Question

The price of a non-dividend-paying stock is $45. The strike price of a six-month American put option is $50. The risk-free rate is 3% (continuously compounded). Which of the following is a lower bound for the option such that there are arbitrage opportunities if the price is below the lower bound and no arbitrage opportunities if it is above the lower bound?

Answer 1

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An arbitrageur should borrow $49.00 at 6% for six months, buy the stock, and buy the put option. This generates a profit in all circumstances.

If the stock price is above $50 in one month, the option expires worthless, but the stock can be
sold for at least $50. A sum of $50 received in six months has a present value of $49.25 today.
The strategy therefore generates profit with a present value of at least $0.75 (50-49.25 $).
If the stock price is below $50 in six month the put option is exercised and the stock owned is
sold for exactly $50 (or $49.25 in present value terms). The trading strategy therefore generates a profit of exactly $0.75 in present value terms.