Question

Stock trades at $50. An at the money call option trades at $4. The maturity of the option is two years. The present value of the strike price over the two years is $45. The stock pays a dividend of 3 dollars in one year and no other dividends prior to expiration. The present value of the dividend is $2. The price of the European call option is $5. The price of the European put is $1.0. Construct a strategy that guarantees riskless arbitrage profits. Provide full details of the strategy and identify the minimum and maximum possible profits.

Answer 1

From the put call parity equation  

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

where c and p are call and put option premiums respectively

K/(1+r)^t is the present value of strike price =$45

, S is the adjusted spot price net of present value of dividends =(50-2)= $48

Here, c+ K/(1+r)^t = 5+45= 50

and p+S =1+48 = 49

As put-call parity do not hold, there is an arbitrage opportunity and prices are not consistent with the lack of arbitrage

The arbitrage portfolio works as follows

1. Today, Borrow $45 for two years and $2 for one year. Sell the call option for $5. In this way, get a total of $52 and buy the stock and put option for $51 today. Keep $1 separately

2. After one year, repay 2 loan with the dividend received of $3

3. After 2 years, amount owed is K the strike price (as $45 is the present value of the Strike price)

If stock price > K, put option will be worthless, call option will be exercised, so sell the Stock at K, repay the loan of $K and thereby $1 kept separately at beginning is the Arbitrage profit.

If stock price < K, call option will be worthless, put option will be exercised, so sell the Stock at K using put option, repay the loan of $K and thereby $1 kept separately at beginning is the Arbitrage profit

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

So, minimum and maximum profit in this case is $1