FRM Part I · FRM Exam Part I · Binomial Trees
A stock trades at 50. Over one year it can rise by a factor of 1.20 or fall by a factor of 0.80. The continuously compounded risk-free rate is 5% per year. What is the risk-neutral probability of an up move in a one-step binomial tree?
The risk-neutral up probability is about 0.628. It equals the risk-free growth factor e^0.05 (1.0513) minus the down factor 0.80, divided by the spread between up and down factors (0.40). This makes the stock's expected return equal to the risk-free rate.
- A0.628Correct
- B0.500
- C0.378
- D0.372
Explanation
p = (e^{rT} - d)/(u - d) = (1.051271 - 0.80)/0.40 = 0.628. The 0.500 figure ignores the growth at the risk-free rate. The 0.372 figure is 1 - p, the down probability. The 0.378 figure uses e^{-rT} instead of e^{rT}.
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