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## Wednesday, May 23, 2012

### fake random walker - compounded rate of return

I feel stock price is a random walker but ln(PriceRelative) is not. Let's denote this ln() value as P.

As a (one-dimension) random walker, a stock price S(at time t) can move from 1000 to 1001.2 to 1000.3 to 999 to 999.5... The steps are cumulative. The current value is the cumulative effect of all incremental steps.

Now look at P. It might take on +0.03, +0.5, -0.01 -0.6, -1, +2... Not cumulative no drift. In fact each value in the series is independent of the previous values. I feel P is a random variable but not Brownian.

If L is a Brownian walker (i.e. follows a Wiener process), then deltaL (incrementatl change in S) is similar to our P.

Q: So which RV is Normal and which is LogNormal in this model?
%%A: I believe S(after i steps) denoted S_i has a log normal distribution but ln(S_i) is normal. More specificially --

Suppose matlab generates 3000 realizations (of the random process). In each path, we pick the i'th step so we get 3000 realizations of S_i. A histogram of the 3000 is a punched bell. However, if we compute ln(S_i) for the 3000 realizations, the histogram is bell-shaped.

Q: So which RV is in a geometric Brownian motion?
%%A: S.

Intuitively, if a RV is in a strictly Brownian motion (not Geometric), then its value at any time has a Normal (not LogNormal) distribution.