GBM + zero drift

I see zero-drift GBM in multiple problems
– margrabe option
– stock price under zero interest rate
For simplicity, let’s assume X_0 = $1. Given

        dX =σX dW     …GBM with zero drift-rate

Now denoting L:= log X, we get

                dL = – ½ σ2 dt + σ dW    … BM not GBM. No L on the RHS.
Now L as a process is a BM with a linear growth (rather than exponential growth).
LogX_t ~ N ( logX_0  – ½ σ2t  ,   σ2t )
E LogX_t = logX_0  – ½ σ2t  ….. [1]
=> E Log( X_t / X_0)  = – ½ σ2t  …. so expected log return is negative?
E X_t = X_0 …. X_t is a log-normal squashed bell where x-axis extends from (0 to +inf) [3].

Look at the lower curve below.
Mean = 1.65 … a pivot here shall balance the “distributed weights”
Median = 1.0 …half the area-under-curve is on either side of Median i.e. Pr(X_t < median) = 50%

Therefore, even though E X_t = X_0 [2], as t goes to infinity, paradoxically Pr(X_t<X_0) goes to 100% and most of the area-under-curve would be squashed towards 0, i.e. X_t likely to undershoot X_0.

The diffusion view — as t increases, more and more of the particles move towards 0, although their average distance from 0 (i.e. E X_t) is always X_0. Note 2 curves below are NOT progressive.

The random walker view — as t increases, the walker is increasingly drawn towards 0, though the average distance from 0 is always X_0. In fact, we can think of all the particles as concentrated at the X_0 level at the “big bang” of diffusion start.

Even if t is not large, Pr(X_t 50%, as shown in the taller curve below.

[1] horizontal center of of the bell shape become more and more negative as t increases.
[2] this holds for any future time t. Eg: 1D from now, the GBM diffusion would have a distribution, which is depicted in the PDF graphs.
[3] note like all lognormals, X_t can never go negative 

File:Comparison mean median mode.svg
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