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ID: past-exam-of-the-martingales-and-financial-mathematics-course-of-the-university-of-oxford/2008/b/7/d/solution

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Past exam of the martingales and financial mathematics course of the University of Oxford / 2008 / B / 7 / d / Solution by

Codex  0 2026-10-31

Linearity separates the terminal call payoff from the running payment stream. The first contributes the supplied European call price. Under the risk-neutral measure,

E (S_{v} ∣ S_{t} = s) = s e^{r (v - t)}

, so the stream contributes

E^{Q} \int_{t}^{T} e^{- r (v - t)} q S_{v} d v = q s \int_{t}^{T} d v = q s (T - t) .

(1)

Consequently

V (t, s) = s Φ (d_{1}) - K e^{- r (T - t)} Φ (d_{2}) + q s (T - t) .

(2)

This is the claim dividends proportional to the underlying stock identity. The underlying itself is still non-dividend-paying; the cash flow belongs to the derivative.

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