Leonardo Fibonacci was a great Italian mathematician who lived in the thirteenth century who first observed certain ratios of a number series that are regarded as describing the natural proportions of things in the universe, including price data.  The ratios arise from the following number series:  1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144 ……
This series of numbers is derived by starting with 1 followed by 2 and then adding 1 + 2 to get 3, the third number. Then, adding 2 + 3 to get 5, the fourth number, and so on.
The ratios are derived by dividing any number in the series by the next higher number, after 3 the ratio is always 0.625.  After 89, it is always 0.618.  If you divide any Fibonacci number by the preceding number, after 2 the number is always 1.6 and after 144 the number is always 1.618.  These ratios are referred to as the “golden mean.”  Additional ratios were then derived to create ratio sets as follows:
Price Retracement Levels
0.236, 0.382, 0.500, 0.618, 0.764
Price Extension Levels
0, 0.382, 0.618, 1.000, 1.382, 1.618
The first set of ratios is used as price retracement levels and is used in trading as possible support and resistance levels.  The reason we have this expectation is that traders all over the world are watching these levels and placing buy and sell orders at these levels which becomes a self-fulfilling expectation.
The second set is used as price extension levels and is used in trading as possible profit taking levels.  Again, traders all over the world are watching these levels and placing buy and sell orders to take profits at these levels which becomes a self-fulfilling expectation.
Most good trading software packages include both Fibonacci Retracement Levels and Price Extension Levels.  In order to apply Fibonacci levels to price charts, it is necessary to identify Swing Highs and Swing Lows.  A Swing High is a short term high bar with at least two lower highs on both the left and right of the high bar.A Swing Low is a short term low bar with at least two higher lows on both the left and right of the low bar.
Fibonacci Retracement Levels
In an uptrend, the general idea is to go long the market on a retracement to a Fibonacci support level.  The price retracement levels can be applied to the price bar chart of any market by clicking on a significant Swing Low and dragging the cursor to the most recent potential Swing High and clicking there.  This will display each of the Retracement Levels showing both the ratio and corresponding price level.  Let’s take a look at some examples of markets in an uptrend.  The same points made by these examples are equally applicable to markets in a downtrend.
Example 1:  Here we plotted the Fibonacci Retracement Levels by
clicking on the Swing Low at about \$71.31 and dragging the cursor to the Swing High at about \$89.83.  You can see the resultant levels plotted by the software.  Now the expectation is that if the market retraces from this high it will find support at oneof the Fibonacci Levels, because traders will be placing buy orders at these levels as the market pulls back.

Example 1.1:Now let’s look at what actually happened after the Swing High occurred.  The market pulled back right through the 0.236 level and continued the next day through the 0.382 level before finding support.  After a few days, the market resumed its upward move.  Clearly buying at the 0.382 level would have been a good short term trade.
Example 2:  Again, the Fibonacci Retracement Levels were plotted on the chart in the same manner as described in Example 1.  Again, we are looking for the market to retrace from the Swing High and find support at one of the Fibonacci levels.
Example 2.1: Now let’s look at what actually happened.  The market again pulled back right through the 0.236 level and continued to pull back until it found temporary support at the 0.50 level (a lot of buyers at this level).  However, once the buying power was exhausted, the market continued to retrace all the way down to the 0.764 level before resuming its upward trend.  In this case, buying at the 0.764 level would have been a good short term trade.

Example 3:  Here’s another example.If the market retraces from the Swing High, where will it find support?

Fibonacci Price Extension Levels
In an uptrend, the general idea is to take profits on a long trade at a Fibonacci Price Extension Resistance Level.  The Price Extension Levels can be applied to the price bar chart of any market by clicking on a significant Swing Low and dragging the cursor to the most recent Swing High. Then by clicking on the Swing High and back down to the retracement Swing Low and clicking there. This will display each of the Extension Levels showing both the ratio and corresponding price level.Let’s take a look at some examples of markets in an uptrend.  The same points made by these examples are equally applicable to markets in a downtrend.
Example 5:Here we plotted the Fibonacci Price Extension Levels by clicking on the Swing Low at about \$38.20 and dragged the cursor to the Swing High at about \$47.67 and then down to the retracement Swing Low.  You can see the resultant levels plotted by the software.Now the expectation is that if the market continues higher it will find resistance at one of the Fibonacci Levels, because traders will be placing sell orders at these levels to take profits on there long trades.
Example 5.1:Now let’s look at what actually happened after the retracement Swing Low occurred.  The market rallied making new highs pausing at the 0.382 level and again at the 1.000 level after a retracement down it rallied again going right through the 1.382 and 1.618 levels.  Taking profits at the 0.382 level would have been premature, but taking profits at the 1.000 level would have made a nice trade.