 Solution

ID: past-exam-of-the-mathematical-models-of-financial-derivatives-course-of-the-university-of-oxford/2023/1/a/solution

Δ = V_{S}

shares and

V - S V_{S}

in the bank account, withdrawing the claim's cash flow

C d t

. A share's total gain includes the continuous dividend yield, so its gain is

d S + D S d t

. Matching the claim's total gain gives

d V + C d t = V_{S} (d S + D S d t) + r (V - S V_{S}) d t .

d V = [V_{t} + (μ - D) S V_{S} + \frac{1}{2} σ^{2} S^{2} V_{SS}] d t + σ S V_{S} d W .

The delta hedge cancels the Brownian motion term. Comparing the remaining drifts yields the Black-Scholes equation with claim dividends, with the stock's own dividend yield included:

V_{t} + \frac{1}{2} σ^{2} S^{2} V_{SS} + (r - D) S V_{S} - r V = - C (S, t), V (S, T) = S .

μ

cancels, as required by risk-neutral pricing.

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