Optimalf 133
We could argue that we were betting only one unit, since 21.20 (our
stake prior to the bet) divided by 20 (the starting value)=1.06. Since
most bets must be in integer form—that is, no fractional bets (chips
are not divisible and neither are futures contracts)—we could bet only
one unit in real life in this situation. However, in these simulations the
fractional bet is allowed. The reasoning here behind allowing the frac-
tional bet is to keep the outcome consistent regardless of the start-
ing stake. Notice that each simulation starts with only enough stake to
make one full bet. What if each simulation started with more than that?
Say each simulation started with enough to make 1.99 bets. If we were
only allowing integer bets, our outcomes (TWRs) would be altogether
different.
Further, the larger the amount we begin trading with is, relative to the
starting value (biggest loss/−optimalf), the closer the integer bet will be
to the fractional bet. Again, clarity is provided by way of an example. What
if we began trading with 400 units in the previous example? After the first
bet our stake would have been:
Stake = 400 +((400/20)* 1 .2)
= 400 +(20* 1 .2)
= 400 + 24
= 424For the next bet, we would wager 21.2 units (424/20), or the integer
amount of 21 units. Note that the percentage difference between the frac-
tional and the integer bet here is only .952381% versus a 6.0% difference, had
the amount we began trading with been only one starting value, 20 units.
The following axiom can now be drawn:The greater the ratio of the amount
you have as a stake to begin trading relative to the starting value (biggest
loss/−optimalf), the more the percentage difference will tend to zero
between integer and fractional betting.
By allowing fractional bets, making the process nonquantum, we obtain
a more realistic assessment of the relationship offvalues to TWRs.The
fractional bets represent the average (of all possible values of the size
of initial bankrolls) of the integer bets.So the argument that we cannot
make fractional bets in real life does not apply, since the fractional bet
represents the average integer bet. If we made graphs of the TWRs at each
fvalue for the+2,−1 coin toss, and used integer bets, we would have to
make a different graph for each different initial bankroll. If we did this and
then averaged the graphs to create a composite graph of the TWRs at each
fvalue, we would have a graph of the fractional bet situation exactly as
shown.