第12章 二叉树模型

8
Valuing the Option



The portfolio that is long 0.25 shares short 1 option is worth 4.367 The value of the shares is 5.000 (= 0.25 × 20 ) The value of the option is therefore 0.633 ( 5.000 – 0.633 = 4.367 )
5
A Call Option (Figure 12.1, page 254)
A 3-month call option on the stock has a strike price of 21.
Stock Price = $22 Option Price = $1 Stock price = $20 Option Price=?
B
1.4147 40
C
E
9.4636
F
A Put Option Example
Figure 12.7, page 263
72 0 60
50 4.1923
1.4147 40
48 4
9.4636
32 20
K = 52, time step =1yr r = 5%, u =1.2, d = 0.8, p = 0.6282
f e
2 r Dt
[ p f uu 2 p (1 p ) f ud (1 p ) f dd ]
2 2
12.4 看跌期权的例子 A Put Option Example; K=52
K = 52, time step = 1yr r = 5%
D
60
50 4.1923
A
72 0 48 4 32 20
12.3 两步二叉树 A Two-Step Example
24.2 22 20 18

19.8

K=21, r = 12% Each time step is 3 months
16.2
22
Valuing a Call Option
Figure 12.4, page 260
24.2 3.2
22
20 1.2823
S0 ƒ
S0u ƒu S0d ƒd
14
Excel应用
运用复制公式的技巧。
12.2 风险中性定价 Risk-Neutral Valuation




When the probability of an up and down movements are p and 1-p the expected stock price at time T is S0erT This shows that the stock price earns the riskfree rate Binomial trees illustrate the general result that to value a derivative we can assume that the expected return on the underlying asset is the risk-free rate and discount at the risk-free rate This is known as using risk-neutral valuation
26
12.5 美式期权 What Happens When the Put Option is American
72 0
60 50 5.0894 1.4147 40 C 12.0 32 20 48 4
The American feature increases the value at node C from 9.4636 to 12.0000. This increases the value of the option from 4.1923 to 5.0894.
16
Original Example Revisited
S0=20 ƒ S0u = 22 ƒu = 1 S0d = 18 ƒd = 0
p is the probability that gives a return on the stock equal to the risk-free rate: 20e 0.12 ×0.25 = 22p + 18(1 – p ) so that p = 0.6523 Alternatively:
23
一般结论
现在，时间单步的长度为 Dt 而不是T
e rDt d p ud r Dt fu e [ pf uu (1 p ) f ud ] r Dt f d e [ pf uBaidu Nhomakorabea (1 p ) f dd ] r Dt f e [ pf u (1 p ) f d ]
Stock Price = $18 Option Price = $0
6
Setting Up a Riskless Portfolio

For a portfolio that is long D shares and a short 1 call option values are
22D – 1

18D

Portfolio is riskless when 22D – 1 = 18D or D = 0.25
7
Valuing the Portfolio(Risk-Free Rate is 12%)



The riskless portfolio is: long 0.25 shares short 1 call option The value of the portfolio in 3 months is 22 ×0.25 – 1 = 4.50 The value of the portfolio today is 4.5e–0.12×0.25 = 4.3670
9
Generalization (Figure 12.2, page 255)
A derivative lasts for time T and is dependent on a stock
S0 ƒ
S0u ƒu S0d ƒd
10
Generalization (continued)

Value of a portfolio that is long D shares and short 1 derivative:
19
现实世界与风险中性世界

必须强调的一点是，p是在一个风险中性的世界中 股价上升的概率（事实上由股价变动所决定）， 而在现实世界中概率事实并不一定是这样的。
我们的 例子 中 p 0.6523 ，当价 格上升 的概 率 为 0.6523 时，股票和期权的预期收益率都等于无风险 利率 12%。假设在现实世界中股票的预期收益率为 16%， p* 是股价上升的概率。那么，有 22 p* 18(1 p* ) 20e0.163/12 则 p* 0.7041。 因此，现实世界中期权的预期收益为： p* 1 (1 p* ) 0 0.7041
2.0257 A
18 0.0
B
19.8 0.0 16.2 0.0
Value at node B = e–0.12×0.25(0.6523×3.2 + 0.3477×0) = 2.0257 Value at node A = e–0.12×0.25(0.6523×2.0257 + 0.3477×0) = 1.2823
12
Generalization
(continued)
Substituting for D we obtain ƒ = [ pƒu + (1 – p)ƒd ]e–rT where
（12.2）
e rT d p ud
（12.3）
13
p as a Probability


It is natural to interpret p and 1-p as probabilities of up and down movements The value of a derivative is then its expected payoff in a risk-neutral world discounted at the risk-free rate
S0uD – ƒu S0dD – ƒd


The portfolio is riskless when S0uD – ƒu = S0dD – ƒd or ƒu f d D （12.1） S 0u S 0 d
11
Generalization

(continued)



Value of the portfolio at time T is S0uD – ƒu Value of the portfolio today is (S0uD – ƒu)e–rT Another expression for the portfolio value today is S0D – f Hence ƒ = S0D – (S0uD – ƒu )e–rT
对股价如何建模？
离散模型 由两点分布 生成算术随机游走 （增量是相加） 由两点分布 生成几何随机游走 （增量是相乘） 描述股价的二叉树模型 衍生产品的二叉树模型 连续模型 由标准正态分布 生成标准布朗运动 生成一般布朗运动 由一般布朗运动 生成几何布朗运动 描述股价的几何布朗运动 伊藤过程
在Excel中对各种过程的模拟。
陕西师范大学 国际商学院
曹培慎
Chapter 12 Binomial [baɪ'nomɪəl] Trees 二叉树模型
Options, Futures, and Other Derivatives, 8th Edition, Copyright © John C. Hull 2012


为期权进行定价的一个有用且常见的方法是构造 二叉树图。这个树图表示了股票价格在期权的有 效期内可能遵循的路径。在这里，我们基本的假 设是股票价格遵循几何随机游走。每一时间步中， 价格上升一定的百分比有一定的概率，价格下降 一定的百分比也有一定的概率。极限情况下，也 就是时间步变得很小的时候，这个模型会导致股 票价格遵循对数正态分布。 在本章，首先简要看一下二叉树图，并解释它们 与风险中性估值原理之间的关系。
12.1 单步二叉树模型与无套利方法 A Simple Binomial Model

A stock price is currently $20 In 3 months it will be either $22 or $18
Stock Price = $22 Stock price = $20 Stock Price = $18
18
Irrelevance of Stock’s Expected Return


When we are valuing an option in terms of the price of the underlying asset, the probability of up and down movements in the real world are irrelevant This is an example of a more general result stating that the expected return on the underlying asset in the real world is irrelevant
e rT d e 0.120.25 0.9 p 0.6523 ud 1.1 0.9
17
Valuing the Option Using Risk-Neutral Valuation
S0u = 22 ƒu = 1
S0=20 ƒ S0d = 18 ƒd = 0
The value of the option is e–0.12×0.25 (0.6523×1 + 0.3477×0)= 0.633



不幸的是，现实世界中很难确定能适用于该预期 收益的准确贴现率。 持有看涨期权头寸比持有相应股票头寸的风险更 大，所以用来贴现看涨期权收益的贴现率应该高 于16%。 不知道期权价值的情况下，我们也不知道这个贴 现率应该比16%高多少。
对于已知价格的期权，还知道其在现实世界中的预 期收益，所以可以计算相应的贴现率。 0.633 0.7041er3/12 r 42.58%