3. A European put and European call both have strike price $17 and expire in six

Question

3. A European put and European call both have strike price $17 and expire in six months. The call is currently selling for $1.30 and the put is currently selling for $1.40. Both options have the same underlying asset which is currently selling at $15. Say that the present value of the strike is PVo($17) 16.20. (a) Show that there is an arbitrage opportunity at the initial time (consider short selling one call, buying one put, buying one underlying asset and borrowing some amount of money) (b) In this problem, does put-call parity hold? Explain your answer.

Explanation / Answer

a)

As per put call parity :

call + PV of bond = Put + Stock

So it says that call + PV of bond will have same payoff as put and stock, so right side of equation should be equal to left. If not there is an arbitrage opportunity.

If there is an opportunity we will buy the cheap side and sell the costlier side and the difference is our profit

So

call = 1.3

PV = 16.2

put = 1.4

stock price = 15

So,

1.3 + 16.2 = 1.4 + 15

17.5 = 16.4

So there is arbitrage opportunity

Sell the higher side and buy the cheaper

So

Short a call, so you will get the premium = 1.3

Also issue a zero coupon bond with strike price ($17) as face value and 6 months expiration, so you will get the PV of the bond = 16.20

So total inflow = 1.3+16.2 = 17.5

Now You buy a put and pay premium = 1.4

also buy a stock at its price = 15

Total outlow = 16.4

Profit = 17.5 - 16.4 = 1.1

b) As i have explained above, put call parity does not hold true in our example due to which there exists the arbitrage opportunity.

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