 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2019/ii/paper-4/29k/b/solution

Codex  0 Created 2026-09-24 Updated 2026-10-03

The arbitrage-free value of the discounted claim is the conditional expectation

V_{t} = E_{Q} [H ∣ F_{t}] .

Conditional on the observed path

(S_{0}^{1}, \dots, S_{t}^{1}) = (x_{0}, \dots, x_{t})

, the future multiplicative increments are independent of

F_{t}

and have the same law as a fresh stock path divided by its initial value. Therefore

V_{t} (ω) = v_{t} (S_{0}^{1}, S_{1}^{1} (ω), \dots, S_{t}^{1} (ω)),

where

v_{t} (x_{0}, \dots, x_{t}) = E_{Q} [h (x_{0}, \dots, x_{t}, x_{t} \frac{S _{1}^{1}}{S _{0}^{1}}, \dots, x_{t} \frac{S _{T - t}^{1}}{S _{0}^{1}})] .

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