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IAI Actuarial Core Principles · Economic Modelling · Models of the term structure of interest rates

In a no-arbitrage model of the term structure, the price today of a zero-coupon bond paying Rs 100 at time T is obtained as which of the following?

The zero-coupon bond price equals the risk-neutral expectation of the payoff discounted along the stochastic short-rate path. The risk-neutral measure embeds the market price of risk, which ensures no arbitrage; real-world expectations or undiscounted payoffs would not.

  1. AThe expected value of the discounted payoff under the real-world probability measure, using the risk-free rate
  2. BThe expected value of the discounted payoff under the risk-neutral measure, using the stochastic short rateCorrect
  3. CThe face value discounted at the expected future short rate under the real-world measure
  4. DThe face value discounted at the current long-term yield of the highest-rated corporate bond
  5. The expected payoff under the risk-neutral measure without any discounting

Explanation

Under no-arbitrage, the bond price is the expectation under the risk-neutral measure Q of the payoff discounted by the stochastic path of the short rate, exp(-integral of r). Using the real-world measure ignores the market price of risk, so it would not guarantee absence of arbitrage. Omitting discounting makes no sense for a time-T payoff.

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