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Outline the advantages and disadvantages of a stop-loss hedging strategy, and explain why using it can result in naked and covered option positions.

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

The idea

A stop-loss strategy hedges a written call by buying the stock when its price rises above the strike K and selling it when the price falls below K. The writer is therefore covered when the option is in the money and naked when it is out of the money. Its appeal is simplicity, because it needs no volatility estimate. In the ideal case the cost of the hedge is about Cost = max(S0 − K, 0). That is below the option price. The flaw is that trades cannot happen exactly at K. Slippage, discrete monitoring and interest make the real cost uncertain and often large, and a price hovering around K triggers repeated trades. Hedging results are therefore poor and highly variable. The exam point is that stop-loss is easy but unreliable, and delta hedging replaces it.

Formulas

Ideal cost of a stop-loss hedge of a written call

Cost = max(S0 − K, 0)

S0
the stock price when the hedge starts
K
the strike price

Assumes every trade happens exactly at K with no interest or costs. Because this is below the option price, it shows the ideal is unreachable rather than that options are overpriced.

A practice question

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

A bank has written a call option with strike K and uses a stop-loss strategy. Which describes the bank's holding in the underlying?

  1. It holds the shares while the price is below K and holds none while it is above K.Why not: Reversed: the writer needs the shares when the call is in the money (price above K).
  2. It always holds a fraction of one share equal to the option's delta.Why not: That is delta hedging, which adjusts the holding continuously rather than switching between all and nothing.
  3. It holds shares equal to the premium received divided by the strike.Why not: No such rule exists; the stop-loss rule is a switch driven only by whether the price is above or below K.
  4. It holds the shares while the price is above K and holds none while it is below K. - correct

The strategy aims to be covered when the call is in the money and naked when it is out of the money. It buys the stock when the price rises through K and sells when it falls through K.

A harder one

A bank uses a stop-loss strategy on a written call with strike USD 50. It starts with no shares (stock at 48). The price then moves to 51, 49, 52, 49 (each crossing the strike). How many trades are made and what is the final position?

  1. Four trades; covered at the endWhy not: The count is right, but the last crossing is downward, which means selling the shares, so the bank ends naked.
  2. Two trades; naked at the endWhy not: Counts only the purchases on upward crossings and forgets the sales on the downward crossings.
  3. Four trades; naked at the end - correct
  4. Three trades; covered at the endWhy not: Miscounts the crossings; the price crosses 50 four times (up, down, up, down).

Buy at the crossing to 51, sell at 49, buy at 52, sell at 49: four trades. The last move leaves the price below 50, so the shares are sold and the position is naked. Each crossing triggers a trade, which is why a price hovering around K causes heavy trading.

The hardest tier

A bank has written 50,000 calls with strike USD 50 on a stock at USD 49 and uses stop-loss. Because of slippage it buys at USD 50.10 on each upward crossing and sells at USD 49.90 on each downward crossing. The price rises through 50, falls through 50, rises through 50 and falls through 50 again and ends at USD 47. Ignoring interest, the total hedging cost is closest to:

  1. USD 10,000Why not: Counts only the loss on the buys (0.10 each) or only on the sells, and misses the other half of each round trip.
  2. USD 40,000Why not: Charges 0.20 on each of the four trades instead of 0.20 per round trip (0.10 per trade).
  3. USD 20,000 - correct
  4. USD 0Why not: Takes the ideal cost for an out-of-the-money start and ignores slippage; the real cost is not zero.

Two round trips, each buying at 50.10 and selling at 49.90, lose 0.20 per share. Total = 2 x 0.20 x 50,000 = 20,000. Without slippage the ideal cost is zero, so the whole cost comes from imperfect execution at the strike.

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