Wiener Process

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If physicists use the Wiener process to describe the movement of molecules, for finance it is interesting because it helps us understand the price behavior of stocks and derivatives.

A Wiener process is a specific version of a Markov stochastic process in which the expected value of the change is zero and the variance is 1 per year.

So it has two key properties:

1. The variable moves according to a normal distribution where
   ( E[\Delta z] = 0 ) and ( Var[\Delta z] = \Delta t ),
   which means ( \Delta z = \varepsilon \sqrt{\Delta t} ), where ( \varepsilon \sim N(0,1) ) (standard normal).
2. The values of ( \Delta z ) over any two small time intervals ( \Delta t ) are independent (the process has no memory; no autocorrelation).

Now let’s move to the limit. For infinitely small time segments, we write:
( dx = a , dt ),
meaning that ( \Delta x = a ) when ( \Delta t ) becomes infinitely small (approaches zero).
This implies that in the limit, the movement of ( z ) becomes “jumpy,” because the smaller ( \Delta t ) becomes, the larger ( \sqrt{\Delta t} ) appears relative to it (see diagrams).

Intuitively, the Wiener process leads to two mind-blowing conclusions:

1. The path is infinitely wrinkled.
   The variable moves up and down so quickly that if we tried to measure the total length of the path, it would be infinite.
2. Because the motion is infinite, the variable crosses every numerical level in its range infinitely many times, no matter how small the time interval is.

Source: Options, Futures & Other Derivatives, John C. Hull

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