Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 211 / 4 / b

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- Solution b

Solution

λ = \frac{μ - r}{σ}, \frac{d Q}{d P} = exp (- λ W_{T} - \frac{1}{2} λ^{2} T) .

By the Girsanov theorem,

W_{t}^{Q} = W_{t} + λ t

is Brownian motion under

Q

. Consequently

d S_{t} = r S_{t} d t + σ S_{t} d W_{t}^{Q},

so discounted stock price is a martingale and

Q

is the Risk-neutral measure for the Black-Scholes model.

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