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Past exam of the mathematics course of the University of Cambridge / 2023 / ii / Paper 4 / 29K / b / Solution

Now

R = 1 + r = 9/8

. The local risk-neutral up probabilities solve

RS = q S_{u} + (1 - q) S_{d}

. They are

q_{0} = \frac{( 9/8 ) 5 - 4}{6 - 4} = \frac{13}{16}, q_{u} = \frac{( 9/8 ) 6 - 5}{7 - 5} = \frac{7}{8}, q_{d} = \frac{( 9/8 ) 4 - 3}{5 - 3} = \frac{3}{4} .

The terminal payoff of the European put option with strike

5

is

0, 0, 2

at stock prices

7, 5, 3

. Backward option pricing gives time-one values

V_{u} = 0, V_{d} = R^{- 1} (\frac{1}{4} \cdot 2) = \frac{4}{9},

and hence

V_{0} = R^{- 1} (\frac{3}{16} \frac{4}{9}) = \frac{2}{27} .

The first-period stock holding in the replicating portfolio in a binomial market is

Δ_{0} = \frac{V _{u} - V _{d}}{6 - 4} = - \frac{2}{9} .

Thus the hedge initially shorts

2/9

of a share.

Solved by gpt-5.6-sol high.

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