Saturday, January 08, 2005

Fibonacci trading

I wonder whether someone believes this stuff - or even whether she or he has some rational arguments why it should work, at least partially.

Quite often, when you read something about the markets, they talk about the Fibonacci retracements. When the name of Fibonacci appears, it's kind of interesting because we know the Fibonacci sequence
  • 1,1,2,3,5,8,13,21,34,55, ...

in which a new element is the sum of the previous two. The asymptotic ratio of the neighbors approaches the Golden mean

  • (sqrt(5)+1)/2 = 1.618034...

whose inverse happens to be the same number minus one

  • (sqrt(5)-1)/2 = 0.618034...

Now it's useful to calculate the square root of both of these numbers, write 1-0.618=0.382, and add the numbers 1/2 and 1, and express everything as percentages:

  • 38.2%
  • 50.0%
  • 78.6%
  • 100.0%
  • 127.2%
  • 261.8%

The last one is the Golden mean squared. Why is it useful? Apparently, many traders use these numbers to predict the turning points of a price, see e.g.

Imagine that the price P of something kind of increases. If it returns a bit, it's a retracement. If the retracement is shallow, people believe that the increase is a trend. Now, if the retracement is bigger, it will probably stop and return to the increase when you return to 50% - to the arithmetic average of the last visible local minimum and the local maximum. Also, it's possible that the turning point is when you return by 38.2% of the difference between the last maximum and the previous minimum. Then you try to calculate all possible turning points by taking the "special" numbers above and multiplying them with the differences between various maxima and minima - and if they cluster, you get a likely turning point. Those with the Golden mean (without its square root) are "strong", I guess.

Do you believe this story? Well, even if it's nonsense, there are apparently many people who use it - and these people put a pressure so that their hypothetical turning points are turning points indeed. For example, euro's latest rally was from \$1.1986 in the summer to the maximum \$1.3664 last week. You can calculate that the 38.2% retracement is at \$1.3023 (very near the current rate) and the 50% retracement is at \$1.2825. If I understand well, the Fibonacci traders will think that one of these two values must be a turning point, and if these two points are broken, euro will have to drop to \$1.1986 again, assuming that the turning points are allowed to be repeated. ;-) If this is also broken (or if the exact rules don't allow it), you must probably make a 161.8% retracement and euro will have to drop to \$1.0949. ;-)

Although I believe that in the world of rational people without fairy-tales, the Fibonacci turning points would be uncorrelated with the real ones (and the "special" values are just some values that look sufficiently different from "small" variations), this fairy-tale sort of seems "self-consistent". For example, if A,B,C are the prices at three turning points and (C-A) = (B-A) * 0.618, then you can also calculate (B-A) = (C-A) * 1.618, and therefore various ways how you relate the points to one another are kind of equivalent, by the defining property of the Golden ratio. ;-) Your opinions about this new kind of science are welcome.


  1. If the Fibonacci traders truely have a method of making easy money (ie. their own personal "money tree" with no losses whatsoever), why do they even bother telling anyone about their trading algorithms in the first place? If these guys are for real, they would be guaranteed to make "easy money" (with zero losses) by writing zillions of naked put options on the dollar/Euro foreign exchange rate at a $1.1986 strike price or lower. The only way I would take these guys seriously is if they are making a huge bet with their OWN money, by writing zillions of those previously mentioned naked put options and/or short selling large amounts of sovereign debt of countries denominated in Euros (ie. federal debt issued by France, Germany, Spain, Greece, Finland, Ireland, etc ...).

    Generally the more well known a trading algorithm is, the less effective it becomes with time. After awhile, the particular trading algorithm becomes no better than investing in an index fund (ie. S&P 500). The "kiss of death" for many trading strategies from the past, was when everybody found out about it and tried to trade using it. Some traders even traded against the particular strategy, such as what many traders were doing right before when the notorious hedge fund Long Term Capital Management (LTCM) collapsed during the Russian ruble default in 1998, by trading against the positions of LTCM's traders. The arbitrage strategies that LTCM were famous for have been less and less effective in recent years, especially in highly liquid markets. It seems like the only places where various arbitrage strategies still seem to work these days, are in less liquid markets (ie. emerging markets, junk bonds, etc ...).

  2. Bizarre and fascinating to this CIP!

    Re Anon's comment though, it's not quite clear to me what this has to do with arbitrage strategies - except that an anti-technical arbitrage strategy might work to the extent that the technical strategy is superstitious.

    It seems to me that if there is anything to this idea, it has to do with human psychology, rather than fundamental market mechanics (if there is any such thing), but I'm not sure.

  3. Fool's Gold?Maybe it's just a selective system using Pascal's triangle, that we hope that system you refer too, works while it might be another that works better.

    Your looking for predictabilty? Using Ramanujan's moduli forms?

  4. Given the tremendous amount of historical data on the prices of stocks and bonds and commodities, checking this should be easy.

  5. I always found it amazing how many finance types do not even bother backtesting their trading theories and algorithms, on historical market data. If they bothered with any backtesting on historical data, it would show that many of their trading strategies are useless once transaction costs are taken into account.

    At times I wonder how long ideas in behavioral finance will "work", or if it will become another "loser" trading strategy. Richard Thaler's hedge fund seems to be putting many behavioral finance theories into practice, though I haven't heard anything really extraordinary about the fund's performance over the years. At least one can give some credit to Thaler for putting his money where his mouth is, by implementing his academic behavioral finance theories into practice as a hedge fund. Only time will tell whether behavioral finance is for real, or if it's effects are too dissipated and too diffuse to have any significant advantage over investing in an s&p 500 index fund.

    From seeing many trading strategies come and go over the years, which eventually became useless after transaction costs, I'm not very optimistic about any trading strategies for the long term. Maybe even one day investing in an s&p 500 index fund will become a "loser" strategy, compared to even investing in US Treasury bills or bank CDs. (In recent years, it turned out investing in bank CDs was better than investing in a money market fund or US Treasury bills).

    The financial markets don't seem to be very good place to make "big bucks" in a short or medium period of time. Longer term there could be another long bull market like the 80's and 90's in America, or it could be a stagnant bear market like the 1930's and 1970's in America. The only other scenarios I can think of offhand would be a total destruction of the financial markets, such as the Bolshevik Revolution in Russia or the hyperinflation in the Weimar Republic of Germany. Anybody who invested during the early 20th century in the financial stock and bond markets of Imperial Czarist Russia, Imperial China, Nazi Germany, Imperial Austria-Hungary, pre-Peron Argentina, etc ... would have the equivalent of medium quality wallpaper today. Before World War 1, America was regarded as the "wild west" and a very unreliable place to invest. (It would have been like the equivalent of investing in Spain under Franco, or Argentina). There were no obvious indications at the time that America would become a world power and a good place to invest throughout the rest of the 20th century. With the benefit of hindsight, the various wars and corrupt dictatorships all over the world pretty much made America into a world power by default. (Other capitalist democracies like England, France, Australia, Canada, Switzerland, etc ... were either too tired and bankrupt and/or too small to be of significance after world war 2).

  6. Dear Gentlemen, thanks for your interesting comments. I have similar impressions like you - if these things work in any sense, it is a matter of psychology, but most likely these things don't work, as the apparent non-stellar success of funds, including the funds of Richard Thaler who is the #1 guru in behavioral finances, seem to suggest.

    But I would still like to understand the math - whether there is any sort of rational explanation why the golden mean is a "more likely" turning point.

    One more thing: it seems to me that if many people believe that XY is a turning point, it WILL become a turning point because these many people will simply start to buy near this level. Is not therefore this approach self-supporting?

  7. I keep returning to this site since the width of issues and lumo's adherence to facts and measures (truth) are refreshing. Thanks, lumo!

    Earlier posts on this issue identifies two types of trading games: self-organising (like Fibonacci trading and possibly business cycles) and 'losing' (disappearing) strategies.

    My guess is that lumo is correct, but I don't know how to prove it; except if it isn't disappearing maybe it must be self-organising, other outcomes absent?

    I have a new question: If the above description is correct, assuming losing strategies initially reaps larger profits, what would be the (possibly self-organising) quota between the returns of one-shot inventive strategies vs Fibonacci type trading?

    Ie, is it beneficient to be inventive instead of dull? I would like to thinks so...

  8. Hmm; to make my question realistic I think one must also add that markets are finite (but growing with time, of course), ie inventive strategies can only make so much money?

  9. It's very well known that a lot of these "anomalies" of higher returns compared to the market S&P 500 index, are frequently found by "data mining". Somebody will take a cursory glance at the time series data of an asset and/or a portfolio of assets, and notice some higher than normal returns over a period of time. People have come up with many imaginative trading theories and algorithms, from data mining alone. The funny thing about data mining is that it can also find "spurious" correlations between two different time series which may very well be coincidences during a particular period of time, but may otherwise be uncorrelated in general. One famous example of this would the correlation of the stock market with the hemline altitude of women's skirts (ie. the hemline indicator). A really silly example would be searching for (via data mining) an asset and/or a portfolio of assets which has a high correlation with the amount of food our host Lubos Motl eats every day. Many of these "anomalous returns" found in this manner by "data mining" seem to dissipate after awhile, once everybody knows about it and trades on it or against it. One can search through old Journal of Finance articles where various authors found some high "anomalous" returns over the S&P 500 index, which later "disappeared" once everybody heard about it. I haven't looked into the history of the Fibonacci strategy, but I wouldn't be surprised if it was the result of extensive data mining.

    In the case of a person and/or a large number of people affecting the market drastically, this usually happens when there is a person "cornering" a market and/or a bubble is formed from "herding". In the former case of somebody "cornering" a market, they essentially have a large amount of money and control over a particular market that they have the equivalent of a quasi "monopoly". The classic case of this is when the Hunt Brothers tried to "corner" the silver market in the late 1970's. They eventually drove up the price of silver to over $50 per ounce in January 1980, until the silver bubble bursted and fell back down to $10 per ounce in March 1980. In 1979 and early 1980, many people were trading in all their silver coins, grandma's silverware, and silver jewelry they could find in their attics.

    When currencies were still under the gold standard, an evil way for really rich folks to manipulate the financial markets was to corner the market for gold. Fisk and Gould tried to corner the gold market in America in 1869, until President Ulysses Grant caught on and intervened to minimize the damage. Perhaps there's some grain of truth to economist Keynes' observation (for which Lenin and Stalin took to heart and put into practice in the Soviet Union):

    "There is no subtler, no surer means of overturning the existing basis of society than to debauch the currency. The Process engages all the hidden forces of economic law on the side of destruction, and it does it in a manner which not one man in a million is able to diagnose."

    For the case of speculative "bubbles", they usually happen when a lot of investors are thinking in the same "groupthink" patterns regardless of how irrational or silly the arguments are. This "herding" type of behavior seems to be based on the "greater fool theory", where one buys an expensive asset with the intention of hopefully selling at an even higher price to another fool. Eventually the bubble bursts when there's no more "fools" left willing to buy the ultra-expensive assets, since there's only a finite number of "foolish" investors. Afterwards asset prices can remain irrationally "low" and/or move "sideways" in a range for a period of time, from many people exiting the markets (ie. once burned, twice shy). One just has to see the historical stock chart of hi-tech companies like Microsoft (MSFT), Cisco (CSCO), Hewlett-Packard (HPQ), Sun Microsystems (SUNW), Intel (INTC), etc ... over the last several years since the dotcom bubble bursted in 2000.

    For the Fibonacci traders to really affect the markets in a drastic manner according to their doctrines, there would have to be a large number of them commanding large amounts of money and financial assets. Otherwise there's no a priori reason to believe that markets will behave in a Fibonacci manner, other than perhaps being a "spurious" correlation found by many years of "data mining" by the Fibonnaci traders.

  10. Thanks! I especially liked the explanation of the bubbles in terms of greater fools. ;-)

  11. What I have noticed over the years is how many financial trading strategies are "marketed" in a similar manner to how "self-help" literature is sold. Whether the actual financial or self-help strategies actually work, is a secondary concern. The advocates of a particular strategy seem to make more money selling their books, newsletters, seminars, videos, consulting services, courses, etc ... than from actually implementing their own strategies in practice.

    After clearing away all the crap and seeing what many individual financial and/or self-help strategies are really all about, that's when I came to the conclusion many are just half-baked ideas which required one to "suspend their disbelief" while imposing the half-baked strategies by fiat decree. Many resemble "snake oil" and in some ways are also a lot like implementing Marxist style "central planning" on one's self and/or on one's "friends". (This is similar to the idea of the "tail wagging the dog"). Everybody knows what eventually happens when half-baked wrongheaded ideas and strategies are implemented in practice. One quote from economist John Maynard Keynes sums things up quite succinctly:

    "The market can remain irrational longer than you can remain financially solvent".

    In the real world, a lot of silly wrongheaded ideas and strategies can hang around for a very long time before they become completely discredited by empirical evidence. Classic examples are ideas like eugenics, communism, fascism, postmodernism, etc ...

    It seems like ideas and/or strategies which do not have a very precise criteria for empirical falsification (in the Popperian sense), tend to hang around for a very long time by their advocates constantly changing their explanations. In this sense, these sorts of quasi "non-falsifiable" ideas and/or strategies aren't much different than practicing astrology. My way of judging how effective an idea and/or strategy is, involves examining how easily it can be falsified in a precise manner. If the idea and/or strategy can't be falsified easily by empirical data, then it may very well be "quasi-tautological" and isn't of much practical use to anyone. (What I mean by "quasi-tautological" is when some theory is so imprecise in it's falsifiability criteria, that it can always be made "true" by changing and massaging the explanations to "adjust" to almost any dissenting empirical data. This is what astrology is like).

    A lot of books in the business section of many bookstores seem to also resemble many financial/self-help ideas and strategies. Caveat Emptor indeed.

  12. Ahh, data mining! Yep, that's probably it; thanks for the excellent info.

    I note that there remains one probably self-organising mechanism, 'herding', though a transitory one.

    I really like that lumo this time is satisfied with the 'fools gold'/'greater fools' explanation. Apparently he feels that the power of fools is as least as great as the power of math?! Both combined in LQG I take it... :-)

  13. Somehow the human brain likes to search for "patterns" in all kinds of empirical data, even if it's just random noise and fluctuations with "nothing" there to start off with. This is how "spurious" correlations are found by data mining. Even if there is an underlying causal mechanism producing the data, there's always the problem of finding out what the independent variables and degrees of freedom are. In most practical real world scenarios, the independent variables can be so obscured and hidden that it becomes a futile exercise in attempting to isolate them, for all practical purposes.

    Another aspect of real world events is that the underlying variables and/or "rules" can change with time, as well as exhibit phenomena like path dependence, process irreversibility, memory hysteresis, mean reversion, etc ... The one thing which can really drastically change the course of events are unexpected surprise "black swan" events such as the invention of the computer, Sept. 11, 2001 terrorist attacks, etc ... (Author Nassim Taleb refers to these unexpected surprise events as "black swan" events, which by definition are events which are a priori not easily predictable from historical empirical data). In most cases of hypothetical "black swan" events which have not happened yet, there is not enough empirical data to determine the parameters of a particular theory or model. At best "black swans" are largely hypothetical conjectures before they actually happen in the real world. History in general seems to be largely made up of numerous "black swan" events happening over long periods of time. Historical financial market data exhibits many "black swan" events when viewed in hindsight, though at the time nobody really knew what was happening.

    Future prognostications (ie. predicting the future and/or "fortune telling") are frequently heavily biased by historical empirical data of the recent past. The human mind frequently extrapolates into the future from what is known in the present and recent past. Before the 20th century, it would have been very difficult to "predict" inventions like the airplane, computer, television, radio, or events like the rise and fall of Nazi Germany, the destruction of the Austrian-Hungarian empire, the rise of America into a superpower, the creation of the state of Israel, Sept. 11, 2001 terrorist attacks, the Nazi holocaust, the rise and fall of communism, etc ... In some ways it takes a vivid imagination to think of possible "black swan" events which may or may not happen in the future. What frequently becomes the "next big thing" in many niches and endeavors, are ideas and/or events which nobody would have thought of a priori.

    "Black swan" events and the hindsight biases inherent in studying history, is what makes me very skeptical about anybody claiming to have a formula to make "easy money" with zero risk. If studying historical empirical data could really predict the future with any accuracy, then historians could get rich from putting their theories and observations into practice in the financial markets. But everybody knows that is not the case.

  14. I don't have anything really intelligent to say but two comments: It always seemed to me that claims about the appearance of the Fiobinacci series in the real world could be traced back to the fact that the golden ration is the root of the first non-trivial polynomial:

    Nobody would notice appearances of "1" or "2" in the real world (well some people already try to relate different 3's...) and roots of linear polynomials are not really interesting. So it should be at least quadratic and have small coefficients. X^2 is too simple as is X^2-1. X^2+1 is not good as it's hard to observe i and X^2+X or X^2-X is not so amusing as well. So, it should be at least X^2+X+1, well that's not real again, so we try X^2+X-1 and here we are. The advanced read might have noticed that we used some anthropic reasoning here.

    [Side remark: If you ever have to teach undergradute algebra and look for problems, here is one of my favourite ones: Show, that the set of all series that obey a(n+2)=a(n)+a(a+1) is a vector space. Determine its dimension. Check if there are any geometric series a(n)=q^n in this vector space. Aha, there are two possible q's. So these geometric series form a basis. Write the element of the vector space with a(0)=0, a(1)=1 in terms of this basis. You obtained the closed form for the Fiobinacci series. Actually this works for all linear recursion relations.]

    Second remark: I enjoyed reading "Financial Risk and Derivative Pricing --- From Statistical Physics to Risk Management" by Bouchaud and Potters. This is the book for theoretical physicists that always thought finance is a more or less trivial application of quantum mechanics. And most references are to

  15. There's a few good books on what I've been talking about in my last few posts.

    "Fooled by Randomness" by Nassim Taleb.

    Taleb has a web site at

    There's a rough draft of his upcoming book about "black swans", as well as several essays on related problems.

    Another good book is "Irrational Exuberance" by Robert Shiller.

    The books "When Genius Failed" by Roger Lowenstein and "Inventing Money" by Nicholas Dunbar, both document the rise and fall of the notorious hedge fund Long Term Capital Management (LTCM) and the personalities behind it.

    From a historical perspective, the books "Devil Take the Hindmost" by Edward Chancellor and "Against the Gods" by Peter Bernstein discuss historical episodes of speculative market bubbles and crashes going back many centuries.

    Both Taleb and Shiller show a healthy skepticism about mathematical models, and how they are used and abused in the financial world.

  16. As a stock trader, I use fibs all the time. I don't care if they are "special" numbers in nature. Enough traders use them that it is a self-reinforcing phenomenon; traders will wait as a stock pulls back from a recent high and look to buy it at a fib retracement level. It frequently works; like anything else in the stock market it fails to work enough times that you can't bet the farm on it. If you are skeptical, bring up a chart of the SP-500 and throw some fib retracement lines on the weekly chart of the SP from the 2000 high to the 2002 low. You will see that the subsequent price action hits these fib levels fairly nicely, giving nice entry and exit signals.