In a binomial risk-neutral stock model, the probability of a downward move is given by which expression?

• A. (u - e^{δ t})/(u - d)
• B. (e^{δ t} - d)/(u - d)
• C. (u - d)/(e^{δ t} - d)
• D. (u - e^{r t})/(u - d)

Answer: (A)

In a binomial risk-neutral model, the stock can move to uS or dS over the time step, and under the risk-neutral measure the expected stock price grows at the risk-free accumulation factor e^{δ t} (δ is the force of interest). Let p be the risk-neutral probability of an up move. Then the expected price after the step is E[S] = S_t [p u + (1 - p) d] = S_t e^{δ t}. Solving p u + (1 - p) d = e^{δ t} gives p (u - d) = e^{δ t} - d, so p = (e^{δ t} - d)/(u - d). The probability of a downward move is 1 - p, which equals (u - e^{δ t})/(u - d).

Thus the downward-move probability is (u - e^{δ t})/(u - d).