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FRM Part I · FRM Exam Part I · Binomial Trees

A modeler builds a one-year CRR step with σ = 10% and Δt = 1, and uses a continuously compounded risk-free rate of 12%. What does the risk-neutral probability formula imply?

The formula gives p of about 1.11, which exceeds 1. This happens because the risk-free growth factor e^0.12 = 1.1275 is above the up factor 1.1052. No-arbitrage requires d < e^(rΔt) < u, so this tree admits arbitrage and is invalid, typically from too few steps or too low volatility.

  1. Ap is about 1.11, above 1, showing the tree admits an arbitrage because e^(rΔt) exceeds uCorrect
  2. Bp is about 0.50, so the tree is valid because CRR sets p equal to one half
  3. Cp is about 0.89, so the tree is valid but the up move is less likely than the down move
  4. Dp is about 1.11, which is acceptable because probabilities in risk-neutral trees need not be bounded

Explanation

u = e^0.10 = 1.1052 and d = 0.9048, while e^0.12 = 1.1275. Then p = (1.1275 − 0.9048)/(1.1052 − 0.9048) = 1.11. A valid tree requires d < e^(rΔt) < u. Here the risk-free asset beats the stock in every state, so an arbitrage exists.

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