Question

. Consider a call and a put option, both with strike price of $30 and 3 months to expiration. The call trades at $5, the put price is $6, the interest rate is 0, and the price of the underlying stock is $29. (1) Suppose the stock does not pay dividends. Is there an arbitrage? If so, write down the sequence of trades and calculate the arbitrage profit you realize in 3 months. If not, explain why not. (2) Suppose the stock will pay a dividend of $2 in 2 months. Assume all other prices are as above. Is there an arbitrage? If so, write down the sequence of trades and calculate the arbitrage profit you realize in 3 months. If not, explain why not.

Answer 1

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Answer:

It is given that the call premium C = 5, put premium P=6, stock price S = 29 and strike price is X and interest rate r = 0 We can use our call parity formula to calculate the arbitrage Part 1) When there is no dividend Put call parity C+X/ exp (interest * time ) = P +S. 5+30/ exp ( 0% * 3/12 ) = 6 + 29 35 = 35 Since both the call side and put side are same that means that there is no arbitrage opportunity available. If there were an arbitrage opportunity then we should be selling the side that has higher value and buying the side that has lower value.

Part 2) When dividend of $2 will be given in 2 months then the present value of the dividend will be $ 2 as interest rate is 0. Again using put call parity formula with dividend C +X/ exp (r*t) = P + C-present value of Dividend 5+30 = 6 + 29 - 2 35 = 33 Here the arbitrage opportunity exists as call side is 35 and put side is 33. The arbitrage value = difference between the call side and put side = $2

Steps to realise the arbitrage profit 1. Sell the call option and earn a premium of $5 2. Sell the security like bond that have zero interest at $30 3. Buy the put option and pay a premium of $6 4. Buy the stock at $29 Total cash flow at the time of initiation = cash outlay-cash inflow = 5+30-6-29 = 0 We will get the dividend in 2 months and that will be our profit i.e. $2 After 3 months no matter where the stock price goes out profit will remain same. For example suppose after 3 months share price will become $10 then Value of call option = 0 and profit due to call option will be $5 as we have sold the call option Value of security that we have shorted at $30 will remain same and profit = 0 Value of stock is now $10 means we will have loss of 29-10 = 19 on the stock Value of put will be $20 as the current price of stock is 10 and strike price is $ 30. We have purchased this option by paying premium of $6 hence net profit from the Option = $20-6 = $14 So net value = profit from selling call option + profit due to selling the security - loss due to share price fall + profit due to put option Net value = 5+0-19+14 = 0 Our profit will remain same as $2 as we are getting this dividend