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FRM Part I · FRM Exam Part I · Binomial Trees

In a binomial tree for options on a futures price, the risk-neutral probabilities are chosen so that the expected futures price at each node equals the current futures price. What is the reason for this choice?

Futures need no initial outlay, so under risk-neutral valuation their expected growth rate is zero and the expected futures price equals the current one. That is why p = (1 - d)/(u - d), with no interest-rate term, and the rate enters only through discounting.

  1. AThe futures contract costs nothing to enter, so its risk-neutral expected growth rate is zeroCorrect
  2. BFutures prices grow at the risk-free rate less the dividend yield
  3. CFutures prices grow at the domestic rate minus the foreign rate
  4. DFutures prices have zero volatility under the risk-neutral measure

Explanation

Entering a futures contract requires no initial investment, so in a risk-neutral world its expected return is zero and the expected futures price stays unchanged. This gives p = (1 - d)/(u - d). Growth at r - q or r_d - r_f applies to index and currency spot prices, not to futures prices.

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