Question

European call and put options with a strike price of $50 will expire in one year. The underlying stock is selling for $51 currently and pays no cash dividend during the 1life of the options. The risk-free rate is 5% (continuous compounding). The price of the call option is $5.00. Please answer the following questions:

1. Please calculate the price of the put option under the put-call parity

2. If the actual price of the put is $2.50, is there an arbitrage opportunity? Please explain.

3. If your answer in 2 is yes, please write down the four transactions needed to realize the arbitrage profit.

Answer 1

1) We can use following formula from put call parity

c = call value = $5
S = current stock price = $51
p = put price
X = exercise price of option =$50
e = Euler’s constant – approximately 2.71828 (exponential function on a financial calculator)
r = continuously compounded risk free interest rate =5%
T = term to expiration measured in years =1 year
  

Solving this, we get p= $1.56

2) Since the actual price of put is $2.50 whereas it should be only $1.56, the arbitrage opportunity exists. The put option is overpriced and we always sell the overpriced product in order to take advantage of arbitrage.

3) Following transactions will ensure that we get risk free arbitrage opportunity

1-Sell the put option, you will get $2.50

2- Short sell the Stock, you will get $51= total proceeds $53.5=51+2.5

3- Buy a call option for $5, remaining amount is $53.5-$5=$48.5

4-Invest remaining amount at 5% for 1 year

At the end of 1 year, the amount will become 48.5*e^(0.05)= $50.98 approximately

After one year, whatever the stock price be, you would have a profit.

If the stock price is higher than $50 after one year, the put option would be out of money and not exercise. You can exercise your call option to buy the stock at $50 (you have $50.98 from risk free bond) and then use it to close you short position on the underlying stock. Profit-$0.98

If the stock price is lower than $50 after one year, the call option would be out of money and not exercised. The put option will be exercised and you will have to buy the stock at $50, you can use the stock to cover you short position on the stock. Profit-$0.98

Kindly let me know if you have any doubt.
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