一道关于replicate payoff of put option 的题，有些不明白的地方，请高手解答～！

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收藏 2012-07-11

有几个地方不是很明白，希望各位朋友能帮帮忙～！

Consider a two-period binomial model.The stock price at time zero is $374.4. The up factor is equal to 1.25, the down factor is equal to 0.625. The risk free rate is 20% per period.
We want to price a European put option that expires at time two. the strike price is $374.4.

这个模型基本的一些数据我已经算出来了

P=9.386
Pu=5.46
Pd=78
Puu=0
Pud=81.9
Pdu=81.9
Pdd=228.15

h= -0.31
hu= -0.28
hd= -1

关键现在是他要 replicate payoff of 1000 option contracts
lending/borrowing cash as necessary at the risk free rate.
Fill in each blank with a number with appropriate sign (+ or -)，
（For stocks, minus sign signifies a short position. For cash, minus sign signifies debt.)

To construct the replicating portfolio that replicates the value of 1000 option contracts, we should begin with a portfolio whose total value at time zero is ___________ dollars,
which consist of __+310_ shares of the underlying stock and  _____________ dollars in cash. Then.....

就第一步这里，portfolio总价值  V=310（374.4）+1000（9.386）=125450 对不对？
这个价值一个是股票，一个是put option，我就搞不懂哪里来的cash，
他说lending 和borrowing都可以，是要让我再借一些钱么？
这老师的课件上起始的total value at time zero 0 好像都是用1000 乘以 option的价格，1000（9.386）=9386，
是这么回事么？

然后接着还有，就是分这个模型股票上升和下降的情况。
Case 1: Stock goes up in the first period.
If this happens, the value of our stock holdings becomes ___________ dollars and the value of our cash holdings become __________,
which bring the total value of our portfolio to  __________ dollars. At this time, we should rebalance our portfolio:
We buy or sell stock as appropriate, so that our position in the underlying stock is changed to __+280__shares. This rebalancing changes the total cash in our portfolio to ________ dollars. And what happens in the final period:

-If stock goes up in the final period, the value of our stock holding becomes______________dollars and the value of our cash holdings becomes__________, which bring the total value of our portfolio to _______ dollars.

-If stock goes down in the final period, the value of our stock holding becomes______________dollars and the value of our cash holdings becomes__________, which bring the total value of our portfolio to _______ dollars.

Case 2: Stock goes down in the first period.
If this happens, the value of our stock holdings becomes ___________ dollars and the value of our cash holdings become __________,
which bring the total value of our portfolio to  __________ dollars. At this time, we should rebalance our portfolio:
We buy or sell stock as appropriate, so that our position in the underlying stock is changed to _______shares. This rebalancing changes the total cash in our portfolio to ________ dollars. And what happens in the final period:

-If stock goes up in the final period, the value of our stock holding becomes______________dollars and the value of our cash holdings becomes__________, which bring the total value of our portfolio to _______ dollars.

-If stock goes down in the final period, the value of our stock holding becomes______________dollars and the value of our cash holdings becomes__________, which bring the total value of our portfolio to _______ dollars.

还有最后一个问题就是
在replicating portfolio 和 hedging portfolio的时候有什么区别？？？

是不replicating portfolio 在起始时间的total value必须是某些特定的数值，
而hedging portfolio的时候stock，option，lend or borrow多少cash 怎么分配都可以？

希望大家帮帮忙，非常感谢～

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bobojin

2012-7-11 04:09:47

B-S equation 已经说明了现金，股票及期权的关系，很基础的东西，再仔细看看text book

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bryansky

2012-7-11 04:14:16

bobojin 发表于 2012-7-11 04:09
B-S equation 已经说明了现金，股票及期权的关系，很基础的东西，再仔细看看text book

是，那是Black schole equation，我这个只是个简单的two-period binomial model，好像还不用black schole吧？

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reduce_fat

2012-7-11 07:29:02

楼主的时间指的是年吗？ h = 1? risk-neutral probability = 0.95424, 有没有 continuous dividend rate? 如果没有就这么算。应该要算一下delta (shares of stocks) 还有 B (borrowing amount) 然后用矩阵方程求解。可以算出你要的cash amount 和 shares of stocks. 1000只是个系数吧，最后算完 put 价格后，可以乘回来。我算的Put (at t = 0) 是 7.65661 和楼主的不一样，但是前面的数字差不多，楼主是不是中间的计算值没有保留所有小数位，还有是不是忘了最后一步还有一个discounting * e^(-0.2)？那个公式不是 option price (put or call) = delta (option) * S + B 吗？辅导书里有讲解，具体哪一部分忘了？我还要继续看。

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bryansky

2012-7-11 16:28:04

reduce_fat 发表于 2012-7-11 07:29
楼主的时间指的是年吗？ h = 1? risk-neutral probability = 0.95424, 有没有 continuous dividend rate? ...

这只是个two-period binomial model, 时间是time 0, 1, 2, 起始时间, 第一阶段,第二阶段.

h 是 hedge ratio 阿 hu 是第二时间阶段股票上升后的 hedge ratio, hd是股票下降后的ratio

1000是指有 1000个option contracts

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bryansky

2012-7-14 17:25:23

在请教了一些同学后终于知道了答案，贴出来供后人参考。
replicating portfolio 就是需要复制和option等价的资产
起始价格就是￥9368，因为是1000contracts，$9.386×1000，
借贷$125450，因为 125450-310×（374.4）=9386
后面的空就根据股票上升和下降的情况，配合第二阶段不同的hedge ratio重新分配资金进行hedge

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