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FRM Part I · FRM Exam Part I · Exotic Options

An asset-or-nothing call and a cash-or-nothing call have the same strike K and expiry. The cash-or-nothing call pays USD K if the asset finishes above K. A trader buys the asset-or-nothing call and sells the cash-or-nothing call. At expiry, what is the payoff of this portfolio?

The portfolio replicates a vanilla European call. Above the strike it receives the asset value and pays K, netting S_T minus K; below the strike both binaries expire worthless. This decomposition shows a vanilla call equals an asset-or-nothing call minus K cash-or-nothing calls.

  1. AS_T - K if S_T > K, otherwise 0, replicating a vanilla European callCorrect
  2. BK - S_T if S_T < K, otherwise 0, replicating a vanilla European put
  3. CS_T if S_T > K, otherwise 0
  4. DS_T - K for all values of S_T

Explanation

If S_T > K, the portfolio pays S_T - K; otherwise both pay zero. This is exactly a vanilla call payoff. The put payoff would need the asset-or-nothing put, and the third option ignores the short cash leg.

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