FRM Part I · FRM Exam Part I · Binomial Trees
A multi-step tree models a non-dividend-paying stock with volatility 20% per year. It uses time steps of Δt = 0.25 years, a continuously compounded risk-free rate of 4%, and Cox-Ross-Rubinstein parameters u = e^(σ√Δt) and d = 1/u. Which is the risk-neutral up-move probability per step?
The up probability is about 0.5252. With u = e^0.1 = 1.10517, d = 0.90484 and a per-step growth factor of e^0.01 = 1.01005, p = (1.01005 - 0.90484)/(1.10517 - 0.90484) = 0.5252.
- A0.4748
- B0.5252Correct
- C0.6787
- D0.5000
Explanation
σ√Δt = 0.2×0.5 = 0.10, so u = 1.10517 and d = 0.90484. The growth factor per step is e^(0.04×0.25) = e^0.01 = 1.01005. Then p = (1.01005 - 0.90484)/(1.10517 - 0.90484) = 0.10521/0.20033 = 0.5252. Using e^0.04 rather than e^0.01 gives 0.6787, which ignores Δt. The value 0.4748 is the down probability, 1 - p.
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