Finding patterns with the Lucas time series

Financial markets are similar to the World Series. Each game is filled with twists and turns with an uncertain outcome until the final pitch is thrown. Trends work the same in free markets as they do in sports contests. They change. Sometimes we can anticipate them. Sometimes we can take advantage of them.

Technical analysis provides us with tools to ultimately take advantage of trend changes. The most overlooked aspect of technical analysis is the time factor. Technicians and traders have difficulty recognizing when a market will reverse. Many technicians pay a great deal of attention to price studies such as stochastics, moving average convergence-divergence (MACD), the relative strength index, moving averages, Bollinger Bands and countless others. These are good tools, but by their nature are lagging, and when their message will play out in real time isn’t stable. MACD and stochastics can display divergences against the main trend for days or weeks. By the time certain moving averages cross over, a good portion of the move may already be through, leading to a whipsaw.

Random walk theorists believe it is impossible to time markets, but cycle analysts come close to identifying reversals. Static cycle theory states that markets will either crest or trough at fixed periods. For example, the four-year Presidential cycle is the most popular sequence followed by the trading community. Analysts suggest a crest will adapt to either a bull or bear market. In bull markets, a cycle will peak past the midpoint, which is known as a right translation. A bear market will peak before the midpoint, which is known as a left translation. This type of analysis is very confusing and four years later many analysts still can’t agree if the true cycle bottom to the current bull market in stocks occurred in October 2002 or March 2003.

Traders need a more practical method of making financial decisions. Fibonacci and Elliott Wave come closest to identifying the structure and shape of financial markets. R.N. Elliott argued that the Fibonacci sequence lays the groundwork for understanding the Elliott Wave Principle. Fibonacci cycle analysts identify major pivots in the market, such as the January 2000 high in the Dow or the bear market bottom in October 2002. They count the number of calendar or trading days from those pivots to determine high probability time windows where a reversal can take place.

Markets tend to turn in all degrees of trend on Fibonacci number bars away from an important pivot, plus or minus one bar. The problem for many Fibonacci analysts is markets don’t always turn on Fibonacci bars (see “Lucas vs. Fibonacci,” below). This doesn’t mean that the Fibonacci-based methodology does not work.

There is another sequence related to Fibonacci known as the Lucas sequence. French mathematician Edouard Lucas (1842 to 1891) uncovered this series of numbers. The significance of Lucas is that as we go higher into the sequence, the relationship of the last two numbers comes closer to a perfect 0.618:1.618 ratio.

Most Fibonacci or Elliott practitioners are unaware of Lucas. So is this another one of those obsolete theories that should be buried in the back of a math textbook?

Understanding how the Lucas series reveals itself in markets can help all market participants make better decisions. These cycles or number sequences offer a superior pattern recognition methodology that works equally well for the institutional currency trader as well as the independent intraday E-mini trader trading from his kitchen table.


The last major correction in the Philadelphia Stock Exchange Gold and Silver Sector Index (XAU) from November 2004 to May 2005 completed a Lucas cycle. The anatomy of the trend included a secondary high on the 76th day of the move and a final leg of 47 daily bars that finally clustered to form a bottom on the 124th day (Lucas number 123 + 1) of the entire move. On a weekly basis, from the January 2004 high to the October 2005 low, the Nasdaq traced out three important corrective phases which completed in 29, 17 (Lucas number 18 minus 1) and 11 weeks. See “Doing the three-step,” above.

Unfortunately, most patterns are not gift wrapped so perfectly. Most legs will have some combination of Fibonacci and Lucas relationships. Like snowflakes or Elliott waves, no two patterns are ever exactly the same. However, they repeat in some fashion, over and over.

Consider fictional Trader A. He prefers a daily time frame and will trade either the Nasdaq futures contract or the QQQQ. “Spring recovery” above, shows the recent February low to April high, which turned out to be the final leg before an important reversal. This chart exhibits many of the characteristics of how the time sequences work in all degrees of a trend. Keep in mind these sequences work either on a low/high, low/low or high/high basis.

Also, there is certainly some art to applying this analysis. Interpreting pivot bars requires a bit of common sense, as in this case the black candle on Feb. 12 closed in the bottom half of the bar. In essence, the first wave started the next day and left a tail on the 13th bar, which is a common relationship for these cycles. The second wave bottomed on the 18th bar of a low-to-low cycle. The main part of the move topped early on the 21st bar off the March low. The final top of this five-wave sequence created a complete 47-day low-to-high cycle, which also clustered with a 29-day move off of the March low.

When two or more relationships cluster at a single point in time, the probability of a change in trend is stronger. The reversal bar peaked on the open and created a large black candle on the 30th bar, but keep in mind that all time windows are plus or minus one bar. The new downtrend left a small pivot on the 12th bar and turned down in earnest on the 13th bar.

Now consider Trader B. This trader is an institutional currency trader who maintains positions on a weekly time frame or longer. The 2005 rally in the dollar traced out a perfect Lucas relationship. This example uses a continuous weekly U.S. dollar chart (see “Backing the buck”), but it is representative of any currency pair.

In his trading rules, W.D. Gann states the safest entry is not trying to catch a bottom but enter on the first retracement. As most Elliotticians know, the third wave (called the C wave), which is the most powerful move, takes place after the first retracement. This is a point on the chart where the risk/reward ratio is most favorable. In this case, sentiment was extremely bearish as a large white candle in the last week of 2004 announced a potential trend change. Using candlestick reversal patterns, the 11-week time window produced a decent harami reversal line that confirmed the low as the white candle covered more than half of the prior black candle. Scaling down to a daily chart would have produced a more precise entry.

For those who use momentum indicators such as RSI or MACD, these weekly indicators were in the process of turning up and the time dimension produced the best risk/reward ratio of the entire sequence. For those who missed that entry, Lucas provided one more chance to get in as there was a small pullback and a white candle created a small low-to-low sequence seven Lucas weeks later. From that point, the move consumed most of the year. Momentum peaked on the 16th week of the third wave. Another pullback took place, but Elliotticians would recognize a five-wave sequence near the high. Note how MACD did not confirm the final leg from 86 to 92.

The problem most traders have is when a divergence appears. Divergences can continue for days or weeks. From the point the June high was taken out, this bearish divergence persisted for three weeks.

This particular leg produced a top when two Lucas relationships collided. The final leg registered 11 weeks while the entire trend traced out the tail in the 46th week, well within the 47-week time window. The trend changed a couple of weeks later when a large black candle closed at the bottom end of the range of the 46 to 47 week time window.

The actual USD/JPY chart shows a similar progression. As we can see, when we combine the time window, the candle pattern as well as the bearish non-confirmation in the MACD, we also have recipe for a high probability short entry. Understanding the market “speaks” in these time sequences adds confidence to a trader pulling the short trigger after a year-long rally.


This methodology holds up. These sequences have been traded on thousands of hours of intraday time charts, down to a one-minute frame on the mini charts of the major averages. They’ve also been tested on a daily scale back to 1998 on the major averages plus the metals, bonds and dollar. The Dow weekly chart has been tracked back to the October 1929 top. Indeed, the entire bear market from the October 1929 high through the July 1932 bottom completed in 143 weeks, which was one week shy of a Fibonacci 144.

It is not only possible to catch turns on the exact bar; it is possible to anticipate them in advance. What works on a one-minute or five-minute time frame also works on a daily, weekly or monthly time frame universally on all of these charts to form a precise pattern recognition methodology.

In May these cycles clustered in four time frames to create a reversal in the Dow less than 100 points from the all-time high. The Dow turned 56 weeks off the April 2005 pivot and the 30th week off the October 2005 low. The turn on May 10 also clustered on the 18th day off the April 17 pivot and 76th day off the Jan. 23 pivot. These points are either plus/minus one or exactly on the correct Fibonacci or Lucas time bar. Static cyclists were calling for a potential four-year crest only after an intense week of selling and could not anticipate this turn in advance. No other cycle methodology can come this close in pattern recognition.

To take advantage of this methodology, traders can combine these sequences with their existing disciplines. The best way to work with the time dimension is to combine it with candlestick analysis. Look for candle reversal patterns, such as morning or evening stars, engulfing, tweezers or haramis. If they line up with an important Fibonacci or Lucas time window, chances are excellent you have a high-probability entry with low risk.

The flip side for those who are trend followers is when price action is not near one of these time windows, odds are good the prevailing trend will continue. Nothing is perfect and markets will do what they want whenever they want, but trading is a game of probabilities. The Lucas series is one more tool that helps put the probabilities in your favor.

Jeff Greenblatt is the editor of the Fibonacci Forecaster, a forecasting service that covers the major indexes as well as gold, bonds, currencies and crude oil. He can be reached at

About the Author

Jeff Greenblatt is the author of Breakthrough Strategies For Predicting Any Market, editor of the Fibonacci Forecaster, director of Lucas Wave International, LLC. and a private trader for the past eight years.