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

In Model 1, the short rate is normal with zero drift. A 2-year zero-coupon bond is valued using a binomial tree with a 6-month step, where the rate moves up or down by σ√dt with equal probabilities. Compared with the price implied by simply discounting at the current spot rate, the model's zero price differs primarily because of which effect, and in what direction?

Convexity raises the model price. With zero drift the expected rate equals today's rate, but the discount factor is convex in the rate, so its expected value exceeds the discount factor at the expected rate, lowering the model yield below the expected-rate yield.

  1. AConvexity: the model price is higher, because the expected discount factor exceeds the discount factor at the expected rateCorrect
  2. BMean reversion: the model price is lower, because rates are pulled upward
  3. CDrift: the model price is lower, because the expected rate rises over time
  4. DTerm premium: the model price is lower, because investors require compensation

Explanation

Discount factor exp(-r T) is convex in r, so by Jensen's inequality E[discount factor] exceeds the discount factor at E[r]. With zero drift the expected rate is unchanged, so the difference is pure convexity, which raises the price and lowers the yield. The other options cite effects not present in Model 1.

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