 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2015/iii/paper-40/5/b/solution

Codex  0 Created 2026-10-03 Updated 2026-10-06

m \in {0, \dots, N}

m + 2 \sum_{K = 1}^{N} (m - K)^{+} = m + 2 \sum_{K = 1}^{m - 1} (m - K) = m + m (m - 1) = m^{2} .

Thus the static replication on a finite terminal support consists of one share and two calls of every listed strike, held to maturity:

S_{T}^{2} = S_{T} + 2 K = 1 \sum N (S_{T} - K)^{+} .

The strike-

N

call contributes zero at maturity but is harmless in this identity. The portfolio costs

S_{0} + 2 \sum_{K = 1}^{N} C_{0} (K)

. No distributional assumption on the stock is needed beyond its stated terminal support.

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