Past exam of the martingales and financial mathematics course of the University of Oxford / 2010 / B / 5 / c

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- Solution c

Solution

For

v \leq t

, the two stock paths agree, so their geometric averages agree. For

v > t

, the future log-price is

lo g S_{u} = lo g S_{t} + (r - σ^{2} /2) (u - t) + σ (W_{u} - W_{t}) .

Split the average's log-integral into

[0, t]

and

[t, v]

. The first part is

t lo g A_{t}

, while integration of the displayed formula gives

v lo g A_{v} = t lo g A_{t} + (v - t) lo g S_{t} + \frac{1}{2} (r - σ^{2} /2) (v - t)^{2} + σ \int_{t}^{v} (W_{u} - W_{t}) d u .

Dividing by

v

and exponentiating proves

A_{v} = A_{t}^{t / v} S_{t}^{(v - t) / v} exp [(r - \frac{σ ^{2}}{2}) \frac{( v - t ) ^{2}}{2 v} + \frac{σ}{v} \int_{t}^{v} (W_{u} - W_{t}) d u] (v > t) .

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