Ch09_IntroductiontoBinomialTrees(金融工程学,华东
2、期权价值计算之二叉树
SdD – ƒd The portfolio is riskless when SuD – ƒu = Sd D – ƒd or
D ƒu fd Su Sd
Fundamentals of Futures and Options Markets, 6th Edition, Copyright © John C. Hull 2007
22 0.25 – 1 = 4.50
The value of the portfolio today is
4.5e – 0.12 0.25 = 4.3670
Fundamentals of Futures and Options Markets, 6th Edition, Copyright © John C. Hull 2007
Fundamentals of Futures and Options Markets, 6th Edition, Copyright © John C. Hull 2007
11.15
Problem 13.2.
Explain the no-arbitrage and risk-neutral valuation approaches to valuing a European option using a one-step binomial tree.
Substituting for D we obtain
ƒ = [ p ƒu + (1 – p )ƒd ]e–rT
where
p erT d ud
Fundamentals of Futures and Options Markets, 6th Edition, Copyright © John C. Hull 2007
Binomial Trees
Binomial TreesA useful and very popular technique for pricing a stock option involves constructing a binomial tree .1. A one-step binomial modelConsider a European call with $21,1/4K T == and 0$20S =. To simplify analysis, we assume that at the end of three months, we have $22$18T S or =. Hence, the option value is either $1 or $0. The option specified in such a way does not involveuncertainty about the value of the portfolio, hence it’s return must equal the risk-free interest rate.Consider a portfolio consisting of a long position in ∆ shares of the stock and a short position in one call option. Ifa.Stock price goes up from $20 to $22, the share are worth 22∆ and value of the option is $1, so the portfolio value is 22∆ - 1;b.Stock price goes down from $20 to $18, the portfolio is worth 18∆, therefore∆−=∆22118hence 0.25∆=. A riskless portfolio therefore consists of long 0.25 shares and short 1 option. In both cases, the value of the portfolio is $4.5. If the risk-free interest rate is 12%/pa, the present value of the portfolio is0.121/4e−×=4.5 4.367Let f denotes the option price, so the cost of the portfolio today is×−=−200.255f ftherefore f = 0.633. This shows that in the absence of arbitrage, the current value of the option must be $0.633.A GeneralisationNow we assume that at the end of T, S T can move up to S0u, u>1, which give payoff f u, or move down to S0d, d<1, which gives payoff f dS0 fS0u f u S0d f dAt the end of the life of the option, the value of the portfolio is either S 0u ∆ - f u or S 0d ∆ - f d , but they must the sameS 0u ∆ - f u = S 0d ∆ - f d or00u df f S u S d −∆=− (1)In this case the portfolio is riskless and must be equal the risk-free interest rate. The present value of the portfolio is(S 0u ∆ - f u )e -rTand the cost of setting up the portfolio is S 0∆ - f , it then follows:S 0∆ − f = (S 0u ∆ − f u )e -rTorf = S 0∆ − (S 0u ∆ − f u )e -rTSubstituting for ∆ from equation (1), we havef = e -rT [p f u + (1 − p ) f d ) (2) whererT e dp u d −=−For instance, in the previous example, u = 1.1, d = 0.9, r = 0.12, T = 0.25, f u = 1, and f d = 0, we can calculate:0.121/40.90.65231.10.9e p ×−==−therefore:()0.120.250.652310.347700.633f e −×=×+×=which is the same as before.It is important to note that in this discussion we do not involve the probabilities of the stock price moving up or down.2Risk-neutral ValuationThough equation (2) is derived without assumption about probabilities of up and down movements, it is natural to interpret p and (1 −p) as the probabilities of an up and down movements in stock price. Thereforep f u+ (1 −p) f dis the expected payoff from the option. With this interpretation of p, equation (2) then states that the value of the option today is its expected future value discounted at the risk-free rate (see the derivation of Black-Scholes solution).On the other hand, with p being the probability of moving up, the expected stock price at time T, E(S T), is given byE(S T) = p S0u + (1 −p) S0dSubstituting from equation (2) for p, this reduces toE(S T) = S0e rT (3) showing that the stock price grows, on average, at the risk-free rate. Setting the probability of the up movement equal to p is equivalent to assuming that the return on the stock equals the risk-free rate. This result is based on risk-neutral principle, which states that expected return on all securities is the risk-free interest rate.Returning to the previous example, let p be the probability of moving up, then p must satisfy22p + 18(1 −p) = 20e0.12×0.25or 4p = 20e0.12×0.25− 18, and hence, p = 0.6523.At the end of the three months, the call option has a 0.6523 probability of being worth 1 and 0.3477 probability of being worth 0, so its expected value is0.6523×1 + 0.3477×0 = 0.6523Discounting at the risk-free rate, the present value of the option is$0.6523e−0.12×0.25 = $0.633.Hence, no-arbitrage and risk-neutral give the same answer.3 Two-step Binomial TreesThe upper number is S; the lower number is f . The start priceis $20 and in each of two time steps may go up or down byDEF24.2 3.219.8 0.0 16.2 0.010%. Each time step is three months long, and the risk-free interest rate is 12%/pa, and K = $21. Clearly, at node D the option price is $24.2 − $21 = $3.2, but at nodes E and F the option is out of the money and its value is zero.At node C, the option price is 0, using the notation introduced before, u = 1.1, d = 0.9, r = 0.12, and T = 0.25, so that p = 0.6523, and from equation (2) at B:e−0.12×0.25(0.6523×3.2 + 0.3477×0) = 2.0257We now calculate the option price at the initial node A. Similarly, we havee−0.12×0.25(0.6523×2.0257 + 0.3477×0) = 1.2823A GeneralisationWe now assume that S0 is the initial price and the time step is δt years. The following notations are self-explained:f u = e-rδt[p f uu+ (1 −p) f ud) (4)f d = e-rδt[p f ud+ (1 −p) f dd] (5)f = e-rδt[p f u+ (1 −p) f d] (6) Substituting (4) and (5) into (6), we havef = e-2rδt[p2f uu+ 2p(1 −p) f ud+ (1 −p) f ud] (7) We explain this in the following diagram:Stock and option prices in a general two-step treeS 0u 2f uuS 0udf udS 0d 2f ddS fFollowing the derivation of equation (3), we can obtain similar result based on the principle of risk-neutral valuation. The values p2, 2p(1 − p), and (1 −p)2 are the probabilities that the upper, middle, and lower final nodes will be reached. The option price is equal to its expected payoff in a risk-neutral world discounted at the risk-free interest rate given in equation (7).4 A Put Option ExampleThe procedures described above are applicable to any deriva-tive dependent on a stock whose price changes are binomial. Consider a two-year European put option on a stock with spotprice S = $50 and K = $52 and r = 5%. We suppose there are two time steps and in each case the stock price either moves up or down by a proportional amount of 20%:72 0 48 432 20So, u = 1.2, d = 0.8, the risk-neutral probability p is given by0.0510.81.20.8rTe d ep u d×−−==−−The possible final stock prices are: $72, $48 and $32, and in this case f uu = 0; f ud = 4, and f dd = 20, from equation (7):f = e−2×0.05×1(0.62822×0 + 2×0.6282×0.3718×4 + 0.37182×20) = 4.1923This result can be equally obtained using equation (6) and working back through the tree one step at a time.5 American OptionsWe now consider the above put option again with the same condi- tions. But it’s now an American. We evaluate it using a binomial72 048 432 20tree. The procedure is to work back through the tree from the end to the beginning, testing at node if early exercise being optimal. The stock prices and their probabilities are unchanged. The values for the option at the final nodes are also unchanged. At B, early exercise leads to negative payoff, so from equation (2) the value of the option is 1.4147. At C, equation (2) gives the value of the option as 9.4636. But early exercise at K = $52 leads to a payoff of $12, which is the optimal value of the option. Finally, at A, equation (2) givese−0.05×1(0.6282×1.4147 + 0.3718×12) = 5.0894But if exercise at A the payoff is $2, which is clearly not optimal. So the value of the option is $5.0894.6 DeltaIn deriving the Black-Scholes equation, we knowV S ∂∆=∂It is an important parameter in the pricing and hedging of options. In general we define the delta of a stock option is the ratio of the change in the price of the stock option to the change in the price of the underlying stock. In the following case:The delta of the above call option is:100.252218−=−Stock price = $22 Option price = $1 Stock price = $18 Option price = $0In the example of European put option, the delta at the end of the first time step is1.41479.46360.40246040−=−− and is either040.16677248−=−− or4201.00004832−=−−at the end of the second time step. Clearly, delta changes overthe two time steps. Thus, in order to maintain a riskless hedge using an option and the underlying stock, we need to adjust our holdings in the stock periodically.7Matching Volatility with u and dIn above discussion, values of u and d are given. But in practice, they are chosen to match the volatility of the stock price. We suppose that the expected return (in the real world) on a stock is µand it volatility is σ. The step length is δt, and the stock price goes up by a proportional amount u or goes down by d. The probability of an up movement (in the real world) is assumed to be q.Stock price change in (a) the real world, and (b) the risk-neutral world The expected stock price at the end of the first time step is S 0e µδt. But on the binomial tree the expected stock price at thisS 0u S 0dtime is:qS 0u + (1 − q ) S 0dEquating the two stock prices:qS 0u + (1 − q ) S 0d = S 0e µδt ort e dq u d µδ−=− (8)As stock return follows an Itô process, its variance over the period of δt is σ2δt. But on the tree in (a), the variance of the return isqu 2 + (1 − q )d 2 − [qu + (1 − q )d ]2Equating the two variancesqu 2 + (1 − q )d 2 − [qu + (1 − q )d ]2 = σ2δt (9) Substituting (8) into (9), we havee µδt (u + d ) − ud − e 2µδt = σ2δt Ignoring high order terms of δt , we can solve the above equation by,u ed e σσ−== (10)In our previous analysis we show that we can replacethe tree in (a) by that in (b) that represents the of risk-neutral. The probability p can be expressed asr t e dp u d δ−=−The expected stock price is S 0e r δt . The variance of the stockprice return ispu 2 + (1 − p )d 2 − [pu + (1 − p )d ]2 = e µδt (u + d ) − ud − e 2µδt Substituting for u and d from (10), we find this equals σ2δt when higher order terms of δt are ignored.This analysis shows that when we move from the real world to the risk-neural world the expected return on the stock changes, but its volatility remains the same (at least in the limit as δt tends to zero). This is an illustration of an important general result known as Girsanov’s theorem.The binomial models presented so far have been unrealistically simple. In real practice, the life of the option is typically divided into 30 or time steps of length δt. In each time step there is a binomial stock price movement. This means that 31 terminal stock prices and 230, or about 1 billion, possible stock prices paths are considered.。
论清华大学经管学院的金融工程(英文版)(ppt 34页)
Discount rate for cash flow of the firm
WACC
re
E DE
rf
1T D
DE
EBITrf D1T
E
E DE
rf
1T
D DE
EBIT1T
DE
Has a change of financial leverage
re any impact on the firm value ?
pizza.
— Merton Miler
14
Probability Distribution of EBIT and EPS
for the Two Companies
State of the EBIT Economy
Company A
EPS
(1 million shares)
Company B
Net
• All liabilities are risk-free
11
A mini - case
Two companies EBIT Capital structure Firm value
A
1 million shares $100 million
$10 million p.a
B
bonds: $40 million, 8% ?
M&M Proposition 1
• Proposition 1:
• Think of the firm as a
Under M&M Assumption, i.e., in the frictionless environment, the total market value of a firm is independent of its capital structure.
北京大学-金融工程-第08章
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构造树图的其他方法和思路
• p=0.5的二叉树图方法（即放弃u=1/d的假 设）
• 控制方差技术
• 适应性网状模型
• 隐含树图
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第八章
期权定价的数法
P=0.5的二叉树图
ue
d e
2 r q
2
Dt 2 Dt
2 r q
2
期权定价的数法
证券价格的树型结构
Su4 Su3 Su2 Su S Sd Sd2 Sd3 Sd4 S S Sd Sd2 Su Su2
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第八章
期权定价的数法
倒推定价法
• 得到每个结点的资产价格之后，就可以在二叉树模 型中采用倒推定价法，从树型结构图的末端T时刻 开始往回倒推，为期权定价 • 值得注意的是，如果是美式期权，就要在树型结构 的每一个结点上，比较在本时刻提前执行期权和继 续再持有到下一个时刻再执行期权，选择其中较大 者作为本结点的期权价值。
第八章
期权定价的数法
79.35 0 62.99 0 50 2.66 39.69 10.31 31.51 18.50 E 89.07 0 70.70 0 56.12 0 44.55 5.45 35.36 G 14.64 28.07 21.93
美式看跌期权二叉树
70.70 0 62.99 56.12 50 A 4.48 2.15 44.55 6.95 0.63 50 3.76 39.69 B 10.35 D 56.12 1.30 C 44.55 6.37 35.36 14.64
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two period binomial tree证明
two period binomial tree证明
两期二项树是在金融衍生品定价中经常用到的工具，特别是在期权定价中。
在证明二项树的情况下，我们通常使用离散时间的风险中性定价原理，以下是两期二项树的基本证明步骤：
假设有一个在两期内的期权，期权在第一期可能有两个状态：上涨（U）和下跌（D）。
在第二期，这两个状态仍然存在。
我们考虑一个风险中性的市场，即在不考虑风险的情况下，市场中所有的资产的期望回报率相同。
1.设定参数：
-设定期初资产价格为S。
-设定上涨因子为U，下跌因子为D，其中U>1，D<1。
-设定无风险利率为r，时间间隔为Δt。
2.构建二项树：
-在第一期结束时，资产价格可能是S*U或S*D。
-在第二期结束时，对于上涨状态，资产价格可能是S*U*U或S*U*D；对于下跌状态，资产价格可能是S*D*U或S*D*D。
3.计算期权价格：
-在第二期结束时，计算期权的支付。
这可能涉及到看涨期权和看跌期权的不同计算方式。
-对于欧式期权，一般取期权在第二期结束时的支付的期望折现值。
4.风险中性定价：
-根据风险中性定价原理，计算一个无套利的期初期权价格，使得在两种状态下的期权价格期望与实际期权价格相等。
以上是两期二项树的基本证明步骤。
这种方法可以推广到更多期数的二项树，并且在实际金融工程中得到广泛应用。
在具体的期权定价问题中，可能需要根据不同的期权类型和特定情境进行调整。
金融工程概论课件 - 期权(二叉树)
z 期权合约及市场运作 z 期权交易策略 z 股票期权的性质 z 期权定价
¾ 二叉树 ¾ BS模型
z 股指期权、货币期权与期货期权 z 期权价格的敏感性及套期保值
期权定价——二叉树 (binomial tree)
z 二叉树用来表示在期权期限内可能会出现的股票 价格变动的路径。
Su
fu
S
f
Sd
40
9.4636
72
0
48
4
32
20
p = e0.05*1 − 0.8 = 0.6282 1.2 − 0.8
fu = e−0.05 (0.6282 * 0 + 0.3718 * 4) = 1.4147
fd = e−0.05 (0.6282 * 4 + 0.3718 * 20) = 9.4636
f = e−0.05 (0.6282*1.4147 + 0.3718*9.4636)
d = 1 = 0.7408 1.3499
p = e0.05 − 0.7408 = 0.5097 1.3499 − 0.7408
e0.05 = 1.0513
e0.3 = 1.3499
期权定价——二叉树 (binomial tree)
z 二叉树定价的一般方法
(ud (=u1)d = 1)
倒推定价法
j
9 支付红利率 9 支付红利额
期权定价——二叉树 (binomial tree)
z 对于不同标的资产
p= a−d u−d
a = e(r−q)Δt a = e(r−rf )Δt
a =1
支付连续股息收益率股票或股指 货币
期货
u = eσ Δt d = e−σ Δt
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发展：20世纪80年代，金融工程学逐渐成熟，成为一门独立的学科
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目录
CONTENTS
金融工程学是一门将金融学、数学、统计学、计算机科学等学科相结合的 交叉学科。
金融工程学主要研究金融市场的运作规律，以及如何利用数学模型和计算 机技术进行金融分析和风险管理。
金融工程学的主要内容包括金融市场分析、金融产品设计、金融风险管理、 金融工程软件等。
添加标题
添加标题
添加标题
统计分析：使用统计方法来分析 金融数据
风险管理：使用定量分析方法来 管理金融风险
蒙特卡洛模 拟：通过随 机抽样进行
数值计算
数值积分： 通过数值方 法求解积分
问题
线性规划： 求解线性约 束条件下的
金融工程 第十一章 二叉树简介
0.120.25
0.633
结论：风险中性定价与无套利方法等价
2.2 现实世界与风险中性世界 计算股价上涨的真实概率 p* 设股票在现实世界中的期望收益率为 16%，则
得 计算现实世界期权价格 f 的困难
不能确定期权在真实世界里的期望收益率，无 法获得贴现利率
11.3 两步二叉树 Two-step binomial trees
即期权的价格为：
代入 D , 即有：
(11.2)
其中
(11.3)
例 P166 ： 在前例中，有
因此
1.3 股票收益期望与期权价格的无关性
股票上涨或下跌一般有不同的概率；
上述期权的定价与股票上涨或下跌的概率无关 根据股票价格给期权定价时，已经将股价的变 化概率考虑在内。
11.2 风险中性定价 risk-neutral valuation
Call option
D
22
B
24.2 3.2 19.8 0.0 16.2 0.0
20 1.2823
A
2.0257 18 0.0
C
E
F
•在第 1 步，
Delta=
•如果在第 1 步，股价上涨，则在第2步
Delta=
Put option
D
60 50 4.192320
B
E
1.4147
现实世界风险中性世界71现实世界realworld设股票在?t时间间隔内收益率记为假设股票价格的年波动率为???年期望收益为???即varr??那么股票在?t时间间隔内波动率和期望收益率为vartr???tsrs???er??tertt?????股票价格的期望与二叉树中股票价格期望一致即二叉树中股票价格波动率应与现实世界的波动率一致即s???可得11112t2020001varvartttssesessss?????????????tes?所以即有利用级数展开并忽略时间间隔的高阶项可得72风险中性世界riskneutralworld假设在风险中性世界里股票的期望收益率等于无风险收益率因此1111成为股票价格的期望其中001
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0.033
0
0
Options, Futures, and Other Derivatives, 4th edition © 2000 by JoShanghai Normal University
•
9.14
Generalization (continued) :
S0 D – f • Hence,
ƒ = S0 D – (S0u D – ƒu )e–rT
Options, Futures, and Other Derivatives, 4th edition © 2000 by John C. Hull Tang Yincai, Shanghai Normal University
• Figure 9.1 (P.202)
Stock price = $20 Option Price=?
Stock Price = $22 Option Price = $1
Stock Price = $18 Option Price = $0
Options, Futures, and Other Derivatives, 4th edition © 2000 by John C. Hull Tang Yincai, Shanghai Normal University
9.1
Introduction to Binomial Trees
Chapter 9
Options, Futures, and Other Derivatives, 4th edition © 2000 by John C. Hull Tang Yincai, Shanghai Normal University
9.2
A Simple Binomial Model of Stock Price Movements
• In a binomial model, the stock price at the BEGINNING of a period can lead to only 2 stock prices at the END of that period
==> D S0-f=0.25*20-0.6=4.4>4.367
Buy call
t=0 -0.600
ST=18 0
ST=22 1
Sell D Shares 5.000 -18*0.25=-4.50 -22*0.25=-5.50
Lend 4.367 at r -4.367
4.50
4.50
Net Flows
• Procedures: Establish a portfolio of stock and option Value the Portfolio
no arbitrage opportunities no uncertainty at maturity no risk with the portfolio risk-free interest earned
Options, Futures, and Other Derivatives, 4th edition © 2000 by John C. Hull Tang Yincai, Shanghai Normal University
9.3
Option Pricing
Based on the Assumption of No Arbitrage Opportunities
•
9.15
Irrelevance of a Stock’s
Expected Return
• When we are valuing an option in terms of the underlying stock the expected return on the stock is irrelevant
9.9
Generalization
• Consider a derivative that lasts for time T and that is dependent on a stock
• Figure 9.2 (P.203)
S0u
S0
ƒu
ƒ S0d
ƒd
Options, Futures, and Other Derivatives, 4th edition © 2000 by John C. Hull Tang Yincai, Shanghai Normal University
A Simple Binomial Model: 9.4 Example
• A stock price is currently $20 • In three months it will be either $22 or
$18
Stock Price = $22
Stock price = $20
Stock Price = $18
SHORT 1 call 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 - 4.367
Options, Futures, and Other Derivatives, 4th edition © 2000 by John C. Hull Tang Yincai, Shanghai Normal University
Options, Futures, and Other Derivatives, 4th edition © 2000 by John C. Hull
Tang Yincai, Shanghai Normal University
•
9.16
Irrelevance of a Stock’s E(R)
Proof: (continued)
• This is because in our formula f = S0D - (S0uD-fu) e-rT
f does not involve the probability of the stock moving up or down
• It does not matter if we say the probability of an increase is 50% or 80% we get the same result
9.13
Generalization (continued) :
Proof with an Example
• This is known as the No Arbitrage methodology
• In our earlier example f=0.633 and D=0.25
• If f<0.633, e.g. f=0.60
or when
D fu fd S0u S0d
Options, Futures, and Other Derivatives, 4th edition © 2000 by John C. Hull Tang Yincai, Shanghai Normal University
9.11
Generalization (continued)
Value the option
Risk-free interest = value of portfolio today
Options, Futures, and Other Derivatives, 4th edition © 2000 by John C. Hull Tang Yincai, Shanghai Normal University
Options, Futures, and Other Derivatives, 4th edition © 2000 by John C. Hull Tang Yincai, Shanghai Normal University
9.5
A Call Option
• A 3-month call option on the stock has a strike price of $21.
• Let’s call pu the probability of an increase in the stock price and pd=1- pu the probability of a stock decrease S0D - f = [pu(S0uD-fu)+ pd(S0dD-fd)] e-kT where k is the appropriate rate for the risk involved
• 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 = 18 * 0.25
• The value of the portfolio today is 4.50e-0.12*0.25=4.3670
9.12
Generalization
(continued) • Substituting for D we obtain
ƒ = [ p ƒu + (1 – p )ƒd ]e–rT
where
erT d p
ud
Options, Futures, and Other Derivatives, 4th edition © 2000 by John C. Hull Tang Yincai, Shanghai Normal University
Options, Futures, and Other Derivatives, 4th edition © 2000 by John C. Hull Tang Yincai, Shanghai Normal University