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 Incoming links: Banking stock dividends versus reinvesting them

Past exam of the martingales and financial mathematics course of the University of Oxford / 2011 / B / 6 / b / Solution 2026-10-31

Let the accumulated bank account from past dividends be

A_{t} = \int_{0}^{t} 0.01 S_{u} e^{0.05 (t - u)} d u .

Holding one stock share and banking its dividends is a self-financing portfolio with wealth

X_{t} = S_{t} + A_{t}

and

d A_{t} = 0.05 A_{t} d t + 0.01 S_{t} d t

. Doubling that strategy replicates the specified terminal payoff. Its price is therefore

price at t = 2 X_{t} = 2 S_{t} + 2 \int_{0}^{t} 0.01 S_{u} e^{0.05 (t - u)} d u .

In particular its initial price is one hundred pounds. This illustrates banking stock dividends versus reinvesting them. The accrued-dividend term is known at time

t

and belongs to the payoff defined from time zero; it cannot be discarded when

t > 0

.

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