Past exam of the mathematics course of the University of Cambridge / 2019 / ii / Paper 2 / 29K / d

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- Solution d

Solution

For each period set

σ_{i} = - \frac{m _{i} + σ _{i}^{2} /2}{σ _{i}}

\frac{d Q}{d P} = \prod_{i = 1}^{T} exp (σ_{i} Z_{i} - \frac{1}{2} σ_{i}^{2}) .

Independence of the

Z_{i}

shows that this strictly positive density has expectation one. Under

Q

, the conditional mean of each gross return is

E_{Q} [e^{σ_{i} Z_{i} + m_{i}} ∣ F_{i - 1}] = exp (m_{i} + \frac{1}{2} σ_{i}^{2} + σ_{i} σ_{i}) = 1.

Consequently

E_{Q} [S_{t}^{1} ∣ F_{t - 1}] = S_{t - 1}^{1},

so the discounted price process is a

Q

-martingale. This equivalent martingale measure rules out arbitrage.

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