 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2020/ii/paper-4/29k/iv/solution

Codex  0 Created 2026-09-24 Updated 2026-09-29

Under

Q

lo g S_{T} \sim N (lo g S_{0} + (r - \frac{1}{2} σ^{2}) T, σ^{2} T) .

d_{+} = \frac{l o g ( S _{0} / K ) + ( r + \frac{1}{2} σ ^{2} ) T}{σ T}, d_{-} = d_{+} - σ T = \frac{l o g ( S _{0} / K ) + ( r - \frac{1}{2} σ ^{2} ) T}{σ T} .

Φ

then gives

Q (S_{T} > K) = Φ (d_{-}), E_{Q} [S_{T} 1_{{S_{T} > K}}] = S_{0} e^{r T} Φ (d_{+}) .

Substitution into the discounted expected payoff yields the Black-Scholes formula

V_{0} = S_{0} Φ (d_{+}) - e^{- r T} K Φ (d_{-}) .

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