Past exam of the mathematics course of the University of Cambridge / 2018 / ii / Paper 2 / 29K / a

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Solution

The price process is a geometric Brownian motion with

\frac{d S _{t}}{S _{t}} = (μ + \frac{1}{2} σ^{2}) d t + σ d B_{t} .

Put

ϑ = \frac{μ + σ ^{2} /2 - r}{σ}

and define the equivalent measure

Q

on

F_{T}

by the exponential Brownian martingale

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

The Cameron-Martin-Girsanov theorem says that

W_{t}^{Q} = B_{t} + ϑt

is Brownian motion under

Q

. Therefore

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

so

e^{- r t} S_{t}

is a

Q

-martingale. This is the unique equivalent martingale measure, since cancellation of the single Brownian risk forces the displayed drift shift.

By risk-neutral valuation,

π_{C} = e^{- r T} E_{Q} f (S_{T}) .

W_{T}^{Q} = T Y

with

Y \sim N (0, 1)

gives

π_{C} = e^{- r T} \int_{- \infty}^{\infty} f (S_{0} e^{σ T y + (r - σ^{2} /2) T}) \frac{e ^{- y^{2} /2}}{2 π} d y .

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