Optimalf 157
Further, the risk involved with a trade is a function of the chronology of the
trade, a fact we would be forced to ignore.
Finding Optimalfvia Parabolic Interpolation
INTERPOLATION
Originally, I had hoped to find a method of finding the optimal f by way
of a single equation like the Kelly formula. In finding the optimalfwe are
looking for that value forfwhich generates the highest TWR in the domain
0 to 1.0 forf. Sincefis the only variable we have to maximize the TWR for,
we say that we aremaximizing in one dimension.
We can use another technique to iterate to the optimal fwith a little
more style than the brute methods already described. Recall that in the
iterative technique we bracket an intermediate point (A, B); test a point
within the bracket (X); and obtain a new, smaller bracketing interval (either
A, X or X, B). This process continues until the answer is converged upon.
This is still brutish, but not so brutish as the simple 0 to 1 by .01 loop
method.
The best (i.e., fastest and most elegant) way to find a maximum in
one dimension, when you are certain that only one maximum exists, that
each successive point to the left of the maximum lessens, and that each
successive point to the right of the maximum lessens (as is the case
with the shape of the fcurve), is to useparabolic interpolation.When
there is only one local extreme (be it a maximum or a minimum) in the
range you are searching, parabolic interpolation will work. If there is more
than one local extreme, parabolic interpolation will not work (see Fig-
ure 4.12).
With this technique we simply input three coordinate points. The axes
of these points are the TWRs (Y axis) and thefvalues (X axis). We can find
the abscissa (the X axis, orfvalue corresponding to the peak of a parabola)
by the following formula, given the three coordinates:
ABSCISSA=X2−. (^5) *
(X2−X1)^2 (Y2−Y3)−(X2−X3)^2 (Y2−Y1)
(X2−X1)(Y2−Y3)−(X2−X3)(Y2−Y1)
(4.15)
The result returned by this equation is the value forf(or X if you will)
that corresponds to the abscissa of a parabola where the three coordinates
(X1, Y1), (X2, Y2), (X3, Y3) lie on the parabola.
The object now is to superimpose a parabola over thefcurve, change
one of the input coordinates to draw an amended parabola, and keep on
doing this until the abscissa of the most recent parabola converges with the