Question

You own one call option contract on Dell Computer with an exercise price of $45 and a remaining maturity of three months. The current price of Dell is $50 per share and the risk-free rate of interest is 5 percent per year with continuous compounding. Dell does not pay dividends and does not expect to do so in the near future. Answer the following questions.

a. Prove that it would never be optimal to exercise your Dell call option contract now if you can sell the contract in the market.

b. What if you can’t sell the option (for whatever reason) and you know (don’t ask me how) that Dell’s stock price will be much lower in the future. Should you exercise the option now and take the profit or wait until the expiration date of the option contract? Provide a detailed and rigorous answer to this question.

Answer 1

a) There are two cases here

i) One wants to hold the Dell Stock after three months : In this case, it is better to not exercise the option and pay $45 right now. By exercising the option after three months, one has to pay $45 only but later and for three months this amount of $45 can be used elsewhere.

ii ) One does not want to hold the stock after 3 months: In this case, it is better to sell the call option in the market rather than exercising it. By exercising, one can benefit by only $5 whereas the option will clearly be more valuable than $5 in the market as it also has some time value.

So, it is not optimal to exercise the call option contract if the option can be sold in the market

b) If one can't sell the option and the price is going to be lower in the future, in the two cases

i) One wants to hold the Dell Stock after three months : one can immediately exercise the contract buy the stock for $45 and sell it in the market at $50 to make a profit of $5 and invest this amount at the risk free rate.

After three months , one can get 5*e^(0.05*3/12) =$5.062892 and buy the stock at a lower price

ii ) One does not want to hold the stock after 3 months:

One can immediately exercise the contract buy the stock for $45 and sell it in the market at $50 to make a profit of $5 as later ,the price as well as the option value will decrease , so it is better to exercise now.

So, it is not optimal to wait until expiry if one knows that the price is going to be lower and the option cannot be sold