  Suppose one buys two very similar stocks and is curious about how much, after
some time T, one of them will contribute to the overall asset, expecting, of
course, that it should be around 1/2 of the sum. Here we examine this question
within the classical Black and Scholes (BS) model, focusing on the evolution of
the probability density function P(w) of a random variable w =
a_T^{(1)}/(a_T^{(1)} + a_T^{(2)}) where a_T^{(1)} and a_T^{(2)} are the values
of two (either European- or the Asian-style) options produced by two absolutely
identical BS stochastic equations. We show that within the realm of the BS
model the behavior of P(w) is surprisingly different from common-sense-based
expectations. For the European-style options P(w) always undergoes a
transition, (when T approaches a certain threshold value), from a unimodal to a
bimodal form with the most probable values being close to 0 and 1, and,
strikingly, w =1/2 being the least probable value. This signifies that the
symmetry between two options spontaneously breaks and just one of them
completely dominates the sum. For path-dependent Asian-style options we observe
the same anomalous behavior, but only for a certain range of parameters.
Outside of this range, P(w) is always a bell-shaped function with a maximum at
w = 1/2.
