第12章二叉树模型1

1、Chapter 12 Binomial banoml Trees二叉树模型二叉树模型Options, Futures, and Other Derivatives, 8th Edition, Copyright John C. Hull 2012n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对数正态分布。n在本章

2、，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风险中性估值原理之间的关系。对股价如何建模？对股价如何建模？离散模型离散模型连续模型连续模型由两点分布生成算术随机游走（增量是相加）由标准正态分布生成标准布朗运动生成一般布朗运动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衍生产品的二叉树模型伊藤过程在Excel中对各种过程的模拟。512.1 单步二叉树模型与无套利方法单步二叉树模型与无套利方法 A Simple Binomial ModelnA stock price is currently $20nIn 3 month

3、s it will be either $22 or $18Stock Price = $18Stock Price = $22Stock price = $206A Call Option (Figure 12.1, page 254)A 3-month call option on the stock has a strike price of 21. Stock Price = $18Option Price = $0Stock Price = $22Option Price = $1Stock price = $20Option Price=?7Setting Up a Riskles

4、s PortfolionFor a portfolio that is long D shares and a short 1 call option values arennPortfolio is riskless when 22D 1 = 18D or D = 0.2522D 118D8Valuing the Portfolio(Risk-Free Rate is 12%)nThe riskless portfolio is: long 0.25 sharesshort 1 call optionnThe value of the portfolio in 3 months is 22

5、0.25 1 = 4.50nThe value of the portfolio today is 4.5e0.120.25 = 4.36709Valuing the OptionnThe portfolio that is long 0.25 sharesshort 1 option is worth 4.367nThe value of the shares is 5.000 (= 0.25 20 )nThe value of the option is therefore 0.633 ( 5.000 0.633 = 4.367 )10Generalization (Figure 12.2

6、, page 255)A derivative lasts for time T and is dependent on a stockS0u uS0d dS011Generalization (continued)nValue of a portfolio that is long D shares and short 1 derivative:nnThe portfolio is riskless when S0uD u = S0dD d ordSuSfdu00DS0uD uS0dD d（12.1）12Generalization (continued)nValue of the port

7、folio at time T is S0uD unValue of the portfolio today is (S0uD u)erTnAnother expression for the portfolio value today is S0D fnHence = S0D (S0uD u )erT 13Generalization(continued)Substituting for D we obtain = pu + (1 p)d erT where pedudrT（12.2）（12.3）14p as a ProbabilitynIt is natural to interpret

8、p and 1-p as probabilities of up and down movementsnThe value of a derivative is then its expected payoff in a risk-neutral world discounted at the risk-free rateS0u uS0d dS0p(1 p )Excel应用运用复制公式的技巧。1612.2 风险中性定价风险中性定价 Risk-Neutral ValuationnWhen the probability of an up and down movements are p and

9、1-p the expected stock price at time T is S0erTnThis shows that the stock price earns the risk-free ratenBinomial 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 ratenThi

10、s is known as using risk-neutral valuation17Original Example Revisitedp 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 = 0.6523Alternatively: 6523. 09 . 01 . 19 . 00.250.12edudeprTS0u = 22 u = 1S0d = 18 d = 0S0=20 p(1 p )18Va

11、luing the Option Using Risk-Neutral ValuationThe value of the option is e0.120.25 (0.65231 + 0.34770)= 0.633S0u = 22 u = 1S0d = 18 d = 0S0=200.65230.347719Irrelevance of Stocks Expected ReturnnWhen we are valuing an option in terms of the price of the underlying asset, the probability of up and down

12、 movements in the real world are irrelevantnThis is an example of a more general result stating that the expected return on the underlying asset in the real world is irrelevant 现实世界与风险中性世界现实世界与风险中性世界n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。n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。n持有看涨期

13、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%。n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。2212.3 两步二叉树两步二叉树 A Two-Step ExamplenK=21, r = 12%nEach time step is 3 months20221824.219.816.223Valuing a Call OptionFigure 12.4, page 260Value at node B = e0.120.25(0.65233.2 + 0.34770) = 2.0257Value at node A = e0.120.25(

14、0.65232.0257 + 0.34770) = 1.282316.20.0201.2823221824.23.219.80.02.02570.0AB一般结论一般结论12.4 看跌期权的例子看跌期权的例子 A Put Option Example; K=52K = 52, time step = 1yrr = 5%504.1923604072048432201.41479.4636ABCDEF26A Put Option ExampleFigure 12.7, page 263K = 52, time step =1yrr = 5%, u =1.2, d = 0.8, p = 0.62825

15、04.1923604072048432201.41479.46362712.5 美式期权美式期权 What Happens When the Put Option is American 505.0894604072048432201.414712.0CThe 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 5.0894.12.6 DeltanDelta (D) is the ratio of t

16、he change in the price of a stock option to the change in the price of the underlying stocknThe value of D varies from node to noden股票期权的D是股票期权价格的变化与标的股票价格的变化之比。n为了构造一个无风险对冲，对一个卖空的期权头寸应该持有的股票数量。n构造无风险对冲有时就被称为D对冲。看涨期权的D是正值，而看跌期权的D是负值。12.7 用用u和和d匹配波动率匹配波动率 Choosing u and d31Choosing u and dOne way of

17、matching the volatility is to setwhere s is the volatility and Dt is the length of the time step. This is the approach used by Cox, Ross, and RubinsteintteudeuDsDs132Girsanovs TheoremnVolatility is the same in the real world and the risk-neutral worldnWe can therefore measure volatility in the real

18、world and use it to build a tree for the an asset in the risk-neutral world34Assets Other than Non-Dividend Paying StocksnFor options on stock indices, currencies and futures the basic procedure for constructing the tree is the same except for the calculation of p12.8 二叉树公式二叉树公式n为了与波动率匹配，我们取 ttuedes

19、sDD r tadpaeudD36Proving Black-Scholes-Merton from Binomial Trees (Appendix to Chapter 12)Option is in the money when j a where so thatnjjnjjnjrTKduSppjjnnec00)0,max()1 (!)!(!nTKSnsa2)ln(20aajjnjjjnjjnjrTppjjnnUduppjjnnUKUUSec)1 (!)!(!)1 (!)!(!)(21210where37Proving Black-Scholes-Merton from Binomial

20、 Trees continuednThe expression for U1 can be writtenwhere nBoth U1 and U2 can now be evaluated in terms of the cumulative binomial distributionnWe now let the number of time steps tend to infinity and use the result that a binomial distribution tends to a normal distribution aajjjnjrTjnjnppjjnneppj

21、jnndppuU*11!)!(!1!)!(!)1 (dppupup)1 (*12.9 增加二叉树的时间步数增加二叉树的时间步数n在实际中应用二叉树时，期权的期限通常会被分割为30步或更多的步数。12.10 使用使用DerivaGem软件软件nDerivaGem可以展示的树形最多为10步，但计算量最多可以达到500步。12.11 对于其他标的资产的期权对于其他标的资产的期权n对于指数期权、外汇期权和期货期权，可以利用二叉树模型分析这些资产的价格。除了计算p的公式变化之外，方法和股票期权的分析方法相同。n公式（12.12）说明期初的价格等于p乘以上升的价值加上1- p乘以下降的价值再以无风险利率贴

22、现所得到的值。支付连续红利收益的股票的期权支付连续红利收益的股票的期权股票指数期权股票指数期权n在前面计算股票指数的未来价格时，我们假设指数的标的股票组合支付的红利收益率为q。在这里，沿用这个假设。因此，股票指数期权的估值和支付连续红利收益率的股票期权的估值非常相似。例例12.1n当前某个股票指数为810，波动率为20%，红利收益率为2%，无风险利率为5%。n用DerivaGem对6个月期执行价格为800的欧式看涨期权估值。外汇期权外汇期权n可以把一种外汇看作是支付外汇无风险利率的一种资产。和股票指数期权的情况一样，可以利用公式（12.13）（12.16），构造外汇期权的二叉树图。n在这种情况

23、下，设定()fr rtaeD例例11.2n当前，一澳元等于0.6100美元，汇率波动率为12%。澳元无风险利率为7%，美元无风险利率为5%。n用DerivaGem估值执行价格为0.6000的3个月期美式看涨期权的价格。期货期权期货期权n买入和卖出期货合约不需要任何成本。因此，在风险中性世界里，期货价格的预期增长率应该为零。n和上述分析一样，可以定义p为期货价格上述的概率、u为期货价格上升的百分比、d为价格下降的百分比。例例11.311.3n当前的期货价格为31，波动率为30%，无风险利率为5%。n利用DerivaGem估值执行价格为30的9个月期美式看跌期权的价格。49The Probabil

24、ity of an Up Movecontract futures a for 1rate free-risk foreign the is herecurrency w a for index the on yielddividend the is eindex wher stock a for stock paying dnondividen a for DDDareaqeaeadudapftrrtqrtrf)()(小结小结n本章简要介绍了基于股票和其他一些资产的期权的定价/估值问题。如果在股票期权有效期内股票价格的变化/运动是按单步二叉树图的方式进行的话，那么有可能构造一个由股票期权和股票本身组成的无风险证券组合。无套利机会的世界中，无风险证券组合的收益率应该等于无风险利率。这有可能要根据股票来定价股票期权。有意思的是：没有必要在每个节点假设股票价格上升和下降的概率。n如果股票价格的运动是按多步二叉树图方式进行的，我们可以分别处理每个单步二叉树图，并从期权有效期的末尾推回到开始位置，获得期权的当前价格。其中只用到了无套利假设，而没有必要假设股票价格在每个节点上升和下降的概率。n另一个为期