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Past exam of the mathematics course of the University of Cambridge / 2022 / ii / Paper 3 / 29K / b / Solution

Set

θ = \frac{μ + \frac{1}{2} σ ^{2} - r}{σ}

and define

Q

on

F_{T}

by

\frac{d Q}{d P} = exp (- θ W_{T} - \frac{1}{2} θ^{2} T) .

The Cameron-Martin theorem for a linear drift says that

W_{t}^{Q} = W_{t} + θt

is a Brownian motion under

Q

. Therefore

S_{t} = S_{0} exp ((r - \frac{1}{2} σ^{2}) t + σ W_{t}^{Q}) .

The discounted stock is an exponential Brownian martingale, so

e^{- r t} S_{t}

is a

Q

-martingale. Thus

Q

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

Solved by gpt-5.6-sol high.

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