不完备市场中未定权益的最优对冲策略

发布时间：2018-04-02 13:22

本文选题：Bessel过程　切入点：最优交易策略　出处：《华南理工大学》2014年硕士论文

【摘要】：经典的无套利理论中因为不存在套利机会，未定权益都可以实现完全对冲，而在不完备市场中存在套利机会，未定权益不能实现完全对冲，一个给定的未定权益可以有不同的方法或途径实现某种意义上的最优对冲，本文以初始资金最小并达到给定最终收益的策略作为最优对冲策略。本文主要研究资产价格模型为借助Bessel过程定义的连续时间马尔可夫过程情形下的最优对冲策略。首先，定义了不完备市场中资产价格的一般模型及交易策略，引入推广夏普比率为多维形式的市场风险价格(MPR)，且在市场风险价格的基础上定义了贴现因子(SDF)，用马尔可夫MPR定义的贴现因子构造对冲价格函数h p。其次，推导出对冲价格函数的伊藤过程表示，并化简为财富过程形式，，该策略即为最优对冲策略，同时根据相应的假设和相关理论获得了对冲价格函数的光滑性。最后，从已存在的测度变换推导出的广义的测度变换，简化了资产价格模型和随机贴现因子(SDF)的倒数的动态特性，从而使对冲价格函数和最优对冲策略的计算更加简便。本文最后分别对以不带漂移项和带漂移项的n维Bessel过程为辅助过程定义的资产价格模型得到了具体的最优对冲策略。通过前面所证的结果，以马尔可夫贴现因子构造对冲价格函数，化简为财富过程形式得到最优策略，并通过和已存在的例子进行对比，证实该策略即为最优对冲策略。
[Abstract]:In the classical no-arbitrage theory, because there is no arbitrage opportunity, the contingent equity can be completely hedged, but in the incomplete market there is arbitrage opportunity, and the contingent equity can not be fully hedged. A given contingent claim can have different ways or means of achieving an optimal hedge in a sense, In this paper, the minimum initial capital and the given final return are taken as the optimal hedging strategies. In this paper, the asset price model is the optimal hedging strategy under the continuous time Markov process defined by the Bessel process. Firstly, the general model and trading strategy of asset price in incomplete market are defined. This paper introduces the market risk price which generalizes Sharpe ratio to multidimensional form, and defines the discount factor based on the market risk price. The discount factor defined by Markov MPR is used to construct the hedge price function h p. The Ito process representation of the hedge price function is derived and simplified to the wealth process form. The strategy is the optimal hedging strategy, and the smoothness of the hedge price function is obtained according to the corresponding assumptions and relevant theories. The generalized measure transformation derived from the existing measure transformation simplifies the dynamic properties of the reciprocal of asset price model and stochastic discount factor (SDF), thus making the calculation of hedge price function and optimal hedging strategy easier. In the end of this paper, the optimal hedging strategies are obtained for the asset price models defined by the n-dimensional Bessel processes with and without drift terms, respectively. The hedging price function is constructed by Markov discount factor, and the optimal strategy is obtained in the form of wealth process. By comparing with the existing examples, it is proved that this strategy is the optimal hedging strategy.
【学位授予单位】：华南理工大学
【学位级别】：硕士
【学位授予年份】：2014
【分类号】：F224;F830.91

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