# The Golden Ratio and Fibonacci Modelling Applied to the S&P 500

Well, here we are again, about to look at some sticky numbers that apply to what has become the grandest mass mania phenomenon ever experienced by mankind, the obsession with financial markets. This essay will focus on examining past inter-market relationships as it pertains to the S&P 500 (SPX) and CBOE Volatility Index (VIX) in the endeavour to construct a model from which the rhythmic signatures of co-relations these markets produce, can be used to forecast price extremity targets and trend changes. The primary thrust of this piece then, is the study of the S&P 500 in an effort to identify progression and regression coefficient factors based on Fibonacci principles to help us understand and trade the beast, based on it's 'harmonic signature'.

In an effort to be expedient, I will not delve into the entire background base of knowledge one should possess about Fibonacci's theories and principles necessary for you to be able to fully enjoy the scope of this discussion. Instead, I would encourage you to refer to my previous paper that deals with this same exercise which pertains specifically to the precious metals complex, attached below, in order to familiarize yourself with the basics surrounding the principles we will employ in this treatise, as well. Although you have no reason to think the model constructs that were arrived at in my previous work regarding Fibonacci's principles applied to the precious metals markets could apply directly, and in proportion, to what could be perceived as unrelated markets, but I can assure you, they do. http://www.lemetropolecafe.com/Pfv1.cfm?pfvID=2938&SearchParam=CAPTAINHOOK

The first task at hand is to formulate a basic construct for the model we will employ for this exercise. In doing so, we will establish a base from which to gauge our observations, in order to refine the model into a functional predictive tool. In the case of the relationship between the SPX and VIX, it makes a lot of sense to first look at averaged outcomes, or the 'mean' values of the ratio between the two, in order to help establish the trend, dimension range, and volatility characteristics, as it pertains to values and time. i.e. the 'harmonic signature'. Thus, the first element we will examine is the 'mean' in the intermediate trend sequences of the VIX / SPX ratio to see if there is an identifiable harmonic signature that we can formulate into progression and regression coefficient factors. (See Figure 1)

**Figure 1**

As one can observe above, the last completed progression sequence mean value
was .04, and it began the move from .02, with the first sequence originating
at .01 and progressing to .02. As a base then, we can conclude that this relationship
is progressing at a coefficient factor of "2", which has statistically
proven to be the **Phi** within the relationship, in terms of defining the
progression sequences. Although this exercise possesses limited utility in
projecting value extremes by itself, it does establish the basis of our model,
from which we can now build on to accomplish primary objective behind constructing
this model, which is to formulate a reasonably accurate predictive tool to
identify extreme values likely to be printed in the VIX and SPX. So, what is
the important information we should take away from the above chart then?

- The
**Phi**progression

If you read my previous essay attached above, you know that constructing a
model of this sort requires not only identifying the progression sequences,
but also delineating the harmonic rhythm of the corrective phases, as well.
The objective in doing so is to identify the corrective coefficient factor,
in order to be able to project likely regression impulses into the future,
within the overall harmonic signature prevalent in the model. This corrective
coefficient factor is labelled **phi**, and once identified, represents
the expected corrective impulse experienced after a progression sequence is
completed, within the overall trend. It should be noted that the trend for
the SPX is down, but the trend for the VIX/SPX ratio is higher.

- The
**phi**regression coefficient factor that defines the mean regression sequences thus far in the advance of the VIX/SPX ratio is ".5", as each progression sequence has retraced 50 % of it's previous advance, and we will now identify**Phi**as "2" and**phi**as ".5" for definitive purposes in our base model, and they will be used as a**predictive/inter-verification tool**throughout the remainder of this study as the definitive 'harmonic signature' for the VIX/SPX ratio's volatility characteristics, in terms of progression/regression sequences, or cycles if you wish.

Unlike the outcome in my previous essay using this model, the identification of Phi/phi as it pertains to the VIX / SPX ratio is not going to be quite so easy as it pertains to the extreme values involved. This makes the utility of using mean values as a predictive tool essentially insignificant on a stand alone basis, with regard to aiding investors in identifying swing points, however it does give a base model to work from.

Now that we have a base model from which to work, it's time to turn up the volume, and take a more precise look at just what is going on inside of the VIX/SPX ratio relationship. More specifically, can we apply the progression and regression coefficients identified using mean values to the extremes experienced at intermediate turn points? In one word, the answer is "yes". But it's going to take a bit of work to fully understand just how the identified Phi/phi coefficients should be applied. In order to come to this understanding, it is necessary to examine the S&P 500's trading characteristics in isolation, to see just what has happened since it has topped, and is now in a significant corrective sequence. (See Figure 2)

**Figure 2**

Above, you have a snapshot of the big picture in terms of where we are in
the S&P 500's corrective sequence, including what I consider to be the
correct Elliott Wave count. As you can see, as far as the big picture is concerned,
thus far the SPX has travelled off of its highs a full 50% of the index's total
point value and retraced **less** than the standard 38.2% Fibonacci measure
of the full sequence. This fact should be quite disturbing to the bulls, as
I believe the market has now topped. So the question then arises, 'does the
50% retrace of the index's total point value constitute a completion of the
corrective sequence for the SPX?' Again, in one word, I believe the answer
is "no". And here's why.

If you break the corrective sequence down into it's component parts based on the wave structure of the aggregate move, you actually get quite a different picture, one that conforms to our model exactly, in terms of the harmonic signature the progression and regression coefficients (Phi/phi) do well to define. When we look at the parameters associated with the volatility characteristics of Wave A down in isolation, it becomes apparent that the SPX is indeed vexing within the identified signature. (See Figure 3)

**Figure 3**

Notice in the above chart that the phi retracement was exactly 50% of the initial impulse down labelled Wave A. Although it may be difficult to ascertain on first glance, the above chart is based on closing prices, as the purpose of this delineation is to show that each of the progression sequences downward so far are characterized by two approximate equal impulses, which can be defined using the Phi coefficient of 2, in that the next impulse lower from the starting point of Wave A is a doubling of the first impulse down from the top. (See Figure 4)

**Figure 4**

There is a clue in the above chart as to what the character of the future
trading pattern in the S&P 500 is going to be. While it is true that future
gyrations should fall within the base model parameters, episodes of increased
volatility will also help form the pattern, as evidenced by the fact Wave D
experienced an initial surge to the 61.8% retrace level, which was not present
in the first retrace sequence comprising Wave B. And now that we have established
that fact, we can progress into some more congruous dimensional measures in
order to more exactly enable our model to project likely turn points with a
higher degree of accuracy. Volatility seems destined to increase as the SPX
progresses into the final leg of this bear market sequence, and we would like
to employ an second examination method as a **verifier **to our base model,
in order to strengthen our conclusions regarding likely turn point values and
timing. Before we move on however, it seems appropriate at this point to summarize
what we have learned in the three charts directly above.

- In examining the downtrend in the S&P 500 in isolation, and just as
with the VIX/SPX ratio's mean value volatility characteristics, our model
has established that the harmonic signature is characterized by the progression
and regression sequence coefficient factors of 2
**(Phi)**and .5**(phi)**respectively, and that volatility is increasing as the overall correction matures.

In order to examine the increasing volatility within the overall corrective
sequence in which the SPX is currently engaged, it would be appropriate for
you to digress for just a moment in order to take a quick glance at Figure
1 once again. Now that you have done this, you will be better able to understand
that we are about to pivot into exponential projection mode, as it pertains
to identifying likely absolute extreme targets within the coming progression
and regression sequences. Within this endeavour, my observations, initially
performing a few basic calculations, led me to the conclusion that the 'Golden
Ratio', widely defined by Fibinacci aficionados as the factor 1.618 for **Phi **and
.618 for **phi, **fit the signature perfectly in the context of an anticipated
rise in volatility. (See Figure 5)

**Figure 5**

Within your observations of the above, one should notice that Golden Ratio
projection does not exceed the base model Phi progression coefficient (2),
rigorously established above, to any large degree. In fact, one has to marvel
at the close proximity of the two values derived from completely separate methods
of calculation, with each acting as a **verifier **for the other. Does it
matter which method we use then, for further projection calculations?

Based on the degree of the retracement in the VIX/SPX ratio off the highs of it's last progression sequence, it would definitely seem appropriate to continue to monitor the changing complexion of the harmonic rhythm using both methods to see if values vault out of base model parameters, as the extent of the correction fell within the extremes of Fibonacci retracement numbers. (See Figure 6)

**Figure 6**

As you can see above, the retracement, characterized by a five-wave affair, has gone to the standard extremity of the Fibonacci doctrine, a 78% correction. Any further deterioration past this point would have to caution bearish assumptions concerning the current downtrend in the S&P 500, at least on a temporary basis. However, based on the work we have done here, one can make a very strong case the retracement in VIX/SPX ratio is now complete for the interim, and that although there may be some transitory topping activity in the SPX, we should soon resume the primary bear market established at the turn back in the year 2000. If this does indeed occur, it will be easy to label this next sequence down as Wave E of Primary Wave A, and the targets using the methods derived in Figure 5 may upset a few of the bulls. (See Figure 7)

**Figure 7**

Although introduced in Figure 5, one should notice the incorporation of **time
cells** into the model, and that these spans have been marking the extremes
and turning points quite accurately for not only the bear phase, but for
the bull phase, as well. Utilizing the coefficient factors derived in Figure
5, the calculations in Figure 7 demonstrate that the SPX should see a bottom
of approximately 585 to 600 in May of 2004, which should correspond to a
print of ~ 80 on the VIX. This would be quite a distance from the VIX's current
proximity of ~20. (See Figure 8)

**Figure 8**

Pictorially, and from a symmetrical perspective, the parabolic bowl captures the essence of the move, both in terms to dimension and time, in that we are approximately half way through the present time cell. Based on the above vigorous statistical examination of the SPX's harmonic signature and the overall appearance of the chart below, it would seem likely that the S&P 500 is about to roll over to complete the sequence (cycle). In fortification of this view, one should notice that next May would mark a two thirds retrace of the previous three consecutive 7.5 quarter time cells off the advance from 600 in the SPX. (See Figure 9)

**Figure 9**

We have now covered quite a bit of material and I have left a lot unsaid for expediency purposes. If one were so inclined there is a very prominent clue in the above material which even today, with the SPX's bear market in it's infancy, would enable you to project it's final nadir and timing using the model constructs. I encourage you to try as an exercise if you are a trader. And if you are an investor looking for a good time to get out of the market, I would be thinking hard on that proposition at present.

One important observation that may not be easily discernable from the above material, is that as the bear market in the S&P 500 progresses, increased volatility will not alter the final trading values that have now been identified by the harmonic signature, only the timing may be altered due to intervention.

In summation, it seems appropriate to list our findings at this time, in order to complete a more readily identifiable picture for you, as follows:

- The
**Phi**progression - The
**phi**regression coefficient factor that defines the mean regression sequences thus far in the advance of the VIX/SPX ratio is ".5", as each progression sequence has retraced 50 % of it's previous advance, and we will now identify**Phi**as "2" and**phi**as ".5" for definitive purposes in our base model, and they will be used as a**predictive/inter-verification tool**throughout the remainder of this study as the definitive 'harmonic signature' for the VIX/SPX ratio's volatility characteristics, in terms of progression/regression sequences, or cycles if you wish. - In examining the downtrend in the S&P 500 in isolation, and just as
with the VIX/SPX ratio's mean value trading characteristics, our model has
established that the harmonic signature is characterized by the progression
and regression sequence coefficient factors of 2
**(Phi)**and .5**(phi)**respectively, and that volatility is increasing as the overall correction matures. - Golden Ratio
**(Phi = 1.618)**calculations were incorporated into the model as a**verifier**, and the deviation from the base model's projected values were found to be insignificant, allowing for an SPX spike low of 585, from the base model's projected nadir of 600, and corresponding to a print of approximately 80 on the VIX. - It has been established through the introduction of
**time cells**into the model that the targets above should be seen sometime in early to mid-May of 2004.

Well, I think that just about covers it, other than to say trading it won't be so easy, as many speculators have already discovered. One must remember that these are long periods of time involved in this process and only proper structure and instrumentation will increase your odds of trading the beast successfully. I believe it is safe to say that most should not partake in these kinds of endeavours, and that only the most skilled and vigilant will enhance their fortunes in the speculation game. If you must however, the above should provide you with an edge and the appropriate perspective to plan your investment strategies well into the future. Speaking of which, I will be sure to return next spring to update the situation and provide you with a roadmap for what to expect as the S&P 500 (markets) make the big turn into Primary Wave B higher.

Good investing,

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