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ID: past-exam-of-the-martingales-and-financial-mathematics-course-of-the-university-of-oxford/2007/b/7/c/solution

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Past exam of the martingales and financial mathematics course of the University of Oxford / 2007 / B / 7 / c / Solution by

Codex  0 2026-10-31

Let

c (u) = Φ (\frac{r + σ ^{2} /2}{σ} u) - e^{- r u} Φ (\frac{r - σ ^{2} /2}{σ} u), c (0) = 0.

The homogeneous function of the Black-Scholes formula gives its at-the-money value at the start date as

S_{T_{1}} c (T - T_{1})

. Before that date, risk-neutral valuation and the discounted stock martingale give

e^{- r (T_{1} - t)} E [S_{T_{1}} c (T - T_{1}) ∣ F_{t}] = S_{t} c (T - T_{1}) .

After the start date the strike price is the realized value

S_{T_{1}}

, so the complete answer is

V_{t} = {S_{t} c (T - T_{1}), C (t, S_{t}; T, S_{T_{1}}), 0 \leq t < T_{1}, T_{1} \leq t \leq T .

At maturity the second line is

(S_{T} - S_{T_{1}})_{+}

. After entry the price depends on the historical strike price as well as the current stock, so that history cannot be discarded when giving the answer for all times.

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