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

Use the tree where the annual short rate is 4% today and moves to 8% or 2% after one year, with risk-neutral probabilities of 50% each. A European call option expires in one year on a one-year zero-coupon bond with face value 100 and a strike price of 95. What is its no-arbitrage value today, which equals the cost of a portfolio of the one-year and two-year zeros replicating it?

The call is worth about 1.46. It pays nothing when rates rise to 8% and about 3.04 when rates fall to 2%, because the bond is then worth 98.04 against a 95 strike. Weighting by 50% and discounting at 4% gives 1.46.

  1. A1.46Correct
  2. B1.52
  3. C2.92
  4. D0.73

Explanation

At expiry the one-year zero is worth 100/1.08 = 92.59 in the up state and 100/1.02 = 98.04 in the down state. The call pays 0 in the up state and 98.04 - 95 = 3.04 in the down state. The value today is 0.5×3.0392/1.04 = 1.461. Option 1.52 forgets to discount, 2.92 omits the probability weight, and 0.73 applies the 50% weight twice.

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