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

A three-period binomial tree for the one-period rate has r0 = 4%. At t=1 the rate is 5% (up) or 3% (down). At t=2 the rate is 6% (up-up), 4% (up-down or down-up, recombined) or 2% (down-down). Each branch has risk-neutral probability 0.5. What is the value today of a three-year zero-coupon bond with face value 100?

The bond is worth 88.93. Roll back from the t=2 nodes (94.34, 96.15, 98.04), taking half of each pair of values and discounting at the node's rate, to get 90.71 and 94.27 at t=1. Then average and discount at 4% to reach 88.93.

  1. A88.93Correct
  2. B92.49
  3. C86.39
  4. D90.71

Explanation

At t=2 the bond is worth 94.3396 (uu), 96.1538 (ud) and 98.0392 (dd). At t=1 up: 0.5(94.3396+96.1538)/1.05 = 90.7111. At t=1 down: 0.5(96.1538+98.0392)/1.03 = 94.2685. At t=0: 0.5(90.7111+94.2685)/1.04 = 88.93. The value 92.49 omits the final discount at 4%. The value 86.39 discounts along the all-up path only. The value 90.71 is the t=1 up-node value.

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