最优停时理论在股票交易问题中应用

论文总字数：12838字

摘 要

股票市场交易包括三个关键步骤:选择一个股票，购买一定数量的股票，最后卖掉他们盈利。这个交易的时机至关重要。根据销售规则可以指定两个预选指标:目标价格和止损限额。本文首先研究一种基于有限状态马尔可夫链耦合的几何布朗运动模型的最优销售规则。这一策略可以通过求解一组两点边值微分方程得到。进而推导出相应的预期目标持有期限、盈利概率和亏损的可能性。在一、二维情况下得到了解析解。接着选取苹果公司股票（代码AAPL）历史数据代入此模型，简单进行实证分析，用数值算例证明该方法的有效性。结果发现本文研究应用于实际交易中成立。

关键词：最优停时理论、几何布朗运动、两点边值微分方程

The Application of Optimal Stopping Time Theory in Stock Trading

Abstract

Trading in stock markets consists of three major steps: select a stock, purchase a number of shares, and eventually sell them to make a profit. The timing to buy and sell is extremely crucial. A selling rule can be specified by two preselected levels: a target price and a stop-loss limit. We are concerned with an optimal selling rule based on the model characterized by a number of geometric Brownian motions coupled by a finite-state Markov chain. Such a policy can be obtained by solving a set of two-point boundary value differential equations. Moreover, the corresponding expected target period and probability of making money and that of losing money are derived. Analytic solutions are obtained in one- and two-dimensional cases. Finally, a numerical example(AAPL) is considered to demonstrate the effectiveness of our method.

KEY WORDS: Optimal selling rule, Geometric Brownian motion, Two-point boundary value problem

目 录

摘 要 Ⅰ

Abstract Ⅱ

第一章 引言 ··············································································· 1

第二章 问题表述 ········································································· 3

第三章 最优停时理论 ··································································· 5

第四章 预计退出时间和概率 ·························································· 7

第五章 一、二维情况下的解析解 ···················································· 9

第六章 一个数值算例 ·································································· 16

第七章 文章总结 ········································································ 19

致谢 ·························································································· 20

参考文献 ···················································································· 21

1. 引言

股票交易包括三个主要步骤: (a)根据某些标准选择股票；(b)根据此种股票的历史资料（如开盘价序列）在适当的时间购买一些股票；(c)持有该股票一段时间，选择合适时机出售，以赚取利润（交易规则一般为T 1）。在统计的角度看待这个问题，其中心是基于历史数据预测，还要考虑误差，最后运用最优停时理论及两点边值微分方程和时间序列分析方法来解决问题。考虑到股票价格，本文的销售规则由目标价格和止损限额所构成，当基础股票价格达到目标价格或止损限额二者任一时，可作出销售（清算）决定。

为了分析和研究某一支股票的性能, 我们可以建立一个数学模型来表征其价格走势。在金融数学上, 股票的价格变化经常被塑造作为几何布朗运动，主要由二个参量决定: 期望回归收益率r和波动率σ。在分析期权定价时, 通常假定这些参数是确定性的。因此, 这样的模型只在相对较短的时间内是好的, 因为它不会对这些参数的随机变化做出响应。

从股票市场的波动可以得出，控制个人股票流动的一个主要因素是一般市场的趋势。如果整体市场上涨, 多数股票上涨;如果一般市场下跌, 多数跟随。总的来说, 市场的动向可以看作是一个主要运动和次要运动的构成。下面表1概述这种分类。

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