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FRM Part I · FRM Exam Part I · Exotic Options

A digital (cash-or-nothing) call on an index pays 1,000,000 USD at expiry if the index is above 4,500 and zero otherwise. At expiry the index is 4,501 for one scenario and 4,499 for another. Near expiry with the index close to 4,500, what is the key hedging difficulty for the seller?

The seller faces a very large and unstable delta near the strike close to expiry, because the payoff jumps from zero to the full amount. This forces large, frequent rebalancing and makes hedging costly, unlike a standard option with a continuous payoff.

  1. AThe delta becomes very large and unstable near the strike close to expiry, making hedging costlyCorrect
  2. BThe gamma is always zero so no rehedging is needed
  3. CThe payoff is linear in the index, so a static position in the index hedges it
  4. DThe option has negative vega at all index levels

Explanation

The payoff jumps from 0 to 1,000,000 across the strike, so near expiry delta spikes and changes sign of its slope abruptly, requiring large, frequent rebalancing. Gamma is not zero and the payoff is not linear. Vega can be positive or negative depending on moneyness, not always negative.

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