先说一个，比如说22.18

一人想买标普的指数基金。他确认一年之后该基金将上涨25%或下跌20%。此人想回避基金跌破1300$的风险，但又不想丧失高于此值的机会。他可以按年7%的无风险利率借贷。

题目问他应该买看涨期权还是看跌期权；行权价几何。

还有该期权公允价值如何。

没给基金现在价值，也没说1300在上涨价和下跌价的什么位置，请大家帮忙看看

[此贴子已经被作者于2006-6-17 23:47:09编辑过]

第7版国内还没有出来吧？不知道LZ是怎么弄到的呢？不过我可以给你22.18的解答如下：

a. Mark would be interested in purchasing a put option on the index fund with a strike price

of $1,300 and 1 year until expiration. This option will compensate Mark for any decreases in value of the index fund below the strike price and places a floor of $1,300 on the net worth of his position.

b. In order to solve a problem using the two-state option model, first draw a stock price tree containing both the current stock price and the stock’s possible values at the time of the option’s expiration.

Next, draw a similar tree for the option, designating what its value will be at expiration given either of the 2 possible stock price movements.

The index fund is trading today at $1,400 per share. It will either increase by 25% or decrease by 20% in one year. If the fund increases by 25%, its value will be $1,750 (= $1,400 * 1.25) per share. If it decreases by 20%, its value will be $1,120 (= $1,400 * 0.80) per share. If the fund falls to $1,120, Mark will exercise his put option for $1,300 and receive a payoff of $180 at expiration. If the fund rises to $1,750, Mark will not exercise his put option, and he will receive no payoff at expiration.

（这里有个二叉树图，我不知道怎么能贴上来）。

If the price of the index fund rises, its return over the period is 25% [= (1750/1400) – 1]. If the price falls, its return over the period is –20% [= (1120/1400) –1]. Use the following expression to determine the risk-neutral probability of a rise in the index fund:

Risk-Free Rate = (Probability_Rise)(Return_Rise) + (Probability_Fall)(Return_Fall)

= (Probability_Rise)(Return_Rise) + (1 - Probability_Rise)(Return_Fall)

0.07 = (Probability_Rise)(0.25) + (1 – Probability_Rise)(-0.20)

Probability_Rise = 0.60

Probability_Fall = 1 - Probability_Rise

= 1 – 0.60

= 0.40

The risk-neutral probability of a rise in the index fund is 60%, and the risk-neutral probability of a fall in the index fund is 40%.

Using these risk-neutral probabilities, determine the expected payoff to Mark’s put option at expiration.

Expected Payoff at Expiration = (.60)($0) + (.40)($180) = $72.00

Since this payoff occurs 1 year from now, it must be discounted at the risk-free rate of 7% per annum in order to find its present value:

PV(Expected Payoff at Expiration) = ($72.00 / 1.07 ) = $67.29

Therefore, given the information Mark has about the index fund’s price movements over the next year, a put option with a strike price of $1,300 and 1 year until expiration is worth $67.29 today.

b. Yes, there is a way for Mark to create a synthetic put option with identical payoffs to the put option described above. In order to do this, Mark will need to short shares of the index fund and lend at the risk-free rate.

The number of shares that Mark should sell is based on the delta of the option, where delta is defined as:

Delta = (Swing of option) / (Swing of stock)

Since the put option will be worth $0 if the index fund rises and $180 if it falls, the swing of the put option is -180 (= 0 – 180).

Since the index fund will either be worth $1,750 or $1,120 at the time of the option’s expiration, the swing of the stock is 630 (= 1,750 – 1,120).

Given this information:

Delta = (Swing of option) / (Swing of stock)

= (-180/630)

= -2/7

Therefore, Mark’s first step in creating a synthetic put option is to short 2/7 of a share of the index fund. Since the fund is currently trading at $1,400 per share, Mark receives $400 (= 2/7 * $1,400) as a result of his short sale.

In order to determine the amount that Mark should lend, compare the payoff of the actual put option to the payoff of delta shares at expiration.

Put Option

If the index fund rises to $1,750: payoff = $0

If the index fund falls to $1,120: payoff = $180

Delta Shares

If the index fund rises to $1,750: payoff = (-2/7)($1,750) = -$500

If the index fund falls to $1,120: payoff = (-2/7)($1,120) = -$320

Mark would like the payoff of his synthetic put position to be identical to the payoff of an actual put option. However, shorting 2/7 of a share of the index fund leaves him exactly $500 below the payoff at expiration, regardless of whether the fund rises or falls. In order to increase his payoff at expiration by $500, Mark should lend the present value of $500 now. In one year, he will receive $500, which will increase his payoffs so that they exactly match those of an actual put option.

Mark should short 2/7 of a share of the index fund and lend $467.29 (= $500 / 1.07) in order to create a synthetic put option with a strike price of $1,300 and 1 year until expiration.

Since Mark receives $400 as a result of the short sale and lends $467.29, the total cost of the synthetic put option is $67.29 (= $467.29 - $400). This is exactly the same price that Mark would pay for an actual put option. Since an actual put option and a synthetic put option provide Mark with identical payoff structures, he should not expect to pay more for one than the other.

[此贴子已经被作者于2006-6-19 12:10:20编辑过]

嗬嗬，多谢路过蜻蜓了。

先回答你的问题，我用的第七版是英文版，国内好像是年初上的？我也不太清楚，别人送的。

然后是关于这道题，关键是那个标普500指数基金现在价值1400是怎么出来的？题中好像没给？不会是现去S&P的网站上查的吧？