The lognormal distribution is assymetric, and hence possibilities of
prices much larger than average are larger than prices much smaller
than average. One kind of expects this kind of random pricing to
return better than a constant price, if you can sell the stock I guess.
But, how I was calculating the standard deviation part of the lognormal
was dumb. I was saying the variance in price was a constant. I don't
know, maybe it is. But in this circumstance, the constant used had no
research behind it, and it need not even be close to what happens in
the real world.
One probability distribution which provides an estimate of standard
deviation as a function of size is the Poisson. Normally used for
situations where counting is involved. Sometimes used in other
situations, where some kind of "natural unit" is present, such that the
standard deviation of an estimate of N, is sqrt( N ).
Okay, I plugged something like that in to my formula, and as the penny
is no longer in circulation, I used a nickel as the natural unit. I
also did a run with the penny as the natural unit.
The Poisson distributions for nickels and pennies are quite different,
the penny distribution is much closer to symmetric. On a fractional
basis (all data divided by the mean, which will be the same for the 2
runs), the penny distribution should also be sharper. The end result
is that the amount of profit seen is quite close to the specified
markup of 30% (a tiny bit higher, could easily not be significant). I
guess the thing to do is to figure out just what unit of currency
people think in terms of, for the expected average price you are
starting from. Here, the price was in the neighbourhood of $1. Do
more people think a dollar is 4 quarters, or 100 pennies?