MINIMUM TRACKING ERROR PROBLEM FOR JUMP DIFFUSION STOCK-PRICE PROCESS
MINIMUM TRACKING ERROR PROBLEM FOR JUMP DIFFUSION STOCK-PRICE PROCESS
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DOI：
10.32219/isms.70.2_143
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发表时间：
2009-09
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影响因子：
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通讯作者：
Y. Tabata
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文献类型：
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作者：
Y. Tabata
本文研究了随机过程中允许跳的股票价格模型的最优化问题。我们表明，该问题的最优随机控制问题下的均方跟踪误差的标准下，在规划的地平线结束。通过求解常微分方程和伴随方程，得到了由风险市场价值和基准投资组合的折现终值乘积所确定的指数基金的最优控制和最优投资比例. 1导言。在不确定性条件下，资本市场的投资绩效往往通过与基准投资组合的比较来评价。一些最广为人知的基准投资组合是标准普尔500（S & P 500）、金融时报股票交易所100股票指数、东京股票价格指数（TOPIX）和日经225指数。然而，如果我们真的想持有这样一个基准投资组合，我们必须下定决心以巨大的交易、管理和信息收集成本来持有它，因为它几乎包括市场上所有或大量的证券（10）。因此，人们强调，具有少量密切跟踪基准投资组合的证券的投资组合对于风险对冲和投资组合管理是非常重要的。这样的投资组合被称为指数基金，在一个恰当的例子。本文将跟踪误差定义为基准投资组合收益率与指数基金收益率之间的均方误差，在离散时间框架下，绿色（3），Meada和Salkin（5）在均值-方差框架的意义上发展了指数基金和前沿投资组合之间的关系。Tabata和Takeda（9）将这一问题转化为一个0-1变量的二次规划问题，并提出了在离散时间静态模型中寻找最小化跟踪误差均方的指数基金的有效方法。另一方面，在现代金融数学理论中，连续时间模型也是一个重要的研究领域。正如R. Cont和P. Tankov（1），Levy过程和其他带跳跃的随机过程在建模市场波动方面越来越受欢迎，无论是风险管理还是期权定价。随机控制在金融问题中的应用，特别是在衍生品定价中的应用，见于Yong和Zhou（11），Kohlmann和Zhou（4），Framstad（2），Oksendal和Sulem（7）等，Mistui和Tabata（4）用Teugel鞅分析了Levy过程，证明了最优套期保值策略的存在性。本文利用随机控制技术，导出了最优指数基金的构造方法.第二节给出了连续时间市场的基本模型。第3节交易
This paper is concerned with a problem on optimization in stock price model that allows for jumps in the stochastic processes. We show that the problem is formulated into the optimal stochastic control problem under the criterion of mean square tracking error at the end of the planning horizon. The optimal control and the optimal proportion invested in the index fund which are specified by the product of the market value of risk and the discounted terminal value of bench mark portfolio are derived by the solution to ordinary differential equations associated with an adjoint equation. 1 Introduction. The performance of investment in capital market under uncertainty is often evaluated in comparison with the benchmark portfolio. Some of the most widely known benchmark portfolios are the Standard & Poor's 500 (S & P 500), Financial Times Stock Exchange 100 Share Index, the Tokyo Stock Price Index (TOPIX) and The NIKKEI 225. However, if we actually wish to hold such a benchmark portfolio, we have to make up our minds to hold it at huge transaction, management and information collection costs, since it includes almost all or large number of securities in the market(10). Thus, it has been emphasized that the portfolio with a small number of securities which closely track the benchmark portfolio is of great concern to risk hedge and to portfolio management. Such a portfolio is called an index fund and in a case in point. So the problem can be regaraded as an optimization problem to minimize the tracking error between the benchmark portfolio and the index fund. In this paper, the tracking error is defined as the mean square error between the returns of benchmark and index fund. In a discrete time framwork, Green(3), Meada and Salkin(5) developed the relation- ship between the index fund and the frontier portfolio in the sense of the mean-variance framework. Tabata and Takeda(9) formulated this problem into a quardratic programming problem with 0-1 variables and proposed the efficient method to find an index fund which minimizes the mean square of tracking error in descrete time static model. On the other hand, much effort has been devoted to the continuous time model in the modern mathematical finance theory. As was pointed out by R. Cont and P. Tankov(1), Levy processes and other stochastic processes with jumps have become increasingly popular for modelling market fluctuations, both for risk management and option pricing purposes. Applications of the stochastic control to a financial problem, especially to the pricing of derivatives are found in Yong and Zhou(11), Kohlmann and Zhou(4), Framstad (2), Oksendal and Sulem (7) and so on. Mistui and Tabata (4) analized Levy process by means of the Teugel's martingale and showed the existence of a unique optimal hedging strategy. In this paper, we derive the optimal index fund based upon the stochastic control tech- nique. In section 2 we give our basic market model with continuous time. Section 3 deals