Question in May 2001 Course 3

[Sam (Retired)]

Could anyone explain the following question to me?

"For a simulation of the movement of a stock’s price:
(i) The price follows geometric Brownian motion, with drift coefficient = 0. 01. and
variance parameter = 0.0004.
(ii) The simulation projects the stock price in steps of time 1.
(iii) Simulated price movements are determined using the inverse transform method.
(iv) The price at t = 0 is 100.
(v) The random numbers, from the uniform distribution on [0,1] , for the first 2 steps
are 0.1587 and 0.9332, respectively.
(vi) F is the price at t = 1; G is the price at t = 2.
Calculate G – F."

The solution says "0.1587 corresponds to -1 standard deviation", but I don't understand. Any other methods to do this question?

Thank you.

[Smash Puny Human]

Geometric Brownian Motion => Y=E^X, where X is ruled by BM. This means that the transistion between states is normally distributed (X-mu)/sigma. Since the values from the simulation are pulled from a uniform distribution, we just look up the values in the normal table .1587 corresponds to -1 (look up 1-.1587: then use the negative of that value) and .9332 =1.5. So (X-.01)/sqrt(.0004)=-1. From here solve for X=-.01, take e^(X)*first state=> e^(-.01)*100=99. The second state follows the same reasoning. e^(.03)*99=103.

[Macroman]

I believe the exam committe has removed Brownian motion from the course #3 syllabus...Someone please correct me if I'm wrong...

[Elisha]

Macroman, you are wrong. It was just taught at a seminar that I attended.