Question

A protective collar involves buying an out-of-the-money put and writing an out-of-the-money call on an underlying asset that you own. Let’s say you own an S&P 500 index security that is currently trading at $30/share. You bought the index at $10 per share so you currently have a big capital gain. You don’t want to sell your shares, but you want to lock in your profits with a protective collar for the next year. You want to make sure you can sell your shares for at least $29/share. The standard deviation of returns on the S&P 500 is 20% and the assume risk free rate is 2%.

a) How much would it cost to buy the put?

b) What strike price should you set on the call so that you make the same premium that you paid for the put (zero net cost)?

c) Draw the net profit diagram for the entire protective collar position (long stock, long put, short call) at maturity (including the premiums). Carefully label in detail all axis, strike prices, payoffs, etc.
 

Answer 1

Part (a)

The value of a put option as per BS formula is as shown below:

Please see the put pricing methodology below.

Please see the table below. Figures in parenthesis, if any, mean negative values. All financials are in $. Adjacent cells in blue contain the formula in excel I have used to get the final output.

Part (b)

After setting up the formula for cll option, we will have to do a goal seek. Please see the snapshot below:

The goal seek gives $ 33 as strike price on the call so that you make the same premium that you paid for the put (zero net cost).

Part (c)

Stock Price	Long Stock	Long Put	Short Call	Net payoff
S	A = S - 10	B = max (K - S, 0) - P = max (29 - S, 0) - 1.6235	C = -max (S - K, 0) + P = - max (S - 33, 0) + 1.6235	A + B + C
10	0	17.3765	1.6235	19
15	5	12.3765	1.6235	19
20	10	7.3765	1.6235	19
25	15	2.3765	1.6235	19
30	20	-1.6235	1.6235	20
35	25	-1.6235	-0.3765	23
40	30	-1.6235	-5.3765	23
45	35	-1.6235	-10.3765	23
50	40	-1.6235	-15.3765	23

And the net payoff diagram is: