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

A risk analyst prices a derivative on a one-period binomial interest rate tree by building a portfolio of a risky bond and cash that matches its payoff in both states. A colleague then argues that the analyst's real-world forecast of the probability of the up-move should be raised from 40% to 60%, and that the derivative's price should rise accordingly. Assuming the replicating portfolio's securities are priced in the market, what is the correct response?

The price is unchanged. A derivative that can be replicated must be priced at the cost of its replicating portfolio, which depends only on state payoffs and traded security prices. Real-world probabilities do not enter the arbitrage-free price, so revising the up-move probability has no effect.

  1. AThe price is unchanged, because the replicating portfolio's cost fixes the price whatever the real-world probability of the up-moveCorrect
  2. BThe price rises, because the expected payoff of the derivative increases with the up-move probability
  3. CThe price falls, because a higher up-move probability raises the discount rate applied to the payoff
  4. DThe price rises only if the derivative's payoff is higher in the up state than in the down state

Explanation

Under no-arbitrage, the derivative must cost the same as the portfolio that replicates its payoff in each state. The replicating weights depend only on the state payoffs and the market prices of the traded securities, not on real-world probabilities. Changing the subjective probability therefore does not change the arbitrage-free price.

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