CM2 April 2018 Q3 (ii)

[Trainee_Act]

Exam Question
The value of an investment asset follows the equation A(t) = exp(Bt), where Bt follows standard Brownian motion.
(ii) Calculate the expected value of this investment at time 5. [2]

Examiner's Report
A(0) = exp(0) = 1, so the students buys 1,000 units of the asset.
E[A(5)] = exp(0.5*12 *5) = 12.182
So the expected value of the investment is $12,182

Can someone explain the middle line of the answer above? I am not sure where the 0.5, 12 & 5 come from.

Any assistance here would be appreciated.

 

[AaronD]

Hi,

I think this is what is happening.

B(t) ~ N(0, 1t)
B(5) ~ N(0, 5)
=> A(t) ~ logN(0, 1t)


E(A(t)) = exp(mu + 1/2 sigma^2 * t)
E(A(t)) = exp(0 + 1/2 1^2 * t)
E(A(5)) = exp(1/2 1^2 * 5)
= 12.182