Question

The price of a European call that expires in six months and has a strike price of $82.50 is $5.5. The underlying stock price is $79.75. The interest rates are 10% per annum for all maturities.

(a) What is the price of a European put option that expires in six months and has a strike price of $82.50?

(b) Explain carefully the arbitrage opportunities if the European put price is $4. Describe the arbitrage strategies. (How to proceed with the arbitrage transaction?)

(c) Show the payoff table for the arbitrage strategies if ST < K or ST > K.

(d) Calculate the arbitrage profit now (at t=0). (Please calculate the answer accurate to the fourth decimal place.)

Answer 1

a) From the put call parity equation (Assuming no dividends in the six months of the option maturity)

c+ K/(1+r)^t = p+S

where c and p are call and put option premiums respectively

K is the strike price of the options

r is the interest rate = 10% per annum

and t is the time period in years = 1/2

S is the spot price

So

5.5+82.50/1.1^0.5 = p + 79.75

=> p =5.5+82.50/1.1^0.5 - 79.75 = $4.41

b)

The arbitrage portfolio works as follows

1. Today, Borrow $78.25 at 10% per annum for 6 months.and sell the call option for $5.5.to get a total cash of $83.75

2. Today, Buy the stock and the  put option for $79.75 and $4 respectively using the money

3. After 6 months, amount payable is 78.25*1.1^0.5 = $82.07

4. After 6 months, If stock price > $82.50, put option will be worthless, call option will be exercised, so sell the Stock at $82.50, repay the loan of $82.07 and get the remaining $0.43 as arbitrage profit.

After 6 months, If stock price < $82.50, Call option will be worthless, use the put option, sell the Stock at $82.50, repay the loan of $82.07 and get the remaining $0.43 as arbitrage profit.

After 6 months, If stock price = $82.50, both Call and put options will be worthless,sell the Stock at $82.50 in the market, repay the loan of $82.07 and get the remaining $0.43 as arbitrage profit.

So,in all situations, one can make an arbitrage profit

c) The payoff table is as given below

	Payoff When
	ST<K	ST>K
Long put	K-ST	0
Long Stock	ST	ST
Short Call	0	ST-K
Borrowing repayment	82.07	82.07
Arbitrage profit	K-82.07	K-82.07

d) Arbitrage profit now = present value of arbitrage profit after 6 months

= 0.43/1.1^0.5

=$0.409989 or $0.4099