How to calculate break even price on delta hedged portfolio?

Question

The underlying currently trades at 10. Suppose you have a portfolio consisting of:

• 1 Call at strike 11 with premium 0.4
• A short position of 25 shares

Calculate the break even prices of this portfolio.

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My attempt to answer:

I was not able to answer this. I examined some cases just to confirm there are 2 break even prices. However, I am only able to use trial and error. I want to know what other methods there are.

So the short position is $+25*10 = +250$. You also paid $100 * -0.4 = -40$ for the single call option. If the value of the shares drop, you can buy them back cheaper and the call expires worthless. If the stock price increases, the loss on the short sale can be compensated by exercising the option. This only happens when the stock moves up or down enough. Small movements lead to a loss. For example, here are some cases:

• Case 1 - No movement:

PnL becomes loss on option premium (40) and no effect on stock (because you can close your position, i.e. buy back stock, at 0 gain/loss).

• Case 2 - Stock moves to 12:

If the stock becomes 12 dollar, you lose 2 dollar per stock on the short position, because you need to buy them back more expensive to close the position. That is, $-2 * 25 = -50$. The call is now in the money and worth $100*(12 - 11) = 100$ but you already spend 40 on the premium, so the profit on the option position is $100-40 = 60$. The profit of the portfolio is then 60 (profit on option) minus 50 (loss on stock): $60 - 50 = 10$. However, I want to find the break even price.

• Case 3 - stock moves to 11:

If the stock becomes 11 dollar, you lose 1 dollar per stock on the short position. That is, $-1 * 25 = -25$. The call is at the money, so there is no reason to exercise. Therefore you lose on the stock and the option premium cost of 40. Total PnL is $-25 - 40 = -60$

• Case 4 - stock moves to 8:

If the stock becomes 8 dollar, you gain 2 dollar per stock in the short position, because you can buy them back cheaper to close the position. That is, $+2 * 25 = +50$. The call is now out of the money and worthless, so you do not exercise and lose the option premium of 40. The profit on the total portfolio is then +50 (profit on stock) minus 40 (loss on option): $50 - 40 = 10$.

• Case 5: stock moves to 9:

Gain 1 per stock: $1 * 25 = 25$. Option expires worthless and the premium cost was 40. Total Pn: $25 - 40 = -15$

Answer 1

Normally, we don't help with anything that looks like a homework problem. However, you've made an attempt to figure this out so here's a very long answer for what takes mere minutes to resolve in a spreadsheet.

The easiest solution would be to set up a spreadsheet. The columns would include

(A) possible future share price

(B) the P&L on the short shares at these various prices, and

(C) the P&L of the call at these various prices.

Then, a final column (D) that sums (B) and (C) and your breakeven points will be where this total equals zero.

You could also look at this algebraically. To the downside, your short shares have to recover the $40 cost of the call. So the question is, at what future price P will the short shares gain $40? That equation would be

-25*(P -10) = +$40

The upside breakeven is a bit more complicated because the price of the short sale of the shares is different than the strike price of the long call. Now the question is, at what price P would the call make up its $40 cost plus the loss due to shorting 25 shares at 10? Those components would be:

(X) -25*(P -$10)

(Y) -$40 + (P - $11)*100

The first equation (X) represents the gain or loss of the short shares at future price P

The second equation (Y) represents the gain or loss of the long call above its strike price of $11. Below $11, the call would be worthless and the loss would be -$40. In a spreadsheet, this would be an if/then formula where there would be two imbedded formulas (if < $11 then... as well as if > $11 then...).

Now, putting it all together, the short shares will have a loss at a price above $10. The long call will have an expiration gain above $11.40 . Since one is negative and the other positive, we have to change the sign of one side. Therefore:

+25*(P -$10) = -$40 + (P - $11)*100

Solve that and you have the upside breakeven.