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ID: past-exam-of-the-mathematical-models-of-financial-derivatives-course-of-the-university-of-oxford/2021/1/a/i/solution

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Past exam of the mathematical models of financial derivatives course of the University of Oxford / 2021 / 1 / a / i / Solution by

Codex  0 2026-10-25

The portfolio value is

Π_{t} = ϕ_{t} S_{t} + ψ_{t} B_{t}

. The self-financing condition means that changes in holdings are financed by trading within the portfolio, with no external infusion or withdrawal. Thus

d Π_{t} = ϕ_{t} d S_{t} + ψ_{t} d B_{t} = (μ ϕ_{t} S_{t} + r ψ_{t} B_{t}) d t + σ ϕ_{t} d W_{t} .

(1)

The holdings are predictable and satisfy the usual stochastic-integrability and admissibility conditions; without admissibility, doubling strategies would spoil the intended no-arbitrage interpretation.

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