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

A risky asset trades at 50 today and will be worth either 60 or 40 in one year. The riskless rate is 5% per year. A derivative pays 20 if the asset is at 60 and 0 if it is at 40. Using a replicating portfolio of the asset and riskless borrowing or lending, what is the no-arbitrage value of the derivative today?

The derivative is worth about 11.90. Buying one unit of the asset and borrowing 40 discounted at 5%, or 38.10, reproduces the payoffs of 20 and 0, so the cost is 50 minus 38.10. Using 50% probabilities or skipping discounting gives the wrong answers.

  1. A11.90Correct
  2. B12.50
  3. C9.52
  4. D10.00

Explanation

The replicating portfolio holds 20/(60-40) = 1 unit of the asset and borrows an amount whose repayment is 40, so the borrowing is 40/1.05 = 38.095. The value is 50 - 38.095 = 11.905. Check: the risk-neutral probability q solves 52.5 = 60q + 40(1-q), so q = 0.625 and the value is 0.625×20/1.05 = 11.905. Option 12.50 forgets to discount. Option 9.52 wrongly uses q = 0.5.

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