资产负债比率对最优投资策略没有影响;(2)在不考虑外生负债时,在最优策略下,投资到风险资产上的资金比例随着风险资产波动率或相对风险厌恶系数的增大而减小,而在考虑外生负债时,并非如此,只有满足一定条件时最优投资策略才是风险资产波动率或相对风险厌恶系数的减函数;(3)不考虑外生负债时,最优值函数是投资时间水平与风险资产预期收益率的增函数,风险资产波动率的减函数,但在考虑外生负债时该结论只在各参数满足一定关系时才成立,否则结论相反。
In the framework of incomplete market for continuous time, this paper considers the expected utility maximization of the ratio of assets and liabilities at the moment of investor terminal. Suppose the financial market consists of one riskless asset and several risky assets, of which the price process of the risky asset is characterized by the geometric Brownian motion; and investors face an exogenous liability characterized by the geometric Brownian motion over the entire investment time horizon. Using the stochastic dynamic programming method, the corresponding HJB equation and verification theorem are given, and the analytical expressions of the optimal investment strategy and the optimal value function are obtained. Further, the sensitivity analysis and numerical examples show that: (1) the expected growth rate of exogenous liabilities and the current ratio of assets and liabilities have no effect on the optimal investment strategy; (2) When exogenous liabilities are not considered, Under the optimal strategy, the proportion of funds invested in risky assets decreases with the increase of risk assets’ volatility or relative risk aversion coefficient, but not when considering exogenous liabilities, and only when certain conditions are met, the optimal investment strategy Is the decreasing function of risk asset volatility or relative risk aversion coefficient; (3) The optimal value function is the increasing function of the investment time level and the expected return rate of risk assets without considering the exogenous liabilities, and the decreasing function of the risk asset volatility However, when considering the exogenous liabilities, this conclusion is valid only when the parameters satisfy certain relations, otherwise the conclusion is the opposite.