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二项期权定价模型(binomal option price model,SCRR Model,BOPM) 二项期权定价模型假设股价波动只有向上和向下两个方向,且假设在整个考察期内,股价每次向上(或向下)波动的概率和幅度不变。模型将考察的存续期分为若干阶段,根据股价的历史波动率模拟出正股在整个存续期内所有可能的发展路径,并对每一路径上的每一节点计算权证行权收益和用贴现法计算出的权证价格。对于美式权证,由于可以提前行权,每一节点上权证的理论价格应为权证行权收益和贴现计算出的权证价格两者较大者。 二项式期权定价模型概述 1973年,布莱克和休尔斯(Blackand Scholes)提出了布莱克-休尔斯期权定价公式,对标的资产的价格服从正态分布的期权进行定价。随后,罗斯开始研究标的资产的价格服从非正态分布的期权定价理论。1976年,罗斯和约翰·考科斯(John Cox)在《金融经济学杂志》上发表论文“基于另类随机过程的期权定价”,提出了风险中性定价理论。 1979年,罗斯、考科斯和马克·鲁宾斯坦(Mark Rubinstein)在《金融经济学杂志》上发表论文“期权定价:一种简单的方法”,该文提出了一种简单的对离散时间的期权的定价方法,被称为Cox-Ross-Rubinstein二项式期权定价模型。 二项式期权定价模型和布莱克-休尔斯期权定价模型,是两种相互补充的方法。二项式期权定价模型推导比较简单,更适合说明期权定价的基本概念。二项式期权定价模型建立在一个基本假设基础上,即在给定的时间间隔内,证券的价格运动有两个可能的方向:上涨或者下跌。虽然这一假设非常简单,但由于可以把一个给定的时间段细分为更小的时间单位,因而二项式期权定价模型适用于处理更为复杂的期权。 随着要考虑的价格变动数目的增加,二项式期权定价模型的分布函数就越来越趋向于正态分布,二项式期权定价模型和布莱克-休尔斯期权定价模型相一致。二项式期权定价模型的优点,是简化了期权定价的计算并增加了直观性,因此现在已成为全世界各大证券交易所的主要定价标准之一。 一般来说,二项期权定价模型的基本假设是在每一时期股价的变动方向只有两个,即上升或下降。BOPM的定价依据是在期权在第一次买进时,能建立起一个零风险套头交易,或者说可以使用一个证券组合来模拟期权的价值,该证券组合在没有套利机会时应等于买权的价 格;反之,如果存在套利机会,投资者则可以买两种产品种价格便宜者,卖出价格较高者,从而获得无风险收益,当然这种套利机会只会在极短的时间里存在。这一 证券组合的主要功能是给出了买权的定价方法。与期货不同的是,期货的套头交易一旦建立就不用改变,而期权的套头交易则需不断调整,直至期权到期。

模型推导过程及案例 二项式过程如下
[pic]
S=20 股票期初价格 q=0.5 股票上涨的概率
[pic] 无风险收益率 u=1.2 股票价格上涨幅度 [pic] d=0.67 股票价格下跌幅度 [pic]

欧式看涨期权期初和期末的价格
[pic]
为计算该欧式看涨期权期初的价格,构造无风险套期保值组合: 以价格S买1份股票,同时卖出m份以该股票为标的物的看涨期权(m为套期保值率)。 如果这个套期保值组合在每种状态下的支付相等,则这个组合为无风险的。 [pic] 让支付相等,得到: [pic] (1) 从上式可以解得看涨期权的份数: [pic] (2) 把例子中的数字代入,得到: [pic] 也就是,为了得到无风险证券组合,需要卖出3.53份看涨期权。

|不确定状态 |证券组合 |期末支付 |
|好状态 |[pic] |1.2*20-3.53*3=13.40 |
|坏状态 |[pic] |0.67*20-3.53*0=13.40 |

因为套期保值证券组合是无风险的,因此,它的期末支付应该等于期初价格乘以[pic],即 [pic] (3)
由上式解得期权的期初价格:
[pic] (4)

把套期保值比率m代入,可得到: [pic] (5)
令:
[pic], 则 [pic]
所以有:
[pic] (6 ) 这里定义的p总是大于0小于1,具有概率的性质,称之为套期保值概率,可以理解为:p是当市场达到均衡时,风险中性者所认为的q值,即股票价格上涨的概率。 作为风险中性者,投资者仅需要投在风险股票上的回报率为无风险收益率。所以有: [pic] (7 ) 得到 [pic] (8 )

继续前面的例子,利用得到的期权公式(6),带入数据得: [pic] [pic] [pic]

在期初的证券组合是买一份股票,卖3.53份看涨期权,其成本为: [pic] 投资回报率为: [pic] 在上述求得看涨期权价格的过程中,有两点至关重要:一是无风险套期保值组合的构建;二是无风险套期保值组合的收益率等于无风险收益率。 看涨期权的定价公式具有以下三个有趣的特征: 1、该公式不依赖于股票价格上涨的概率q。这使得,即使投资者对q的预期不一致,只要他们对别的参数的估计一致(包括u、d、S、K和[pic]),他们就会有一样的定价公式。原因在于,我们不是在绝对意义上给期权定价,而是以标的股票价格计算期权的价格。而上涨和下跌的概率已经包含在股票的定价中,这就是说,我们依据股价给期权定价时,不必再一次考虑这些概率。 2、该公式的获得不依赖于个体投资者的风险偏好。所需要的假设仅仅是无套利。 3、该公式依赖的唯一随机变量是标的股票。

两期二项式期权定价

[pic]

[pic]
利用单期期权定价公式(6)式得到一期末的价值:
[pic] (9) [pic] (10)
再次利用(6)式得到期初的期权价格:
[pic] (11)

看涨期权定价的完全二项式模型: 期末的一般支付形式为 [pic] (12) T为总的时间区间数,n是股票价格上涨的次数。每个支付的概率的一般形式为二项式分布: [pic] (13)
看涨期权定价的完全二项式公式为:
[pic] (14) 为了更好的观察(14)式,将(14)求和中为零的项去掉。以a表示支付为正的最小正整数,即 [pic] (15) (14)可以变形为: [pic] (16) 分成两部分: [pic] [pic] (17) 因为
[pic][pic]
其中,[pic] 经过代换,(17)可以变形为:
[pic] (18)

-----------------------
S=20

q

1-q

uS=24

dS=13.4

[pic]

[pic]

1-q

q

C

[pic]

[pic]

1-q

q

S-mc

套期保值证券组合的到期支付

dS=13.4

uS=24

1-q

q

S=20

[pic]

q(1-q)

q(1-q)

[pic]

[pic]

[pic]

[pic]

股票价格的变化

欧式看涨期权的支付

[pic]

[pic]

[pic]

q(1-q)

[pic]

q(1-q)

[pic]

[pic]

[pic]

1-q

q

c

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