i am very confused by question F on page 90 of the CFA Derivs book. The questions deals with the situation where the underlying stock is priced at 40, and a call option with a strike price of $40 is selling for $7. You buy the stock and sell the put. What is the payoff when the stock price is $52 at expiration?

The book’s solution is 52-Max(0,52-40)=40. I understand the Max piece of it – that is calculating the payoff of the call option. However, I can’t seem to be able to interpret the formula as a whole. In addition,I am confused by the fact that in this case, the person is writing an option, rather than buying it. The reading itself did not cover calculation of payoff for a case such as this. Would someone please be able to help?

I assume that this is a typo and that you’re buying the stock and selling a call option, not a put option.

There are two parts to the payoff: the payoff on the stock and the payoff on the option. The payoff on the stock is easy: $52. The payoff on the option is Max(0, S_{0} – X) when you are _ long _ the call option, so it’s the negative of that when you’re short the call option. (The payoff of a short option is always the negative of the payoff of the corresponding long option: if the buyer wins, the seller loses.) Thus, the payoff on the portfolio (covered call) is $40 (= payoff on stock + payoff on option = $52 – Max($0, $52 – $40)).

Just wondering, when you calculate the payoff from the stock, why don’t you substract the original purchasing price of the stock 40? If the stocks goes to $52, at expiration, the payoff from the stock should be 52-40, no?

In a covered call, if the price of the underlying rises above the strike price of the call option, the option is exercised, you receive the strike price in cash, and you hand over the underlying. There is only one cash flow.

Can I say, similarly, in a protective put, if the price of the underlying falls below the strike price of the put option, the option is exercised and the cash flow is the strike price (the price at which the underlying is handed over to put seller) ?

Is the profit -$5 or $7? Would you include the capital gain of the stock as a loss?

I think it is $7. Bought the stock for $40, sold it for $40 plus premium from the option? You wouldn’t need the $52 dollars in the calculation, would you?

Your payoff is just the option premium of PLUS $7 received from the buyer of the call since you sell a covered call (if you sell a call you limit your upside risk by buying the stock at strike).

If you simplify it: stock moves from 40 to 52 (your payoff both upside and downside is theoretically unlimited with a stock) but in this case it is PLUS $12. However, at expiration the underlying (on which you wrote the call) also has un unlimited payoff on the downside (not on the upside, from a sellers point of view) so when the buyer of the call exercise his/her option you have to buy the underlying in the market for $52. However, since you bought the stock at the same time as you wrote the call your payoff is MINUS $12. This leaves you with a payoff of PLUS $7 (PLUS 12 MINUS 12 PLUS 7 = 7), which is equal to the option premium received. However, if you hadn’t bought the stock and just sold a naked call than your payoff would be MINUS $5 ONLY if the stock is exercised at $52 at time of expiration.

If, however, it should be, as written in the question, that it is a written put, than your payoff would be PLUS 12 PLUS 7 = PLUS 18, because the payoff is limited on the upside by the premium received.

Just to be clear on the language: the premium paid is not considered part of the payoff. The term “payoff” means the cash flow at the expiration of the option. The premium will be included in computing the profit, but not in computing the payoff.

For example, if you bought a call option for $8, with a strike price of $50, and at expiration the stock is selling for $65, then the payoff is $15 (= $65 – $50): the cash flow at expiration is -$50 (you exercise the option and buy the stock at the strike price) plus $65 (you sell the stock at the market price). The profit is $15 – $8 = $7.

Make sure that you understand these terms. On the exam, if they ask for the payoff, one of the wrong answers will be the profit; if they ask for the profit, one of the wrong answers will be the payoff.