NO.PZ202208260100000802
问题如下：
If Kleinert's clients observe that the one-year put option with a €100 exercise price is trading at €2.50, which of the following statements best describes how Kleinert's clients could take advantage of this to earn a risk-free return greater than 0.37% over the year.选项：
A.Kleinert should purchase the put option and also purchase approximately 0.23 shares per option to match the hedge ratio.
B.Kleinert should purchase the put option and purchase 50% of the underlying shares given the 50-50 chance the stock will fall and the put option exercised.
C.Kleinert should purchase the put option and purchase 47% of the underlying shares to match the risk-neutral probability of put exercise.
解释：
Solution
A is correct.
If the put option can be purchased for less than the no-arbitrage price, then a potential arbitrage opportunity is available. In this case, Kleinert's clients should purchase the underpriced put option and buy h* units of SparCoin's stock. The hedge ratio, h*, is calculated as:
Note that the negative hedge ratio implies that both the put option and underlying are purchased or sold to create a hedge. This initial purchase of the put option and stock will cost:
€2.50 + 0.2276 × €105.25 = €26.45.Should the stock price decrease, the value of this portfolio will be:
The strategy generates a risk-free return of (€26.83 – €26.45)/€26.45 = 1.44%, which is greater than the 0.37% return on other available risk-free investments.
中文解析
由上面一问我们可以知道，该看跌期权的无套利价格是2.78，现在市场上看跌期权的价格是2.50。根据低买高卖的套利原理，我们应该买入该看跌期权，对应的如果构成hedgedportfolio，需要long stock。然后根据公式计算h为0.2276份。
此时，一份的put和0.2276份stock可以构成一个hedged portfolio.
该组合初始价值为：€2.50 + 0.2276 × €105.25 = €26.45
计算当股价下跌的时候，组合新的价值V1为€26.83.
此时可以计算得到return为1.44%，是大于题干所说的高于0.37%的。
当然，就选出答案来说，只需计算出h即可。
h是负数为什么要long stock呢
