GEOMETRIC RANDOM WALKS
Consider the following model:
where 
is a sequence of independent normal variables, and rt, the return, is computed as
is a sequence of independent normal variables, and rt, the return, is computed as
Returns are therefore normally distributed, and the return over each interval of length 1 has mean µ and standard deviation σ. How can we express future prices if returns are determined by the equations above?
Suppose we know the price at time t, St. The price at time t + 1 can be written as
This last equation is very similar to the equation for the arithmetic random walk, except that the price from the previous time period appears as a factor in all of the terms.
The equation for the geometric random walk makes it clear how paths for the geometric random walk can be generated. As in the case of the arithmetic random walk, all we need is a way of generating the normal random variables 
. We start with an initial price S0, which is known. ...
. We start with an initial price S0, which is known. ...Get The Theory and Practice of Investment Management: Asset Allocation, Valuation, Portfolio Construction, and Strategies, Second Edition now with the O’Reilly learning platform.
O’Reilly members experience books, live events, courses curated by job role, and more from O’Reilly and nearly 200 top publishers.
