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magic456 · 2025年06月06日

put call parity和bsm模型矛盾

  1. CFA II
  2. Derivative

* 问题详情，请 查看题干

NO.PZ201702190300000304

问题如下：

For the Alpha Company option, the positions to take advantage of the arbitrage opportunity are to write the call and:

选项：

A.

short shares of Alpha stock and lend.

B.

buy shares of Alpha stock and borrow.

C.

short shares of Alpha stock and borrow.

解释：

B is correct.

You should sell (write) the overpriced call option and then go long (buy) the replicating portfolio for a call option. The replicating portfolio for a call option is to buy h shares of the stock and borrow the present value of (hS- - c-).

c = hS + PV(-hS- + c-).

h = (c+ - c-)/(S+ - S-) = (6 - 0)/(56 - 46) = 0.60.

For the example in this case, the value of the call option is 3.714. If the option is overpriced at, say, 4.50, you short the option and have a cash flow at Time 0 of +4.50. You buy the replicating portfolio of 0.60 shares at 50 per share (giving you a cash flow of -30) and borrow (1/1.05) x [(0.60 x 46) - 0] = (1/1.05) x 27.6 = 26.287. Your cash flow for buying the replicating portfolio is -30 + 26.287 = -3.713. Your net cash flow at Time 0 is + 4.50 - 3.713 = 0.787. Your net cash flow at Time 1 for either the up move or down move is zero. You have made an arbitrage profit of 0.787.

In tabular form, the cash flows are as follows:

中文解析：

根据题干信息可知，当前的市场上关于Alpha公司的看涨期权是被高估的，因此套利操作下我们可以卖出被高估的买进被低估的，因此正如本题问题中表述的已经卖出了看涨期权，然后需要的操作是买入一个合成的看涨期权。

看涨期权的合成相当于借钱买股票，因此本题选B。

看涨期权c

put call parity 合成c=p+s-x/e^rt

BSM 模型则是 long stock +short bond，为什么会差一个put，这不是矛盾了吗

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1 个答案

李坏_品职助教 · 2025年06月06日

嗨，从没放弃的小努力你好：


如果用put call parity， c=p+ s- x/e^rt， 这个公式的long stock系数是1，而我们上面的方法里，long stock系数是h。并且borrow的金额也不一样，既然stock和borrow都不一样了，那差一个put也是正常的。 所以这两个方法都是合理的，只是无套利定价的方法更符合题目要求。

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虽然现在很辛苦，但努力过的感觉真的很好，加油！

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