FRM Part I · FRM Exam Part I · Binomial Trees
A 1-year option is valued on a 4-step Cox-Ross-Rubinstein tree with volatility of 30% and a continuously compounded risk-free rate of 4%. The tree uses u = e^(σ√Δt) and d = 1/u. What is the risk-neutral probability of an up move, p, per step?
The risk-neutral up probability is about 0.496. With four steps, Δt is 0.25, u is 1.1618, d is 0.8607, and the per-step growth factor is e^0.01. The probability is (1.01005 minus 0.8607) divided by (1.1618 minus 0.8607).
- A0.496Correct
- B0.500
- C0.504
- D0.598
Explanation
Δt = 0.25, so u = e^0.15 = 1.1618 and d = 0.8607. e^(0.04×0.25) = 1.01005. p = (1.01005 - 0.8607)/(1.1618 - 0.8607) = 0.1493/0.3011 = 0.496. Using the annual growth factor e^0.04 = 1.0408 instead of the per-step factor gives 0.598. The value 0.504 reverses the numerator.
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