Continuous-Time Markowitz's Model with Transaction Costs
Continuous-Time Markowitz's Model with Transaction Costs
复制标题
DOI：
10.1137/080742889
复制
发表时间：
2009-06
期刊：
影响因子：
--
通讯作者：
M. Dai;Z. Xu;X. Zhou
中科院分区：
文献类型：
--
作者：
M. Dai;Z. Xu;X. Zhou
研究了一个具有股票、债券和交易费用成比例的连续时间马科维茨均值-方差投资组合选择问题。这是一个奇异随机控制问题，固有地具有有限的时间范围。通过一系列的变换，这个问题变成了所谓的双障碍问题，这是一个在物理学和PDE文献中研究得很好的问题，具有两个时变的自由边界。这两个边界定义了买、卖和不交易区域，在时间上被证明是平滑的。这反过来又通过Skorokhod问题描述了最优策略的特征，即试图在非贸易区域内保持一定的调整后的债券存量头寸。揭示了最优策略与无交易成本策略显著不同的几个特征。结果表明，在时间上存在一个临界长度，它与股票超额收益和交易费用有关，但与投资目标和股票波动率无关，因此，如果计划期限短于该临界长度，则可能无法实现预期的终端收益（而在没有交易费用的情况下，任何预期收益都可以在任意时间段内达到）。进一步证明，任何遵循最优策略的人都不应该在股票到期时间短于上述临界长度的点上购买股票。此外，当到期日临近时，投资者购买股票的可能性更小，而卖出股票的可能性更大。这些特点，虽然与广泛接受的投资智慧一致，但表明规划范围是投资机会的一个组成部分。
A continuous-time Markowitz's mean-variance portfolio selection problem is studied in a market with one stock, one bond, and proportional transaction costs. This is a singular stochastic control problem, inherently with a finite time horizon. Via a series of transformations, the problem is turned into a so-called double obstacle problem, a well-studied problem in physics and PDE literature, featuring two time-varying free boundaries. The two boundaries, which define the buy, sell, and no-trade regions, are proved to be smooth in time. This in turn characterizes the optimal strategy, via a Skorokhod problem, as one that tries to keep a certain adjusted bond-stock position within the no-trade region. Several features of the optimal strategy are revealed that are remarkably different from its no-transaction-cost counterpart. It is shown that there exists a critical length in time, which is dependent on the stock excess return as well as the transaction fees but independent of the investment target and the stock volatility, so that an expected terminal return may not be achievable if the planning horizon is shorter than that critical length (while in the absence of transaction costs any expected return can be reached in an arbitrary period of time). It is further demonstrated that anyone following the optimal strategy should not buy the stock beyond the point when the time to maturity is shorter than the aforementioned critical length. Moreover, the investor would be less likely to buy the stock and more likely to sell the stock when the maturity date is getting closer. These features, while consistent with the widely accepted investment wisdom, suggest that the planning horizon is an integral part of the investment opportunities.