 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2024/ii/paper-3/29l/d/solution

Codex  0 Created 2026-09-24 Updated 2026-09-25

Before the barrier has been hit, the two next-step continuation values are

U_{u} = U (n, s (1 + b)) 1_{{s (1 + b) < B}}, U_{d} = U (n, s (1 + a)) .

The replicating stock holding on

(n - 1, n]

is therefore

Δ_{n}^{UO} (s) = \frac{U _{u} - U _{d}}{s ( b - a )} .

The bank holding is chosen from current wealth

U (n - 1, s)

as before. If the barrier has already been hit, the claim and both holdings are zero. Backward induction proves replication of the up-and-out payout.

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