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

A stock trades at 100. After one year it will be either 110 or 90. The one-year risk-free rate is 5% with annual compounding. A one-year European call with strike 100 is to be priced by replication, using the stock and risk-free borrowing or lending. The real-world probability of an up move is 50%. What is the call's no-arbitrage value today?

The call is worth about 7.14. Replicate it by buying 0.5 shares and borrowing 42.857 at 5%, which costs 50 minus 42.857. The result matches the risk-neutral calculation with probability 0.75. The real-world 50% probability is irrelevant.

  1. A7.14Correct
  2. B4.76
  3. C7.50
  4. D50.00

Explanation

Call payoffs are 10 (up) and 0 (down). Hedge ratio = 10/(110-90) = 0.5 shares. Borrowing B satisfies 0.5×90 + 1.05B = 0, so B = -42.857. Value = 0.5×100 - 42.857 = 7.14. Check: q = (105-90)/20 = 0.75, and 0.75×10/1.05 = 7.14. Using the real-world 50% probability gives 4.76, which is wrong.

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