Question

What are the upper and lower bounds for the price of a two-month put option on a non-dividend-paying stock when the stock price is $27, the strike price is $30, and the risk-free interest rate is 5% per annum? What is the arbitrage opportunity if the price of the option is $1? What are the net profits?

Answer 1

max.(Ke^-rT -S₀ ,0)<p <Ke^-rT  

Lower bound:- 30*e-^0.05*(2/12) - 27 = $2.75

Upper bound: $29.75

Since, $1 < $2.925,

An arbitrageur should borrow $28 at 5% for one month, buy the stock, and buy the put option. This generates a profit in all circumstances.

If the stock price is above $30 in one month, the option expires worthless, but the stock can be sold for at least $30. A sum of $30 received in one month has a present value of $29.75 today. The strategy, therefore, generates profit with a present value of at least $1.75

If the stock price is below $30 in one month the put option is exercised and the stock owned is sold for exactly $30 (or $29.75 in present value terms). The trading strategy, therefore, generates a profit of exactly $1.75 in present value terms.