基于二叉树模型的期权定价

目录

摘要 ....................................................... 1 ABSTRACT ................................................... 2 第一章 绪论 ............................................... 3

1.1背景介绍 ....................................................... 3

1.2 本文的主题 ................................................... 3

第二章 预备知识 ........................................... 4

2.1 期权 ......................................................... 4 2.2二叉树方法 .................................................... 4

2.2.1 方法概述 ........................................................ 4

2.2.2 二叉树方法的优点和缺点 .......................................... 6 2.2.3 风险中性定价 ................................................... 6

2.3 Black-Scholes 期权定价模型 .................................. 7

2.3.1模型来源 ......................................................... 7

2.3.2风险中性定价 ..................................................... 7 2.3.3模型假设 ......................................................... 8 2.3.4Black-Scholes期权定价公式 ........................................ 8

第三章 本论 ............................................... 9

3.1期权定价的二叉树模型 .......................................... 9

3.1.1参数确定 ......................................................... 9 3.1.2资产价格树形 .................................................... 11 3.1.3通过树形倒推 .................................................... 11 3.1.4代数表达式 ...................................................... 12

3.2 例子模拟计算和结果分析 ...................................... 12 3.3 模型改进——三叉树 ........................................... 15

第四章 结论 .............................................. 17 谢辞及参考文献 ............................................ 19

谢辞 ............................................................. 19

参考文献 ......................................................... 20

附录 ...................................................... 22

计算过程中涉及算法 ............................................... 22

摘要

Black-Scholes 期权定价模型为期权定价尤其是欧式期权定价提供了良好的解析结果，而Black-Scholes 公式是此模型的核心，但是此公式并不能很好地求解出在很多衍生模型例如亚式期权以及美式期权中的解析解。二叉树方法作为一种数值方法，同时也是图论中一种重要方法，应用于期权定价问题中，它有了更特别的演变。本文利用二叉树方法计算期权定价的数值解，用二叉树方法迭代多次，求出较为准确的期权价格。通过B-S公式得出的结果与二叉树方法得到的结论对比，分析二叉树方法模拟的优点和缺点。同时，我们还要研究二叉树模拟的步数与预测结果和精度间的关系，从而更加深入了解二叉树方法。然而，我们在模型中设立了许多条件，这些都使模型离真实情况越来越远，我们必须不断发展模型，完善模型。三叉树方法正是二叉树方法的合适补充。

关键词：二叉树方法，Black-Scholes 模型，风险中性定价

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ABSTRACT

Black-ScholesFormulaisthecoreofBlack-ScholesOptionPricingModelwhichprovidesapracticalmethodforoptionpricing.Ithasanalyticalsolutionswithgoodpropertiesinsomespecialsituations,forinstance,Europeanoptions.However,theanalyticalsolutionisdifficulttofindinmanyderivativemodelslikeAsianoptions American

option.Asasortoftypicalstatisticalsimulationmethod,Binomial treeplaysveryimportantrolesinGraph

Theory

and

othersignificantacademicfields.When it applies to the option price,binomial tree method has much more special use.Themainideaisthat we put the binomial tree into effect,reapply this method and get numericalresultsofoption price.By comparing the results of Black-Scholes formula with the results of binomial tree method,we come to the advantages and disadvantages of both method. Meanwhile,the study of the steps of binomial tree method is also included to get its relationship with the method’s results and accuracy,which leads us to understand this method deeply and rightly.However,we set many extra conditions,which pushes the situation further away from the real situation.The simple binomial tree method is supposed to be improved constantly in case the finance market changes ceaselessly. Ternary tree is a good supplement for the binomial tree.

Keywords:Binomial tree method,Black-Scholesoptionpricingmodel,Risk-neutral valuation

and

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