Question

A four-month European put option on a non-dividend-paying stock is currently selling for $2. The stock price is $45, the strike price is $50, and the risk-free interest rate is 12% per annum. Is there an arbitrage opportunity? Show the arbitrage transactions now and in four months.

Answer 1

PV of strike price = 50*e^(-rT) = 50*e^(-12%*4/12) = 48.04

Net profit now = PV - stock price = 48.04 - 45 = 3.04

Put option price of $2 is less than $3.04, so there is an arbitrage opportunity here.

PV of strike price of 50 (if it is received 4 months later) will be 50/(1+(1%)^4 = 48.05

FV of 48.05 now will be 48.05/(1+1%)^4 = 46.17

Borrow 46.17 at 12% p.a. for 4 months, buy the stock and buy the put option. If, after 4 months, the stock price is greater than 50, the option won't be exercised but the stock can be sold for minimum $50. This will generate a profit of 50 - 48.05 = $1.95 in PV terms. If stock price is less than 50 in 4 months' time then put option is exercised and the stock can be sold for 50. Again, there is a profit of $1.95 in PV terms.