short call option

CFA Level II
#1

SHORT CALL

Why is payoff negative when the call option is in the money S>X?

payoff = max(0, S-X) ??

why is my logic so screwed I understand the diagram with P/L as y axis

This graph uses payoff as y axis, in the CFAI books they use ‘value’ and it shows negatives.

#2

In this case, the payoff is negative to the writer (short position), because the option is in the money to the call holder. The long position (holder) will have a positive payoff.

The payoff is negative for the call writer (person who shorted the call). This can be represented by inversing the payoff function of the long position in the call. Remember, the long position in the call option has a payoff = max (0, S-X), and the short position has an inverse function which would be the payoff = min (0, X-S) [this is the same as -Max(0, S-X)] . Zero sum game.

Quick note: I’m using X as the strike, where they are using K.

#3

The combination of a long call and a short call is a zero-sum portfolio: whatever the long gains, the short loses, and vice versa. (Think of writing a call option and selling it to yourself. Charge a high premium: I hear that the buyer’s a sucker.) When the call is in the money the long gains (dollar for dollar), so the short loses (dollar for dollar).

Being able to draw payoff diagrams for options (long, short, calls, puts) is a valuable skill; I encourage you to hone it. It will be especially useful at Level III.

#4

Much appreciated thank you. The y axis is ‘value from writer’s perspective’ and zero sum is a good way to think.

So just repeating

  • call payoff = max (0,S-X)
  • put payoff = max(0,X-S)

but in a short call payoff = - max(0, S-X )

thanks

#5

Correct. If you ever get stuck, solve it from the long position. Then, adjust your answer if needed.

For example: payoff for a call writer when the strike price is 5 and the stock price is 15.

Well, for the long position, the payoff is max(0,15-5)= 15-5=10.

Since the call writer has the opposite position and we know this is a zero sum game, the writer’s payoff must be -10. You’ll see that this is the same as min (0, X-S) and -max (0,S-X).

Hope this helped.