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

In a one-period binomial rate tree with risk-neutral probability of 0.5 per branch, the current short rate is 5.00%. A trader prices a one-year zero (face 100, maturing after the next period) and a two-period zero using the same tree. Which statement best describes how the tree must be calibrated to avoid arbitrage?

The tree must be calibrated so its model prices reproduce the observed market prices of zero-coupon bonds. That ensures no arbitrage relative to the current curve. Pricing uses risk-neutral probabilities, not real-world ones, and expected future short rates need not equal today's rate.

  1. AThe tree rates must be chosen so that model prices of the observed zero-coupon bonds equal their market pricesCorrect
  2. BThe up-state rate must equal the forward rate and the down-state rate must equal the spot rate
  3. CThe expected future short rate must equal the current short rate
  4. DThe probabilities must be set equal to the real-world probabilities of rate rises

Explanation

Arbitrage-free calibration requires the tree to reproduce the current observed term structure of bond prices (the drift is fitted accordingly). Setting expected future rate equal to current rate ignores convexity and the term premium and generally does not match market prices. Real-world probabilities are not used for pricing.

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