The MATH behind the random walk of stocks

We are looking at a percentage wise change of the stock values. Let us say the stock S changes by some amount D S in the time interval D t. The relative change is the ratio of the value change and the value of the stock D S / S (You get the change in percents by multiplying this number by 100).

This ratio is rising continuously by some amount r , like the interest on savings, multiplied by the time interval:

D S / S = r D t

In addition you have random fluctuations (so-called Gaussian fluctuations). If D x is such a fluctuation with a size of +1 or -1 on the average we can write the whole equation for the stock evolution as follows

D S / S = r D t + s D x

The parameter s is the strength - or amount - of the fluctuation during the time span D t. s  is the volatility of the stock. This equation determines the change of the stock value with time.


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