Question

Assume the risk-free is zero and the market prices for Apple stock options that expire in 1 year are as follows:

Strike Price $350.  $450

Call Premium $25. $75

Put Premium $15. $60

a) Specific the components and present cost of a long synthetic one-year forward agreement on Apple stock using only the options with strike price $350 in the table. The position should have a payoff at expiration that is identical to a 1-year forward agreement with a forward price of $350.

b) Using any of the option in the table, does an arbitrage opportunity exist? If so, specify the exact components of the portfolio you need to capture the arbitrage, and the exact arbitrage profit.

Answer 1

A - In this Question we have to create an Option strategy that would pay off similar to a Payoff in case of A Forward price agreement at $350.

Since this is a long position we are committing to purchase the stock at a certain price one year forward. So we are concerned with price rising. Hence either we enter into an agreement to purchase the stock 1 year forward at 350$ or we purchase a call option to have a right to be able to purchase the stock 1 year forward at the strike price that is 350$.

We will have to pay a premium of 25$ in that case which will be treated as a cost.

Now we have to check whether the payoff at expiration will be the same.

Assuming Price of Apple after 1 year is 400$.

In case 1- Forward rate Agreement - We are obliged to purchase the stock at 350$ and eventually we can sell the stock in the market at 400$ giving us a payoff of 50$

In case 2- Options- We have the right to purchase the stock at 350$ no matter the price. Since the price is over 350$ we will exercise the option and purchase the share for 350$. Again making a profit of 50$.

Hence We will use a Call Option with a Strike Price of 350$ at a cost of 25$

2-

Yes an Arbitrage Opportunity exists.

In order to capture the Arbitrage via Options we will:

Purchase the Call Option at a Strike Price of 350$ costing us 25$

Sell the Call Option at a Strike price of 450$ giving us 75$.

This will cap our profit to 100$ in case price goes above 450$

However further downside below 350$ is not capped for which we will purchase a put option.

Sell the Put option at 350$ giving us 15$

Purchase the Put Option at 450$ costing us 60$

So total Profit =75-25+15-60= 5$ This is our profit from the premium

Now assume the price at the end of 1 year is 550$

Payoff from different options will be as follows:

Purchased call Option at 350$ = 200$

Sold Call option at 450$ = -100$

Put Option will not give any Payoff since prices did not go below 450$

Total profit = 100$

Assume price is at 250$ after 1 year

Call options would not give nay payoff since price did not go above 350$

Put sold at 350$ = -100$

Put purchased at 450$ = 200$

Total profit = 100$

Assume Price is 400$ after 1 year. Payoff:

Call Purchased at 350$ = 50$

Call sold at 450$ = 0

Put Sold at 350$= 0

Put Purchased at 450$= 50$

Total profit = 100$

So Arbitrage Profit = 5$