Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 211 / 1 / f

Table of contents
- Solution f

Solution

Normalize the self-financing strategy

η

by defining

L_{t} = \frac{η _{t}}{π _{t - 1}^{η}} .

Part e makes this well-defined, and

π_{t}^{L} = \frac{η _{t + 1} \cdot P _{t}}{π _{t}^{η}} = 1.

Moreover,

ξ_{t}^{L} = \frac{η _{t} \cdot ( P _{t} + δ _{t} )}{π _{t - 1}^{η}} - 1 = \frac{π _{t}^{η}}{π _{t - 1}^{η}} - 1.

Both

K

and

L

have constant unit price and predictable dividends. Their difference has zero price and predictable dividend

ξ^{K} - ξ^{L}

. If that dividend were nonzero with positive probability, taking its known sign would produce an arbitrage. Therefore

ξ^{K} = ξ^{L}

, proving

ξ_{t}^{K} = \frac{π _{t}^{η}}{π _{t - 1}^{η}} - 1.

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