Past exam of the mathematics course of the University of Cambridge / 2005 / ii / Paper 3 / 27J / b

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- Solution b

Solution

At gross rate

R = 5/4

, the conditional up probabilities of the equivalent martingale measure are

q_{0} = \frac{( 5/4 ) 15 - 12}{30 - 12} = \frac{3}{8}, q_{u} = \frac{( 5/4 ) 30 - 36}{45 - 36} = \frac{1}{6}, q_{d} = \frac{( 5/4 ) 12 - 10}{16 - 10} = \frac{5}{6} .

All lie strictly between zero and one. The four terminal path probabilities are

1/16, 5/16, 25/48, 5/48

in the order good-good, good-bad, bad-good, bad-bad.

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