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Work out the profit and loss on a position in options, using the payoff function to explain it.

FRM® Part I · Financial Markets and Products · Options Markets · FRM1.FM.12.b

The idea

Work from the payoff at expiry, then subtract the premium paid or add the premium received. For one share, a long call pays max(S − K, 0) and a long put pays max(K − S, 0), where S is the share price at expiry and K the strike. The writer's payoff is the negative of the holder's. For the buyer, Profit = Payoff − Premium. For the writer, the profit is the premium received less what the holder is paid, so Profit = Premium − Holder payoff. Setting the buyer's profit to zero gives the breakeven price. Breakeven = K + Premium for a long call and K − Premium for a long put. A buyer's loss is capped at the premium, while a short call has unlimited loss and a short put loses at most K minus the premium. For combinations such as spreads, covered calls or straddles, add the payoffs and the net premium of every leg. Multiply by the shares per contract, usually 100, and by the number of contracts. Forgetting either multiplier is a common error.

Formulas

Profit on a long call at expiry

profit = max(S − K, 0) − premium

S
the share price at expiry
K
the strike
premium
the premium paid

Per share: multiply by the shares per contract, usually 100, and the number of contracts. Breakeven is K + premium; the writer's profit is the negative.

Profit on a long put at expiry

profit = max(K − S, 0) − premium

S
the share price at expiry
K
the strike
premium
the premium paid

Breakeven is K − premium. The writer's worst loss is K less the premium, at a share price of zero.

A practice question

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

An investor buys a call option with strike 50 for a premium of 3. At expiry the stock is at 52. What is the profit per share?

  1. Loss of 1.00 - correct
  2. Gain of 2.00Why not: Ignores the premium paid; the payoff of 2 must be reduced by the 3 premium.
  3. Gain of 5.00Why not: Adds the premium to the payoff instead of subtracting it.
  4. Loss of 3.00Why not: Assumes the option expires worthless and loses the whole premium, but the call is in the money and pays 2.

Payoff = max(52 - 50, 0) = 2. Profit = payoff - premium = 2 - 3 = -1 per share, a loss reduced from the full premium.

A harder one

An investor buys 5 put contracts (100 shares each) with strike 60 at a premium of 4.20 per share. At expiry the stock is at 51. What is the total profit?

  1. USD 4,500Why not: Ignores the premium paid; the payoff of 9 per share must be reduced by 4.20.
  2. USD 2,400 - correct
  3. USD 480Why not: Uses only one contract (100 shares) instead of 5 contracts, i.e. 500 shares.
  4. Loss of USD 2,100Why not: Uses the call payoff max(S - K, 0) = 0, so only the premium is lost.

Payoff per share = max(60 - 51, 0) = 9. Profit per share = 9 - 4.20 = 4.80. Total = 4.80 x 100 x 5 = USD 2,400.

The hardest tier

A trader buys a straddle on 10 contracts of 100 shares each: a call and a put, both with strike 100, costing 5.50 and 4.50 per share. At expiry the stock is at 88. What is the total profit?

  1. USD 2,000 - correct
  2. USD 12,000Why not: Ignores the total premium of 10 per share paid for both options.
  3. USD 7,500Why not: Deducts only the put premium (4.50) and forgets that the call premium was also paid.
  4. Loss of USD 10,000Why not: Assumes both options expire worthless and ignores the put's payoff of 12 per share.

The call expires worthless; the put pays 100 - 88 = 12. Premium paid is 5.50 + 4.50 = 10. Profit per share = 2, so total = 2 x 1,000 shares = USD 2,000.

In the app: 6 practice questions on this outcome, spaced review that brings it back before you forget it.

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