 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2006/ii/paper-1/28i/solution

The diagram has common multipliers

u = 5/3, d = 2/3

on every branch, with gross riskless return

R = 4/3

. The martingale condition

p u + (1 - p) d = R

gives

p = \frac{R - d}{u - d} = \frac{2}{3}, 1 - p = \frac{1}{3} .

Since

d < R < u

, these probabilities lie strictly between zero and one at each node, and the market is arbitrage-free. All valuations below use discounted conditional expectations under this measure.

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