| One of the first cognitive skills babies
learn is to identify certain shapes, like circles, squares,
and triangles. It is kind of funny, but technically-oriented
traders follow in these very infant footsteps when we
examine charts. A big part of technical analysis involves
identifying certain recurring shapes on price charts.
While I have pretty much outgrown circles, squares,
and triangles, one shape that endlessly fascinates me
today is the mighty parabola. My 4" thick Webster's
dictionary has a complicated definition for a parabola
that only a mathematician would love, but I like to
think of them as just plain old constantly accelerating
The proper definition of a slope is equally convoluted
unless you have a doctorate in applied math, but the
no-nonsense carpenters' definition is easy to understand,
rise over run. If you are laying piping or installing
a roof, the slope is critical. A roof that rises one
vertical foot for every ten feet of its horizontal distance
has a slope of 1/10, a rise of one over a run of ten.
Parabolas, or constantly accelerating slopes, are ubiquitous
in the financial markets. Once you start looking for
them they start to pop up everywhere, kind of like Elvis
The most famous example I am aware of
today is the enormous NASDAQ bubble of the late 1990s.
If you examine a graph
of the entire decade, you will note that the NASDAQ
was barely climbing in the early 1990s, it started accelerating
in the mid-1990s, and by the late 1990s it was shooting
vertical in a mania frenzy. The NASDAQ carved a massive
parabola! Gold left a similar parabolic slipstream in
its own wake during its gargantuan 1970s
These parabolas are so fascinating because they are
always unsustainable in financial markets. A price can
rise gradually in the early years of a major move for
quite a long time, but when the really big moves erupt
in the final mania years the parabola has already sown
the seeds for its own destruction. Once a parabola goes
vertical it becomes a kind of mathematical black hole,
its own weight guaranteed to implode it back in on itself.
If you imagine a decade-long financial parabolic pattern,
the final year may have a 50%+ gain. Well, if you extrapolate
this vertical rise out, the resulting numbers are just
plain silly. After 5 more years, this market would be
7.6 times higher. After 10 more years at +50% per year,
the market would be 57.7 times higher! Obviously this
is just absurd because far before these numbers are
reached all the capital on earth would have been exhausted.
Thus parabolas, when they hit their final vertical
zone, are absolutely guaranteed to fail in the near
future. Any other outcome is mathematically impossible.
As investors and speculators we have to remain aware
of this macro view on parabolas. If we have a decade-long
chart that is accelerating vertically with huge annual
gains, then we are witnessing a mania and need to sell
before the inevitable reckoning, usually a crash, arrives.
Parabolas are certainly not limited to strategic charts
though. Like the markets themselves, parabolas are totally
fractal in nature. A fractal is an identical shape or
pattern that appears at all different scales. The whole
art of technical analysis relies on this fractal nature
of price charts. Just as you can draw support and resistance
lines on any scale of chart from a few hours to a few
decades, parabolas too can be big or small but certainly
present at any scale.
I have been pondering parabolas and fractals of late
as I analyze price charts. I have been wondering if
parabolic shapes offer important trading clues of when
a market has become overbought or oversold in an intermediate
timeframe. When a parabolic formation, either bullish
or bearish, goes vertical, do the probabilities swing
in our favor for making trades in the opposite direction?
In order to analyze this thesis, we are going to take
a look at parabolas in the HUI this week. The HUI is
the premier unhedged gold-and-silver stock index, and
probably the best performing sector on the planet over
the last several years. The HUI is up a massive 614%
bull to date from its November 2000 lows to its latest
interim top in December 2003. Our subscribers, my partners,
and I have been blessed with enormous realized profits
trading this largely overlooked bull market.
Our first graph outlines the HUI's bull
market to date and reveals large parabolas, both bullish
and bearish, jumping out all over the place in the raw
HUI price data.
Before we delve into the HUI parabolas, please check
out the reference parabola in the lower right corner
of this chart. The dotted red line, of course, is the
parabola, with its constantly accelerating upslope.
But the black lines that frame it are revealing. While
they look like a curve, they are actually six perfectly
straight black lines drawn at sequentially increasing
Thus one price, increasing its slope a
half-dozen times or so over a major upleg or downleg,
will create a nice approximation of a true curved parabola.
This illustration is important because it helps show
that a very parabolic shape can form in a price chart
with only a relatively few changes in slope over the
life of an intermediate trend. I call this approximation
a linear parabola and recently used it in analyzing
a vicious silver
correction earlier this year.
The HUI's bull market to date has formed at least six
major parabolas, all labeled above. We rendered two
identical sets of parabolas in this chart, one directly
superimposed on the blue HUI price data with a second
matching copy pasted in a little higher. While these
two sets of parabolas are the same, it is useful to
view them both superimposed over the HUI to see how
well it conforms to them as well as independently from
the HUI so we can get a better sense of the index's
true underlying parabolic nature.
Parabolas are multiplying like rabbits in this chart,
but what is the whole point of this exercise? I suspect
that if we can recognize the terminal vertical stage
of each intermediate-term parabola in time, it can help
us determine when the HUI has a high probability of
rising or falling. We can then buy or sell accordingly
based on whether a bullish or bearish parabola is reaching
its unsustainable vertical stage and therefore hinting
that an intermediate trend change is imminent.
A bullish parabola is one that is carved in a rising
market, a HUI upleg. Parabolas 1, 2, 4, and 5 above
are all of the bullish variety. Now if you look carefully
at the lower set of parabolas actually superimposed
directly on top of the HUI, an interesting phenomenon
jumps out as these bullish parabolas go vertical. In
all four bull-to-date examples, the HUI entered a correction
without fail soon after the time the bullish parabolas
shot vertical and hence became mathematically unsustainable.
It is interesting that this observation has a fractal
bent too. Whether a bullish parabola lasted a long time
like last year's massive number 5 or a short time like
the number 1 that kicked off this bull market, once
the HUI started shooting vertical a normal healthy pullback
loomed. Vertical price movements, even over the intermediate
term, just cannot be sustained. When an intermediate
bullish parabola is shooting vertical, it is probably
prudent for us speculators to sell HUI longs, throw
short, buy puts, or at the very least ratchet up our
During corrections following these vertical-stage bullish
parabolas, inverted bearish parabolas form. Parabolas
3 and 6 above are perfect examples. In both cases the
HUI reached a new bull-to-date interim high before correcting.
These corrections, however, tend to start out slowly.
Once a new high has been reached people are very hesitant
to be anything but euphoric so early-stage corrections
often witness relatively modest declines.
As the correction picks up steam though, its downslope
accelerates. As the latest interim highs fade in the
distance and folks grow more nervous, selling intensifies
considerably. The price decay curve tends to ride an
inverted parabola, falling with a constantly accelerating
downslope. Eventually people get downright scared and
the price drops vertically, the final stage of a bearish
parabola. But just as normal parabolas are not sustainable
once they shoot vertical, neither are inverted ones.
When these bearish parabolas fall vertical, odds are
the correction just witnessed is pretty much over. When
investors and speculators recognize these inverted parabolas
reaching their vertical maturity stage, they can add
new long PM stock positions, close out any shorts, or
buy call options. Probabilities definitely favor the
HUI launching a new bullish upleg once an inverted bearish
parabola shoots vertical in a correction, and the sharp
decline this past April is a perfect example.
Since the markets are ultimately just a study in probabilities
anyway, it makes sense to watch for these intermediate-term
parabolas. They are not precise trading signals by any
means, but they seem to do a fairly good job in signaling
the general season, whether a major upleg or downleg
is getting long in the tooth and hence bets in the opposite
direction may be profitable in the months ahead.
The HUI parabolas help us understand when the probabilities
for a major intermediate trend change are ballooning,
all because parabolic ascents and descents become inherently
unstable and unsustainable in their terminal vertical
This idea is pretty simple, but I have been struggling
with how to measure it. As an active speculator, I want
to trade on hard empirical data that I can quantify
and compare, not just the subjective "I have a
feeling this is getting too vertical here". I suspect
the HUI parabolas would become the most useful, not
to mention the most emotionally neutral, if there was
some way to precisely measure them.
After weeks of thinking about this problem, I am disappointed
to admit that I haven't found an elegant solution. One
of the reasons I am writing this essay is to solicit
feedback for superior ideas to measure these unfolding
parabolas. So if you have a better idea than this admittedly
sorely lacking one I am going to present, please drop
me an e-mail. If you solve this parabola measurement
puzzle I would be honored to discuss the solution in
a future essay, thanking you by name and granting you
all the glory.
Parabolas are constantly accelerating slopes, and a
slope is a rise over run. While not satisfied with this
interim idea, the best I have been able to muster is
measuring HUI parabolic slopes as percentage gains over
time. Unlike building a roof where both the rise and
run are in feet, in charts the rise is in price and
the run is over time. Time and price are not synonymous
so this isn't really a pure slope comparison.
After playing with all kinds of different timing mechanisms
to measure the X-axis of time, the one that struck me
as the most interesting was a 15-trading-day percentage
change in the HUI close as an approximation of slope.
15 days is three trading weeks, which seems to feel
about right given how long interim topping and bottoming
processes at the apexes of parabolas typically take
to unfold. If a parabola can shoot vertical to a big
gain or loss over three trading weeks, then perhaps
it is vertical enough to anticipate a trend change.
Our final graph attempts to quantify this
initial attempt to measure parabolic slopes. The HUI
slides over to the right axis, and the 15-day percentage
change in the HUI is graphed on the left axis in red.
While the raw red squiggly line looks like the electrocardiogram
of Alan Greenspan the day gold crosses $1000, it actually
hides some provocative data. While this can't be the
ideal slope measurement, it still illustrates the general
validity of this parabola trading principle.
Since this graph is busy even by my loose standards
of busy-ness, we didn't draw in the parabolas again.
But once you see them as we did in the first graph,
it is really hard to miss them. Kind of like those computer-generated
pictures hidden in white and black dots that look like
random visual noise at first, once you finally see the
image hidden within the first time you can't help but
focus on it instantly when you look again.
The key to understanding this chart is the blue HUI
line. We took the greatest trading opportunities in
this entire HUI bull market to date, including all the
interim highs and lows signaled by the parabolas above,
and we drew vertical blue lines through them. In hindsight
these were the best moments to trade, both long and
short, to maximize profits during the HUI's bull market.
Each vertical blue line marking great moments of opportunity
for HUI trades intersects the violent red 15d percentage
change line. These intercepts reveal where this 15d
approximation of the HUI's slope was meandering at each
opportune moment to trade the index.
Per the parabola theory, the 15d slopes of the HUI
should be extremely high near major interim tops and
extremely low near major interim bottoms, marking the
final vertical stage of major parabolas. This indeed
proved to be the case as the graph above reveals. This
HUI slope chart is easiest to interpret if we start
our analysis at major interim highs in the HUI.
The five red percentage numbers on top of this chart
show where the HUI 15d percentage change slope approximation
happened to be at or near major interim highs in the
HUI. The blue numbers underneath these percentage changes
show the offset, in days, between the 15d HUI slope
top and the actual interim top in the HUI itself. A
-1d number indicates that the 15d HUI slope topped one
day before the HUI itself, while a +1d indicates the
slope topped one day after the HUI.
It is interesting that every major HUI interim top
happened when the 15d slope of the HUI was extremely
high, running from a 23% gain over the past 15 trading
days alone on the low side to a massive 35% on the high
side. Can you imagine the HUI blasting up 35% higher
in only 15 trading days? Wow! The average 15d HUI slope
at these major tops was 28%, very high. Thus if we see
a situation in the future where the HUI has advanced
by 25%ish or more in only three weeks, odds are its
current parabola is shooting vertical and is hence unsustainable
so a healthy correction is due.
The offsets between 15d HUI slope tops and the actual
HUI tops are revealing as well. First, note that the
three biggest HUI uplegs to date, parabolas 1, 2, and
5 from the first graph, all ended with fresh new bull-to-date
highs, slopes nearing vertical, and trivial offsets
of only plus or minus a single trading day. So it looks
like the maximum 15d HUI slopes do indeed occur very
close to major new bull-to-date highs signaling the
end of each major upleg.
But if we get stuck in a largely sideways trading range
where the uplegs and corrections are fairly minor, like
in late 2002 and early 2003, then the offsets balloon
quite a bit. Thus the parabolas, or at least this imperfect
15d approximation of the HUI's slope, are not as useful
in a sideways grinding market between major uplegs and
Shifting our attention to the major interim lows in
the HUI, the 15d HUI slope numbers and offsets are still
useful but less precise. Of the five greatest buying
opportunities in the HUI since its bull launched, the
average 15d HUI slope during these times weighs in at
-19%, but the range is pretty broad running from -12%
to -25%. In light of this data, if the HUI is correcting
and an inverted bearish parabola is forming, once the
HUI trades down 15%+ in only three weeks odds are that
a major new upleg is imminent.
The offsets are all over the map at these major interim
lows in the HUI, with its 15d slope tending to bottom
anywhere from six weeks to one week before the HUI itself.
On the bright side, I looked through all of these offsets
individually, both on the high and low side, and found
that generally even buying or selling as appropriate
at the 15d HUI slope high or low would have been very
profitable. Usually the difference between the HUI on
the 15d slope extreme day and the actual interim extreme
day in the index itself was trivial, within a few percent
While this particular approximation of the HUI parabolas
is certainly not precise enough to define a hard and
fast trading tool, it does still seem useful for helping
us understand when probabilities swing in our favor
for major uplegs or corrections.
If the HUI is in an upleg shaped like
a parabola, that is accelerating higher, and the index
has rocketed up 25% or more in only three trading weeks,
then odds are that particular upleg is waxing too euphoric
and a correction is imminent. In these situations, if
other trading indicators concur, speculators can liquidate
HUI longs, throw short, buy HUI
puts, or at the very least ratchet up their trailing
stops in anticipation of weakness.
Conversely if the HUI is languishing in
a correction shaped like an inverted parabola, that
is accelerating lower, and the index has plummeted by
15% or more in only 15 trading days, then odds are that
particular correction is growing too fear-laden and
an upleg is imminent. During these exciting times like
this past May, if other trading indicators concur, investors
can buy PM stocks while speculators can buy stocks,
close shorts, or buy
If you are interested in trading these
major uplegs and corrections in the HUI, which have
been very profitable bull to date, please consider subscribing
to our acclaimed monthly Zeal
Intelligence newsletter. We have been painstakingly
analyzing and actively trading this entire 614% HUI
bull to date and have been blessed with some awesome
realized gains in past uplegs and unrealized gains in
this current upleg. We will continue to refine our analysis
and tools and alert you to potentially stellar HUI-related
trading opportunities going forward.
The bottom line is intermediate parabolas formed on
price charts have great potential for helping speculators
define probabilities weighing whether an intermediate
trend change is imminent. When you see any multi-month
trend accelerating vertically to the upside or downside,
chances are this move is unsustainable and the inevitable
reversal is looming.
While I don't particularly like this clumsy 15d HUI
slope approximation described here, I am confident that
there is some elegant way to measure these parabolas
empirically and perhaps even distill these measurements
into a promising trading tool. Please let me know if
you figure it out!
Adam Hamilton, CPA
October 15, 2004
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