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Recognise the payoffs on option and forward contracts and compute them.

FRM® Part I · Financial Markets and Products · Introduction to Derivatives · FRM1.FM.4.d

The idea

At maturity a long forward with delivery price K pays S − K, where S is the final price of the underlying. A short forward pays K − S. Forward payoffs can be positive or negative, and they are linear in S. A long call pays max(S − K, 0) and a long put pays max(K − S, 0). The writer of either option pays the negative of that amount, so option payoffs are kinked at the strike. Profit is the payoff minus the premium paid for the buyer, and the premium received minus the payoff for the writer. The usual mistakes are using the call formula for a put, flipping the sign for the writer, and forgetting the premium when asked for profit rather than payoff. Multiply by the contract size at the end.

Formulas

Payoff to a long forward at maturity

Long forward payoff = S − K

S
the price of the underlying at maturity
K
the delivery price

Linear and can be negative. The short receives K − S.

Payoff to a long call at expiry

Long call payoff = max(S − K, 0)

S
the price of the underlying at expiry
K
the strike price

This is payoff, before the premium. The writer's payoff is its negative.

Payoff to a long put at expiry

Long put payoff = max(K − S, 0)

S
the price of the underlying at expiry
K
the strike price

Strike minus price, the reverse of the call. Applying the call formula to a put is the usual error.

A practice question

From the question bank, with the reason each wrong option is wrong.

A European call option has a strike price of 80. At expiry the underlying price is 95. What is the payoff to the holder of the call, before deducting the premium?

  1. 0Why not: Uses the put payoff max(K - S, 0) = max(80 - 95, 0) = 0.
  2. -15Why not: Gives the payoff of the writer, the short side, rather than the holder.
  3. 95Why not: Reports the value of the underlying rather than the payoff over the strike.
  4. 15 - correct

Call payoff = max(S - K, 0) = max(95 - 80, 0) = 15. The holder exercises because the option is in the money.

A harder one

An investor writes a European put with a strike of 60 and receives a premium of 3.20. At expiry the underlying is at 52. The writer's profit per unit is closest to:

  1. -8.00Why not: Reports the payoff and forgets the premium received.
  2. -4.80 - correct
  3. -11.20Why not: Subtracts the premium (-8.00 - 3.20) instead of adding it; the writer receives it.
  4. +3.20Why not: Assumes the put expires worthless and the writer keeps the premium, but it is in the money at 52.

Put payoff to writer = -max(60 - 52, 0) = -8. Profit = -8 + 3.20 = -4.80.

The hardest tier

An investor buys 1,000 calls at a premium of 4.50 and 1,000 puts at a premium of 3.75, all with a strike of 100 and the same expiry, on one share each. At expiry the share price is 86. The investor's net profit, ignoring interest, is closest to:

  1. USD 14,000Why not: Reports the put payoff of 14 per share times 1,000 and ignores both premiums.
  2. USD 22,250Why not: Adds the premiums to the payoff instead of deducting them.
  3. USD -8,250Why not: Assumes both options are lost, so only the premiums matter; the put is in the money.
  4. USD 5,750 - correct

Call payoff = 0; put payoff = 100 - 86 = 14. Premiums = 4.50 + 3.75 = 8.25. Profit per share = 14 - 8.25 = 5.75; times 1,000 = 5,750.

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