From: John Conover <john@email.johncon.com>

Subject: Re: Fractal "bubbles", Re: Business News from Wired News

Date: 30 Dec 1998 07:30:56 -0000

John Conover writes: > > BTW, most bubbles in the simulation lasted about 4 years-some less, > some more, depending on how one judges what a bubble is-I just let the > simulation run long enough to get a big one to impress you. The > chances of a big one increase with the square root of time. The > chances of a zero crossing of the real average go down with the square > root of time, and the chances of finding a long run length bubble goes > down with the reciprocal of the square root of the bubble's run > length. The deviation of the fractal from its real average diverges > with the square root of time, and on average, the fractal's maximum > divided by its minimum will be a factor of two in any year, and the > range, ie., maximum - minimum, scales with the square root of time, > for values larger than a year. The stocks on the NASDAQ, NYSE and AMEX > have these characteristics to many orders of magnitude precision, (but > there are minor discrepancies, so don't jump to conclusions-its not ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ > exploitable, anyhow.) > So, what are the minor discrepancies? Attached is a PDF graph of the frequency distribution of the daily marginal returns for the NYSE, DJIA, and S&P 500, indices for the last 27 years. If these had the frequency distribution of a random walk fractal, they would be a perfect Gaussian/normal bell curved distribution. Although it is close, it is not. There are more small fluctuations than can be accounted for, less moderate ones, and more big ones. The smooth line is a least squares best fit Gaussian/normal frequency distribution to the data. The 3 rough lines are the data. What are the consequences? There is little consequence created by there being more small fluctuations than can be accounted for. Too few moderate fluctuations, also, have few consequences-it tends to lead to volatility predictions that are very slightly conservative, (ie., stocks would have less volatility than one would estimate.) That there are more big "hits" in a portfolio value can be a significant issue-if it were a true Gaussian/normal distribution, three sigma hits would occur, on average, every 741 trading days. The attached graph states that this is optimistic by about a factor of two, (ie., they would occur about twice as frequently as calculated.) For the casual investor, such precision issues can probably be disregarded, but to the likes of derivative fund managers, (who make their living off of Black-Scholes' concepts which depends on it being a perfect Gaussian/normal curve,) it can be the kiss of death. Here's why. In derivative hedge funds, volatility is leveraged by investing in multiple volatile financial instruments, concurrently, in such a manner that the fund's volatility is less than any of the individual financial instrument's volatility, (to an EE, it is like adding noisy voltage sources together-the DC adds linearly, the noise root mean square, so you end up with more DC, and less noise, at the same time.) That's why they are called hedge funds-they are not as volatile as what one is trying to hedge. What fund operators depended on is knowing the likelihood of a big hit in ANY financial instrument in the portfolio, quite precisely-two or more big hits occurring at the same time, and he is probably out of business-having to liquidate to cover losses, (this is what happened in August/October this year to several hedge funds.) Bear in mind that the the three sigma hits account for only 0.1% of fluctuations, and the theory agrees with metrics to within a factor of two, on the 0.1%, (there were 14 in the last 27 years-theory says there should have been 7, out of the 6905 daily fluctuations.) So, what causes the discrepancy with theory? There is a lot of speculation on that-but no one really knows. One suggestion is that there is a structural issue, (ie., regulation issue, political issue, or something,) that causes modulus 4 years, minus 1, (in this century-we just had one,) to be a down year with a negative glitch in the last half of it, lowering the entropy, (ie., making the markets less random than the theory says they should be.) But if this is true, why in off election years, instead of election years? What structural issue occurs only on modulus 4 years minus 1? Yet another reason offered is that day-to-day fluctuations in equity values are not a 50/50 probability for an up, or down movement, ie., there is persistence, (ie., there is a slight propensity for market movements to do tomorrow what they did today.) A detailed analysis of the attached graph would say that instead of a 50% chance of an up movement tomorrow, there is a 58%, but only if it moved up today, (likewise for down movements.) This is a fractal, also, but a more sophisticated one, (one with a Hurst exponent of 0.58,) which occur frequently in nature, also, (mountain ranges have Hurst exponents of about 0.6, for example.) The proponents of this argument point out that the 50/50 assumption is not quite right-the assumption that people make arbitrage decisions by disregarding all of the past is just not intuitive-since people do remember whether they got burned on a stock yesterday, which would influence their decision today, (the persistence would just decay exponentially at a 0.58 * 0.58 ... rate in time.) This would account for the low entropy we see in the markets, and the increase in the number of three sigma hits. It, also, has a nice mathematical infrastructure, in that instead of adding volatility root mean square, it would be (a^1.72 + b^1.72 ...)^0.58 instead of (a^2 + b^2 ...)^0.5 as in the random walk model. But it doesn't explain the 4 years minus 1 phenomena, which the fractalists point out is on feeble metrics and empiricals-there were only 23 of them this century-and fractal analysis requires a lot of data. More than we have acquired in the century, (remember the adage quoted in early 1997 that years that end in 7 are down years? All such years in the century were down years, except 1997, as it would turn out.) The most sophisticated explanation is that there are chaotic, (ie., non-linear dynamical system,) effects. Very elaborate simulation techniques, (from the Santa Fe Institute, using neural networks, with each node representing an investor doing equity value arbitrage,) have produced equity values that have the modulus 4 year minus 1 phenomena, and the shape of the glitches every 4 years look remarkably like what is seen in the stock markets. Unfortunately, if it is a chaotic system, it is doubtful that a deterministic causality, (ie., cause and effect,) could ever be found to explain the phenomena-many argue it to be theoretically impossible. So, although it may be intellectually satisfying, it may be difficult, or impossible, to exploit. (However, short term probabilistic forecasts, may be possible, but like the weather, long term forecasts are not seen as being viable.) The modeling must be seen in a different perspective, however; with a simple random walk fractal, a more sophisticated fractal model with persistence, and a chaotic model, all being a continuum-the formulas used are all related. Strike the non-linear terms from the chaotic model, you have a fractal model with persistence. Set the persistence to zero, and you have a random walk model. John

-- John Conover, john@email.johncon.com, http://www.johncon.com/

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