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FRM Part I · FRM Exam Part I · Exotic Options

A dealer hedges a short position in a down-and-out call (barrier below the strike) with the stock price just above the barrier. Which statement about dynamic delta hedging is correct?

Near the barrier, the down-and-out call's delta becomes very large and positive, because its value collapses to zero when the barrier is hit. This steep value change makes dynamic hedging costly and unstable, which is why static replication is often preferred.

  1. ADelta falls toward zero as spot nears the barrier, so the hedge needs almost no shares
  2. BDelta is the same as that of the equivalent vanilla call, so the hedge is unaffected by the barrier
  3. CDelta becomes very large and positive as spot nears the barrier, because the option value drops to zero at the barrier, so the hedge is costly and unstableCorrect
  4. DGamma is negligible near the barrier, so rebalancing is infrequent

Explanation

The knock-out call has positive value above the barrier but is worth zero once the barrier is touched, so its value changes steeply with spot near the barrier. This gives a large delta and large gamma, making dynamic hedging costly and prone to error. The vanilla delta ignores this discontinuity.

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