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FRM Part II · FRM Exam Part II · The Art of Term Structure Models: Drift

In a one-period binomial tree, a 1-year zero-coupon bond is priced using a short rate of 4.00% (continuously compounded here for simplicity, so the discount factor is exp(-0.04)). A 2-year zero-coupon bond has a price today of 92.00 per 100 face. After one year the 1-year rate is either 5.00% (up) or 3.00% (down), each with risk-neutral probability 50%. Assume the 2-year bond's value after one year is 100 x exp(-r) for the realized rate r. What is the model price of the 2-year bond today, and does it equal the market price of 92.00 (approximately)?

The model price is the discounted risk-neutral expectation of the bond's value: about 96.08 expected after one year, discounted at 4% gives roughly 92.3, above the market 92.00. So the model overprices the bond, and the drift would need recalibrating to match the market.

  1. AAbout 92.17, which is above the market price of 92.00, so the model overprices the bondCorrect
  2. BAbout 92.00, exactly matching the market price
  3. CAbout 92.17, but this is meaningless because no-arbitrage requires a price above 100
  4. DAbout 91.83, which is below the market price of 92.00, so the model underprices the bond

Explanation

Up value = 100*exp(-0.05)=95.123; down value = 100*exp(-0.03)=97.045. Expected value = 96.084. Discount at exp(-0.04)=0.96079 gives 96.084*0.96079=92.32. Recomputing: 96.084*0.96079 = 92.317. The model price is about 92.32, which is above 92.00. The closest and correct description is that the model overprices the bond relative to market, so drift would need adjusting.

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