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Assess the risks that naked and covered option positions carry, and describe them.

FRM® Part I · Valuation and Risk Models · Option Sensitivity Measures: The “Greeks” · FRM1.VR.16.a

The idea

A naked option position is a short option with no offsetting holding in the underlying, while a covered position pairs the short option with the underlying. A naked call writer earns only the premium but faces a loss that is theoretically unlimited, because the stock can rise without bound. A naked put writer's loss is large but capped at the strike less the premium, since the price cannot fall below zero. A covered call (long stock, short call) caps the proceeds at the strike plus the premium received. It still bears nearly the full downside of the stock, cushioned only by the premium. By put-call parity its payoff has the same shape as a short put at the same strike. The usual mistake is to call a covered call risk free, since covering removes the rally risk, not the market risk.

A practice question

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

A dealer writes a European call option on a stock and holds no shares and no other position in it. Which description of this position's risk is correct?

  1. Its loss is capped at the strike price less the premium, because the stock cannot fall below zero.Why not: Describes the naked put writer, whose loss is bounded; a naked call writer has no upper bound on the stock price.
  2. Its profit is unlimited as the stock rises, while its loss is capped at the premium.Why not: Reverses the position: this is the profile of the call buyer, not the writer.
  3. Its risk is eliminated because the premium received exceeds the option's intrinsic value at inception.Why not: Premium received at inception does not protect against later price rises; the short call remains fully exposed.
  4. Its profit is capped at the premium received, while its loss is theoretically unlimited as the stock price rises. - correct

A naked call writer collects a fixed premium but must deliver stock at the strike whatever the market price. With no shares owned, a rally costs the writer the full difference between the market price and the strike, with no ceiling.

A harder one

A bank writes 10,000 European calls with a strike of USD 40 for a premium of USD 3.00 each and holds no shares. At expiry the stock is USD 48. Ignoring discounting and costs, what is the bank's profit or loss on the position?

  1. A loss of USD 80,000Why not: Ignores the premium received: 10,000 x (48 - 40) = 80,000 is only the payoff, not the net result.
  2. A loss of USD 110,000Why not: Adds the premium to the payoff instead of subtracting it from the loss: 80,000 + 30,000.
  3. A profit of USD 30,000Why not: Assumes the call expired worthless and keeps only the premium, but at 48 the call is in the money and is exercised.
  4. A loss of USD 50,000 - correct

Payoff owed = 10,000 x (48 - 40) = 80,000. Premium received = 10,000 x 3.00 = 30,000. Net = 30,000 - 80,000 = -50,000. With no shares in hand, the bank must buy at the market price to deliver.

The hardest tier

A dealer writes 100,000 European calls with a strike of USD 50 on a stock priced at USD 49, for a premium of USD 4.00 each, and the buyer exercises when the stock finishes at USD 60. Ignoring discounting, which pair of results is correct for (naked position, covered position in which the dealer bought 100,000 shares at 49 at the outset)?

  1. Naked: loss of USD 600,000; covered: gain of USD 500,000 - correct
  2. Naked: loss of USD 1,000,000; covered: gain of USD 100,000Why not: Ignores the premium of 4.00 in both cases: -(60 - 50) and +(50 - 49) per share.
  3. Naked: loss of USD 1,400,000; covered: loss of USD 300,000Why not: Subtracts the premium instead of adding it: -(10 + 4) and (1 - 4) per share.
  4. Naked: loss of USD 600,000; covered: gain of USD 1,500,000Why not: Credits the covered writer with the full stock gain of 11 plus the premium, forgetting that the shares are delivered at 50.

Naked per share: 4 - (60 - 50) = -6, i.e. -600,000. Covered per share: premium 4 + (50 - 49) from delivering shares bought at 49 at the 50 strike = +5, i.e. +500,000. Owning the shares removes the cost of a rally but caps the gain at the strike.

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