The PHI-ellipse is a unique Fibonacci trading tool that can only be drawn by a computer.

As mentioned in Chapter 3, PHI-ellipses identify underlying structures of price moves and circumvent price patterns. When a price pattern changes, the shape of the PHI-ellipse circumventing the respective market price pattern changes, too. Nearly all market price moves follow the pattern of a PHI-ellipse.

PHI-ellipses are related to the Fibonacci ratio PHI in a similar way as the Fibonacci summation series, as well as corrections and extensions (see Chapter 4).

Generally speaking, the ratio of major axis a to minor axis b defines the shape of an ellipse. Ellipses are turned into PHI-ellipses in all cases where the ratio of major axis to minor axis ex = (a + b) is a member number of the PHI series.

To make PHI-ellipses work as tools for chart analysis, we have applied a (proprietary) transformation to the mathematical formula that describes the shape of the ellipse. We still consider the ratio of major axis a to minor axis b of the ellipse, but in a Fischer-transformed way, in mathematical terms ex = (a + b)*.

We introduce PHI-ellipses as instruments for investments counter-trending to market action. We observe whether a price move stays within a PHI-ellipse and invest accordingly if a price move breaks out of a PHI-ellipse at the very end.

Historical charts show that almost all price moves in commodities, futures, stock index futures, or stocks can be circumvented with a PHI-ellipse. However, finding the correct PHI-ellipse is an art. It takes skill, experience, patience, and trust in the analysis to effectively use PHI-ellipses as Fibonacci investment tools.

It is impossible to forecast the final shape of a PHI-ellipse at the beginning of a price move. As we later prove on various examples,

PHI-ellipses may follow one after the other symmetrically. Small PHI-ellipses may be followed by long PHI-ellipses, or PHI-ellipses may be connected with each other, and so on. The challenge is to correctly interpret price moves and select PHI-ellipses accordingly. Once investors learn how to identify the appropriate market pattern, working with PHI-ellipses becomes easier.

The remainder of this section is divided into two parts. We first examine the shape and the slope of PHI-ellipses, and include approaches for attaching and overlapping PHI-ellipses. In the second part, we address entry and exit rules for the generation of trading strategies from PHI-ellipses.

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