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

A one-period binomial interest rate tree has a current six-month rate of 4.00% (semiannual compounding). In six months the rate is either 5.00% (up) or 3.00% (down), each with risk-neutral probability 50%. What is the value today of a 1-year zero-coupon bond with face value 100, using the tree? (Discounting at each node uses the semiannual rate at that node.)

The bond is worth about 96.12. Discount 100 back one period at each node's semiannual rate (97.561 and 98.522), average with equal risk-neutral probabilities to get 98.0415, then discount one more period at the 2% root rate.

  1. A96.04
  2. B96.12Correct
  3. C96.20
  4. D96.27

Explanation

Value at 0.5 years: up node = 100/1.025 = 97.561; down node = 100/1.015 = 98.522. Expected value = 0.5×97.561 + 0.5×98.522 = 98.0415. Discount at 2.00% per half-year: 98.0415/1.02 = 96.12. Option 96.04 comes from discounting 100 at 4% for a year (100/1.04^... mixing), 96.27 uses the wrong compounding.

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