GEOMETRIC RANDOM WALKS

Consider the following model:
247
where 248 is a sequence of independent normal variables, and rt, the return, is computed as
249
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
250
251
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 252 . We start with an initial price S0, which is known. ...

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