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 σ

^{2}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 σ

^{2}t , σ^{2}t )
E LogX_t = logX_0 - ½ σ

^{2}t ..... [1]
=> E Log( X_t / X_0) = - ½ σ

^{2}t .... 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<X_0) > 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

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<X_0) > 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

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