期权套利

Option Arbitrage

专题: Finance / 金融
难度: L5
来源: OpenQuant

题目详情

Jamison 持有标的 $S$ 上的两个看涨期权，初始价格 $S_0 = 24$ ，执行价 $K=21$ 。其中一个是欧式期权，另一个是美式期权。欧式期权只能在到期时行权，而美式期权可以在任意时刻行权。

欧式期权价格为 $3.21$ ，美式期权价格为 $3.15$ 。此外，你还可以交易贴现因子为 $Z_0 = 0.9$ 的债券。标的不支付股息。

请问套利机会是什么？请用你在初始时刻收到的现金流来作答。

Jamison has two call options on the underlying $S$ with initial price $S_0 = 24$ and strike $K=21$ . One option is European and the other is American. A European option is one where you can only exercise it at expiry, while an American option can be exercised at any time.

The European option is valued at $3.21$ while the American option is valued at $3.15$ . You also have access to bonds with discount rate $Z_0 = 0.9$ . The underlying pays no dividends.

What is the arbitrage? Give the answer in the form of the initial credit you receive.

解析

对于不支付股息的股票，美式看涨期权和欧式看涨期权应当具有相同价值。如果行权美式期权可获得 $S-K$ ，那么在不存在套利的情况下，它的价值至少不应低于欧式看涨期权。这里可以卖空欧式看涨、买入美式看涨。若到期时看涨期权处于实值，两者的价值都为 $S-K$ ，最终净值为 $0$ ，但初始时保留了 $0.06$ ；若到期时处于虚值，两者价值都为 $0$ ，同样可以保留这 $0.06$ 。

Original Explanation

The American and European calls on non-dividend paying stocks should have the same value. If you exercise the American option and gain $S-K$ , it must be worth at least the European call for there to be no arbitrage. We will short the European call and go long the American call. If the call expires in-the-money, then both options have value $S-K$ and we obtain $0$ , but we keep our $0.06$ . If the calls expire out-of-the-money, then both options have value $0$ and we once again keep our $0.06$ .