Question

5. The price of a stock is $40. The price of a one-year European put option on the stock with a strike price of $30 is quoted as $7 and the price of a one-year European call option on the stock with a strike price of $50 is quoted as $5. Suppose that an investor buys 100 shares, shorts 100 call options, and buys 100 put options. a. Construct a payoff and profit/loss table b. Draw a diagram illustrating how the investor’s payoff and profit or loss at expiation.

Answer 1

We are given the following information:

Security	Stock	Call	Put
Price	40	5	7
Strike Price		50	30
Position	100	-100	100

• We are long the stock and the put and short the call so position 100 for stock and put is positive and -100 for call
• If the price of stock is lower than 40 then we will make a loss as we purchased for 40 and if the price is higher than 40 then we gain
• Now if the price at the time of expiry is greater than 50 then the call buyer will exercise it but otherwise we will pocket the premium
• If the price is below 30 then we will exercise the put option otherwise we will lose the premium paid

Keeping this in mind following is the table for payoff and profit for various assumed prices:

Price at Expiry	Payoff				Profit
	Stock	Call	Put	Total	Stock	Call	Put	Total
0	-4000	0	3000	-1000	-4000	500	2300	-1200
20	-2000	0	1000	-1000	-2000	500	300	-1200
25	-1500	0	500	-1000	-1500	500	-200	-1200
30	-1000	0	0	-1000	-1000	500	-700	-1200
35	-500	0	0	-500	-500	500	-700	-700
40	0	0	0	0	0	500	-700	-200
45	500	0	0	500	500	500	-700	300
50	1000	0	0	1000	1000	500	-700	800
55	1500	-500	0	1000	1500	0	-700	800
60	2000	-1000	0	1000	2000	-500	-700	800
65	2500	-1500	0	1000	2500	-1000	-700	800

• Lets understand a few values:
  • Stocks:
    • When stock price is 25, payoff is negative on 100 shares and equals 100 x (25-40) = -1500
    • When stock price is 40, payoff is 0 on 100 shares and equals 100 x (40-40) = 0
    • When stock price is 55, payoff is positive on 100 shares and equals 100 x (55-40) = 1500
    • Stock payoff and profit is the same
  • Call:
    • When stock price is less than 50 for eg 25, payoff is 0 because the call is not exercised by the buyer and the entire premium is our profit =100 x 5 = 500
    • When stock price is 50, payoff is 0 because the whether call exercised or not exercised there is no benefit to the buyer and the entire premium is our profit =100 x 5 = 500
    • When stock price is greater than 50 for eg 65, payoff is -100 x (65-50) = -1500 because the buyer exercises the call and we have to oblige as we are short on it and the premium is reduced from the payoff to calculate our profit =-1500+100 x 5 = -1000
  • Put:
    • When stock price is less than 30 for eg 25, payoff is 100 x (30-25)=500 because the put exercised by us and the premium is reduced from the payoff to calculate our profit =500-100 x 7 =-200 because we paid the premium to get this option
    • When stock price is 30 or more, payoff is 0 because the whether put is exercised or not we have no benefit and the entire premium is our loss =100 x -7 = -700
• Total payoff is calculated by summing up the payoff for each security at any price
• Total profit is calculated by summing up the profit for each security at any price

We need to show the payoff and profit diagrams for each security and then the total payoff and profit diagram too: