 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2024/ii/paper-4/29l/c/solution

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

For

s < t

e^{- r t} S_{t} = e^{- rs} S_{s} exp (- \frac{1}{2} σ^{2} (t - s) + σ (W_{t} - W_{s})) .

The increment is independent of the Brownian filtration at time

s

, and its exponential factor has mean one by the Gaussian moment-generating function. Hence the conditional expectation of the discounted time-

t

price is the discounted time-

s

price.

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