Mandelbrot’s Observations and Empirical Proof
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- Fractals and Scaling in Finance
| Mandelbrot’s Observations and Empirical Proof By Jim Goulding Benoit Mandelbrot, the father of fractals, has one of the greatest mathematical minds on earth. I’ve been studying one of his books for…oh…mmmm…well…going on three years now. Its mind numbingly complicated; filled with enough mathematical equations to make your head spin. It’s worth it though, because I knew he was onto something due to his honesty. He’s completely against the theories that are consistently put out there to sucker people into believing they are buying the holy-grail of financial price prediction. (I know the people who sell these theories, personally, because I’ve been sucked in many times.) If you relish your money, read what he has to say, in this paper, about price prediction. There are several reasons why I wrote this paper. One reason is to introduce you to his world of thinking. Although he doesn’t give a specific trading system that you can import into your analytical trading software, he does write about what you should investigate and what you shouldn’t investigate. Secondly, I want to warn you that if the word fractal is tied to an indicator or software program, don’t buy it or even waste your time with it (if it is free) because there are many bad fractal programs. More specifically, this paper covers what Benoit has written concerning certain “indicators” and “trading systems”. I’ve also included some of his observations about the markets. Meaning, some of the things he thinks deserve extra attention. However, I’m going to write a separate paper on that subject because it's deserving. I’m leaving all math equations out of the paper… because…well…I don’t understand any of them. (I’ve provide reference pages from his writing if you’d like to look them up.) What I do know about his equations is that they are empirical proof that unequivocally discount many financial price-predicting models and probability-theory models. One of the greatest things I’ve realized, from his work, is what not to study. This paper includes a lot of that. Indeed you may see some things in here that I have personally recommended and Benoit is indeed discounting the method. It took me a long time to extract exactly what Benoit was saying in his writings. He writes so technically. And, as I said before, I am not a math person. Theories and abstractions maybe, but equations? Fergetit. What brought me to the writing of this paper is very important. I was implementing the installation of MATLAB which is a sophisticated analytical program for financial instruments. The reason I wanted MATLAB was to run a Mandelbrot theory called the Multifractal Model of Asset Returns (MMAR) and apply it to our markets. What I found, while setting up the software, was an enormous amount of assumptions the software made about analyzing price data. Assumptions that have been, empirically, proven wrong. MATLAB is very expensive. And, it’s well regarded in many industries as cutting edge. It has lots of bells and whistles. But, much of it is useless due to the assumptions made from the writers of the software. I’ll give them this much, they admitted it in their documentation. However, while reading the documentation, if you didn’t know what stationary, nonstationary series process, Fat tails, and Long memory were, you’d of trusted this software to give you signals to get in and out of the market. Unfortunately, 99% of traders trust the software / system, because they want the holy-grail. If it was the holy-grail, you know one thing, it wouldn’t be packaged and sold to you. So, beware of what you don’t know, concerning trading system/software/indicator claims, etc. Learn from my mistakes. Understand that there’s a heck of a lot of people selling tents and shovels. As you read this paper, don’t freak out because of the terminology. What’s important is what he’s saying about a specific term. Terms that are identifiable within software trading systems and indicators. (Terms that people make unfounded claims about, so you’ll be sold on their system.) Example of a term in this paper: If you see the term Gaussian, then there will be an explanation later. It may be written by me or Mandelbrot. The terms make more sense after you’ve read the whole statement and/or the notes I’ve made after the statement. [My notes will be encased in brackets.] Fractals and Scaling in Finance“Gaussianity1, random walks, and martingales are attractive hypotheses, but disagree with the evidence concerning price variation.”2 [In a paper Mandelbrot wrote, called M1963,3 he proved that the price behavior of financial data is clearly not Gaussian. This conclusion, that the price behavior of financial data is clearly not Gaussian is a main theme throughout his work. Without going into major background, you need to understand that price prediction has been based on the theory that prices are Gaussian, since the Gaussian theory was discovered, in 1900, by the mathematician, Bachelier. M1963 proved the theory wrong. However, people still base everything on the theory that prices are Gaussian. Read on, you’ll be a pro at these terms by the end of this paper. More on Random walk and Martingales later. They’re important because many software/trading systems contains these terms. Their sucker terms because they don’t work.] “Scaling can take two forms, uniscaling and multiscaling, and the proper model of foreign exchange rates appears to be a multifractal one.” 4 [I note the statement above, because it may come in handy at a later date. I’ll abbreviate these types of notes as LD, which stands for Later Date. If for some reason you are studying a system in currency trading, you’ll know that multifractal scaling is important.] On Risk & Probability of Ruin theory: “In effect, much of the statistics places an equal degree of confidence in the Gaussian distribution. For example, many refined theorems concerning risk and the probability of ruin only hold in a universe in which the Gaussian holds with absolute exactitude.” 5 [In M1963, he proved that the price behavior of financial data is clearly not Gaussian. So he’s saying that Risk & Probability of Ruin theory doesn’t hold water because they are based on Gaussian distribution of prices. This simply blows me away. The street is over run with probability software and PhDs claiming you must run Risk & Probability of Ruin statistics! No…No you don’t have to run them. They don’t work.] “…changes in fast moving interest rates are surely not independent.” [This is one of those things that I find extremely important and have no idea why. So, I toss it in for use at a LD.]6 Concerning ARCH7-like processes and Brownian motion: “I take the liberty of challenging the proponents of the Brownian motion and ARCH-like processes to examine how the graphic outputs of their algorithms compare with the actual data. I have no doubt about the outcome.”8 [There are other examples of these two processes later in this paper. ARCH stands for “Autoregressive Conditional Heteroscedasticity”. You’ll see many different bastardizations of this model. You’ll also see the words Autoregressive and Heteroscedasticity used by software sales-people to help sell a program. And, you should see the pretty pictures it makes! Of the terms mentioned above (Brownian motion and ARCH) ARCH is of particular importance. Many trading theories out there are based on the math of ARCH. MATLAB has a whole book on ARCH and a whole software package. It’s useless.] “Hyperbolic distribution is analytically convenient.”9 [Another big term thrown around in the trading system world, hyperbolic distribution. If you see it, run away.] Hydrology and finance are similar. 10 [fascinating] Fourier series is not a model, only a versatile representation. “A satisfactory statistical fit is of no use in science unless the fitting parameters are consistent in time and have no intrinsic meaning.” 11 “…ARMA12 is not a model, only a versatile representation. A satisfactory statistical fit is of no use in science unless the fitting parameters are consistent in time and have no intrinsic meaning….they have no predictive value whatsoever. 13 [He is saying the same thing about Fourier and ARMA in the same sentence. I simply separated them into two statements.] “The ARCH representations and its variants. The comments that follows are addressed to readers the readers already acquainted with this common “fix” to Brownian motion. In spirit, the ARCH model is closely related to models that inject a trading time, such as the 1967 model based on subordination (M & Taylor 1967 or the M1972 model. The fit of an ARCH-like model is likely to be good or even excellent, if only because there is no upper bound on the number of parameters. However, all the reservations concerning ARMA extend to ARCH….That is, ARCH analysis fails, by its very nature, to be faithful to the long-term component that the multifratcals involve, and that the eye sees in the financial data.” 14 [Mandelbrot’s emphasis] “Interpolation and extrapolation. Yet another deep difference between ARCH and the M 1972 model shows up when models based on daily data have to be interpolated or extrapolated to higher frequencies. In ARCH, analytical procedures are not available.”15 Benoit comments that interest in Pareto’s law for distribution of personal income regarding the scaling distribution related to the law, should dominate economics.16 [He’s stating that Pareto’s law should be revisited. He also goes on to introduce this law into statistical economics as an alternative to what‘s being used now in statistical economics. He also writes about scaling distribution, which is very important in his work. More on this below.] “…I propose to show that the scaling distribution literally cries out for our attention under many circumstances. Those circumstances include (1) taking seriously the simplified models based on maximization or on linear aggression (2) taking a cautious view of the origin of the economic data or (3) believing that the physical distribution of various scarce mineral resources and of rainfall is important in economics”17 “We shall see that the most important feature of the scaling distribution is the length of its tail, not its extreme skewness.”18 “…Pearson’s measures of skewness and kurtosis – are meaningless.”19 [I’ve run across Pearson’s and Kurtosis’ measurements when analyzing trading systems, with backtesting results. Many statisticians swear by these two measurements. Benoit purports that they are useless.] “The only criteria of trustworthiness is replicability in time.” 20 [By far, this is his most pertinent statement in the book. If there is a system that you want to trade, after backtesting it and running it through the most sophisticated software programs out there, run it for a while and see if it actually works in real-time, day after day.] Here are some terms that Mandelbrot writes about that I’ll list for the particular reason that you should avoid anything to do with these. In other words, if claims are being made about a trading system or indicator that include the following terms, avoid them. They are worthless.
On probabilists: Mandelbrot calls, Adrei Nikolaievich Komogorov (1903-1987), the greatest probabilist of the century. 21 On Elliott Wave: [Sorry folks, you won’t like this. Even I like EW. However…] “Elliott’s work fails the requirements of objectivity and repeatability: in his own words [Ralph Elliott’s], “considerable experience is required to interpret [it] correctly” and “no interpretation [is] valid unless made by [him or his direct licensees].”22 On Randomness: Benoit states that randomness must be broken down into three categories to properly understand it. As he writes, “…different states of randomness must be distinguished and faced. There is wild randomness exemplified by distribution with infinite variance. There is also an intermediate possibility exemplified by the lognormal: it is slow randomness – a term deliberately selected to imply what it says.”23 [What he is saying is that randomness in the market can’t be taken as a whole. And, that’s exactly what statisticians do when measuring the market with their ‘tools’. He writes that as physics must distinguish between different states of matter, so must probability theory. So, when taking probability theory into account, ask if the software you are using breaks down the randomness into three categories or does it look at randomness as a whole. The odds are the software does not take this into account. Therefore, the software is useless. However, Benoit’s work does take this into account. I’m looking for the software that includes this feature, concerning probability theory studies. I’ll cover his theory on Randomness that he thinks should be implemented, in another paper.] On Spectral Analysis: “A major but unrecognized “blind spot” of the spectral analysis; spectral whiteness is insensitive to change of trading time, therefore misleading.” 24 [Benoit is referring to analysis that involves some of the following:
What would be considered a good wavelet? One that simulates Fractal Brownian Motion.] On Lognormal Distribution: “A hard look at the lognormal reveals a new phenomenon of delocalized moments. This feature implies several drawbacks, each of which suffices to make the lognormal dangerous to use in scientific research….the lognormal’s wondrous properties are irrelevant and thoroughly misleading; it is not the statisticians best friend, perhaps even their worst one.” 25 [Again, this is discussing a term. That term is, Lognormal. This term is thrown around as a price predictor. It isn’t.] On Martingales: [They don’t work as price prediction models.26] Well…that’s it for now. Although a paper like this can be difficult to comprehend due to the unfamiliar terminology, it’s worth reading through and keeping for later reference. Especially when you run into the situation where someone is trying to sell you an indicator or a trading system that has complicated terms attached to it. Simply search this paper for term. If you don’t find it, search the web and limit the search to ‘.edu’. See what you come up with. My guess is that the term can be connected to the terms within this paper; therefore, disproving the snake oil you are being sold. References: Fractals and Scaling in Finance, by Benoit B. Mandelbrot. Statistical analysis of financial time series under the assumption of local stationarity, by Clémençon Stephan. A Multifractal Model of Asset Returns, by Benoit B. Mandelbrot. 2 Fractals and Scaling in Finance, p 50 3 M = Mandelbrot, and 1963 is the year he published the paper. You could look this paper up in the Internet by typing in “M1963”. With the quotation marks. 4 Fractals and Scaling in Finance, p 61 5 ibid, pp 62-63 6 ibid, p 63 7 All right, get ready for this term. ARCH stands for “Autoregressive Conditional Heteroscedasticity”. 8 Fractals and Scaling in Finance, p 66 9 ibid, p 69 10 ibid, p 72 11 ibid, p 72 12 auto-regressive moving averages 13 Fractals and Scaling in Finance, p 72 14 ibid, p 73 15 ibid, p 73 16 ibid, p 79 17 ibid, p 80 18 ibid, p 80 19 ibid, p 93 20 ibid, p 95 21 ibid, p 115 22 ibid, p 115 23 ibid, p 115-16 24 ibid, p 163 25 ibid, p 255 26 ibid, p492 Yüklə 57 Kb. Dostları ilə paylaş:
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