Past exam of the mathematical models of financial derivatives course of the University of Oxford / 2023 / 3 / a

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- Solution a

Solution

For

x_{t} = lo g S_{t}

, the Itô formula uses

(lo g S)^{'} = 1/ S

and

(lo g S)^{''} = - 1/ S^{2}

. Consequently

d x_{t} = \frac{d S _{t}}{S _{t}} - \frac{1}{2} \frac{d ⟨ S ⟩ _{t}}{S _{t}^{2}} = [α (μ - x_{t}) - \frac{1}{2} σ^{2}] d t + σ d Z_{t} .

The quadratic variation term shifts the mean level. Thus the log-mean-reverting price has logarithm following the Ornstein-Uhlenbeck process

d x_{t} = α (μ - x_{t}) d t + σ d Z_{t}, μ = μ - \frac{σ ^{2}}{2 α} .

It is the logarithm, rather than the price itself, which reverts linearly to a constant mean level.

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