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FRM Part II · FRM Exam Part II · Arbitrage Pricing with Term Structure Models

A risk manager prices an option on a bond using a one-period binomial tree and a replicating portfolio. A colleague argues the price should be raised because the bond's true probability of an up move is higher than the risk-neutral probability used. Which response is correct?

The colleague is wrong. Replication weights depend only on the state payoffs, the bond's current and future prices and the risk-free rate. The bond's current price already reflects real-world probabilities and risk premia, so the option price does not change with the true up probability.

  1. AOnce the underlying bond's current price is taken as given, the true up-move probability does not affect the replication price, because replication already ties the option to that price.Correct
  2. BThe price should be raised, because a higher true up probability raises expected payoffs and replication must reflect expected payoffs.
  3. CThe price should be lowered, because a higher true up probability means the replicating portfolio holds fewer bonds.
  4. DThe price depends on the true probability only if the option is out of the money.

Explanation

The replicating portfolio's weights depend only on the state payoffs, the bond's state prices and the risk-free rate. The bond's observed price already embeds the true probabilities and risk premium. Any option price different from the replication cost would create arbitrage, so the true probability does not change it.

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