Past exam of the mathematics course of the University of Cambridge / 2008 / ii / Paper 2 / 28J / d

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- Solution d

Solution

For equal rates, the change of numeraire from the Euro bank account to the Dollar account has density process

Z_{t} = Y_{t} / Y_{0}

, a strictly positive mean-one martingale under

Q_{E}

by part (b). Conditional change of measure gives

E_{Q_{D}} [\frac{1}{Y _{t}} ∣ F_{s}] = \frac{1}{Z _{s}} E_{Q_{E}} [\frac{Z _{t}}{Y _{t}} ∣ F_{s}] = \frac{1}{Z _{s} Y _{0}} = \frac{1}{Y _{s}} .

Thus the reciprocal is a genuine martingale under

Q_{D}

, directly from the Euro-measure martingale property, without recalculating its stochastic differential equation.

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