A stock is presently selling at a price of $50 per share. After 1 time period, its selling price will be ...?

Question

either $150 or $25. An option to purchase y units of the stock at time one can be purchased at the cost of cy. Assume that the strike price is $125.

a) what should c be in order for there to be no sure win?

b) If c=4, explain how you can guarantee a sure win

Answer 1

We have to make a few simplifying assumptions to solve the problem.

(1) There is no chance that an ex-dividend date will occur before expiration.

(2) There is no chance that a short stock position will have to be covered prior to expiration.

(3) Transaction costs are insignificant and not included.

(4) Long stock plus short option combinations are not being considered since they would require unrealistic option pricing.

With those assumptions, the only way to make a profit with the stock at $25 is if we have a short position in the stock, so we immediately know that we short some number of shares, and in order to make a profit with the stock at $150 we have to have enough options to overcome the $100 per share loss the short stock position will experience if the shares go to $150. Since the lower limit for the option price is zero, the maximum profit per share possible from the options is $25 per share. Consequently we need options on at least four times as many shares as the number of shares we sold short.

Also, since the with profit from the short stock position is $25 if the stock closes at $25, we know the cost of the options must not exceed $25 per share shorted.

<<<a) what should c be in order for there to be no sure win?>>>

To be sure there is no sure win the option needs to be priced at $6.25 ($25.00 / 4) or higher.

<<<b) If c=4, explain how you can guarantee a sure win>>>

Sell n shares short

buy options on 5n shares

For example, short 100 shares for $5,000 and buy options on 500 shares for $2,000.

Your net credit to open the position is $5,000 - $2,000 = $3,000

If the stock is at $25 after one time period the options will expire worthless and it will cost $2,500 to buy the stock to cover.

If the stock is at $150 the options will have an intrinsic value of $25 per share, or $12,500, and it will cost $15,000 to cover the short stock position. That makes the total cost to close both positions $15,000 - $12,500 = $2,500.

Thus with the stock at either $25 or $150 it will cost $2,500 to close the position. Since the initial credit was $3,000 the net profit will be $500.