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

In a binomial short-rate tree used for pricing, why is the value of a bond at an earlier node computed by backward induction using risk-neutral probabilities?

Backward induction with risk-neutral probabilities gives arbitrage-free prices because, under those probabilities, every security earns the short rate in expectation. This is a pricing device, not a statement about real-world likelihoods, and the probabilities need not equal 0.5.

  1. ABecause it makes expected bond returns equal to the risk-free rate, so prices are consistent with no arbitrageCorrect
  2. BBecause real-world probabilities are unobservable and so cannot affect bond prices in any model
  3. CBecause risk-neutral probabilities always equal 0.5 in any tree
  4. DBecause it ensures the short rate is mean reverting

Explanation

Under risk-neutral probabilities, discounted expected values at the short rate give arbitrage-free prices, with bonds earning the risk-free rate in expectation. Probabilities need not equal 0.5, and mean reversion is a separate feature of the rate dynamics.

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