Question

A 3-month European put option on a non-dividend-paying stock is currently selling for $3.50. The stock price is $47.0, the strike price is $51, and the risk-free interest rate is 6% per annum (continuous compounding). Analyze the situation to answer the following question:

If there is no arbitrage opportunity in above case, what range of put option price will trigger an arbitrage opportunity? If there is an arbitrage opportunity in the above case, please provide one possible trading strategy to take advantage of this opportunity and show your trading results.

Answer 1

Strike Price =$51

Current Stock Price =$47

You can buy stock at $47 and Buy a put option with strike $51 at Put Premium =X

Total Investment =47+X

Future value at 6% continuous compounding =(47+X)*(e^(0.06*(3/12)))=(47+X)*1.015113

At expiration , exercise the option to sell at strike price =$51

There will be no arbitrage opportunity if the Future Value of initial investment = $51

1.015113*(47+X)= 51

47.71+1.015X=51

1.015X=3.29

X=3.29/1.015=$3.24

If the Put Option Price is less than $3.24, there will be ARBITRAGE OPPORTUNITY.

In this case price is $3.50, hence there is no arbitrage opportunity.

Assume that Put Option Price =$3.00

Trading strategy:

Take Loan of $50(47+3) at 6% interest

Buy one share at $47 and one option at strike =$51

After 3 months amount to be returned for loan =1.015*50=$50.75

Amount Received from selling share at strike price=$51

Profit per option =51-50.75=$0.25