Mr. Sogi points out that prices seem to chart out along a line and that they would not have done this if they followed a random walk. My question is: If prices are DERIVED from a random walk, are trends then possible? My rationale for asking this comes from:

1. A random walk is very easy to model. Finance students use binomial trees to get an intuition of option pricing. Even the Nobel Prize winning Black-Scholes model uses a special case of this (infinitesimal time steps.)

2. Although the stock price distribution in these models obviously is wrong, the insight it offers into derivative pricing is worthwhile. But what if we for instance use the state of the world as the underlying and then derive a stock price thereof?

I am experimenting on this through a game theoretical framework. First, a price settles where everyone agrees through their beliefs and pockets. With belief I mean that everyone has his own private opinion on how the stock might perform. Instead of the standard set up of probability distributions and risk adjusted returns etc, I simply set up the market in two camps; for instance bulls vs. bears, contrarians vs. trend followers, and so on. With pocket I refer to the budget constraint one faces. Second, any participant will be willing to push the price somewhat in their own disfavor. So, through the market’s bargaining process the beliefs will indirectly be adjusted as a new price settles. This could perhaps be used to explain bubbles and crashes — and any linear paths between such highs and lows? I have posted a preliminary model, with a simulation.



 The Chair has issued a challenge for anyone who can prove of disprove the existence of linear trend lines as suggested by Jim Sogi.

The first issue in doing a market study is to develop an adequate definition of what a trend is. Given that the idea is widely and perhaps first used in Technical Analysis it is good to start there. Tom DeMark, a widely known and respected TA guru, defines a trend line as the line connecting two bottoms in a price series. This, of course assumes one has a good definition of a bottom. He defines a bottom as a daily low which has the property that the low of the previous day and the low of the next day are higher than the bottom low. Thus it takes three days to define a bottom day. The most recent bottom cannot be known until one day after it has happened.

The idea of a trend line is to find the most recent bottom and then go back to the next most recent bottom. The whole pattern takes at least 6 days to work out and can be much longer because there can be an arbitrary number of days between the two most recent bottoms. Our own John Bollinger has confirmed that this is essentially what his understanding is of how trend lines are used by practitioners.

The theory of a trend line based on lows is that it acts as support. In other words the market will tend to stay above the line more often than would be expected by chance. It should be noted that nothing in the above definition presupposes an uptrend or a down trend. In an uptrend the second low point is higher than the first low point. And the difference per day defines the slope of the line. In a down trend the second low is lower than the first.

There is another type of trend line which is based on the highs of the day. A high point is defined as the high day between two adjacent days, whose highs are both lower. Again two high point days are required to draw a well defined trend line.

Mathematically it is always possible to draw a line between two points, so one should not be surprised to find trend line patterns in random data as well as real market data. The real challenge is to test whether they occur more frequently or less frequently. More importantly do trend lines have any predictive value either as measured by higher probability of a successful trade or a higher average return?

The study looked at 1800 trading days of SPY, the S&P ETF. This period started 100 days ago and went back 1800 days. It should be noted that the SPY was down 1.5 points during this period resulting in an average daily return of 0.00% to 5 decimal places. The study was further classified into four categories based on whether it was a high point or low point trend and whether it was an up or down trend:

Low  Up
Low  Down
High Up
High Down

The measure of profitability was the simple next day close to close return for the trade.

For trends based on Low points we have:

Trend   n     # Up   % Up      Avg Profit     Total Profit
 Up     204    109    53.4        -0.212        -43.23
Down    202    115    56.9        +0.031          6.33

For trends based on High points we have:

Trend    n     # Up   % Up      Avg Profit     Total Profit
 Up     182     86    47.3        -0.143        -25.98
Down    225    106    47.1        -0.050        -11.22

Most of the results showed about 200 pairs of trend points. This means that the typical trend point pair took about 9 days to form and thus had about 3 days in between points. TA practitioners would expect more days to be up after a trend point pair is put in place. There appears to be weak support for this idea because for both cases based on Low points the % Up seems slightly favorable. However this is belied by the fact that the avg profit for Up trends is actually strongly negative. Thus the preferred strategy is probably to fade the appearance of and up trend pattern.

For High Points the expectation is that the market has reached an extreme. Thus we expect it to fall back. In fact all the above does qualitatively agree with that premise because both the % Up and the Avg Profit are negative.

The key to all of this is to test the results for significance. Often one has noted that when a trend line is broken there is sometimes a dramatic drop. The import of this is not whether it is true or not. Rather if true then it implies a negatively skewed distribution. Thus the standard normal / log-normal assumptions may be too far off. For something like this then that argues that it would be better to choose a bootstrap test for significance so that we do not have to worry about the normality of our data.

That is the subject of Part 2.

Dr. McDonnell is the author of Optimal Portfolio Modeling, Wiley, 2008

Jim Sogi writes:

 Thanks Phil. The problem with defining a 'trend line' is that a rather random number of bars may or may not form the trend 'line'. More than two points would be needed to define a trendline as any two points forms a line, so it become totally random as to which points when only two define it. Then when not all lines touch the support line, then it turns random again. By the way, I was wrong. Random walks DO regularly form 'trend lines' to the naked eye.

One idea is to follow the a variant to the solution to the math problem Buffon's Needle  which determines the probability of a needle of a given length touching two parallel lines when its thrown down. The problem must be restated to determine the probability of a needle of fixed or variable length touching three or more points on a grid of timeseries points. Then the time series could be randomly simulated from actual data, and probability determined or randomly generated and compared to actual data. Here is a nice trailhead with the R code to visualize the problem.

If this code could be altered to solve the above variation, this might help solve this problem.

The solution may require limiting the time period to some defined time period such as a day, week or month so that the 'straw' has a defined length and require that the touches, touch within 1 point of an interval low to give a little slack as you do when eyeballing. However, there appear to be solutions to Buffon's Needle allowing for various or random length needle's. Perhaps Buffon's 'points' could also be lengthened to be short parallel lines that make up the time series, and use the formula to determine the probability that the needle (trendline) crosses three or more time series points to create the trendline.



 I don't understand why prices seem to chart out along a line along their tops or bottoms of the bars in a line. A random walk would not and does not do that. What is the function that makes it happen?

Victor Niederhoffer replies:

What we need to do is write a little essay on the linear thing of Jim's. He asks "why lines drawn between highs or lows tend to be straight in markets." Its a very good question. It reminds me of the work that The Professor and his student Chris Hammond did to test whether there were turning points or bear and bull markets in the Dow. An anonymous donor will give $500 for the best answer for this. Committee of me and Doc Castaldo and Jim Sogi to decide. I would point out that the above duo concluded there was no such thing as bull and bear markets, that the turning points were completely consistent with chance. My working hypothesis is that the same thing is true here.

Phil McDonnell responds:

One possible idea is the cobweb theory. In this, the declines come back to the demand curve which is a 45 degree line supporting successive bottoms. The upper bounds (the tops) are delineated by a parabola. The whole thing can be described by a difference equation. The following site describe the math and have some graphics. The graphs with the rising 45 degree line and the overlaid parabola above it are the most interesting.

Dr. McDonnell is the author of Optimal Portfolio Modeling, Wiley, 2008



As the S&P 500 is up more than 20% from its lows. It took only 12 calendar days for the S&P 500 to move up 20% from the March 6 low. In this regard, faster may not be better. There has been a strong positive correlation (R squared = 0.51) between the length of time of the initial 20% advance and the ultimate magnitude of the advance before the next 20% decline. The predicted top for this advance is 846, but the margin of error is wide.

First           Total
   date Calendar calendar
   20%     days    days    Gain  log(gain
  above   after  bottom  bottom   bottom
 bottom  bottom  to top  to top  to top)
 01/05/88      77    1000     71%    0.23
 02/06/91     118    2839    304%    0.61
 11/02/98      25     533     68%    0.23
 05/21/01      60      61     22%    0.09
 11/08/01      48     108     25%    0.10
 08/15/02      22      29     24%    0.09
 11/04/02      25    1827    105%    0.31
 10/14/08       4       4     24%    0.09
 11/28/08       7      46     27%    0.11



 I drive a lot. Easily putting 35,000+ miles/year on my SUV. As a result, I see a lot of billboards. Over the last several months I have noticed that a large number of billboards that are either blank or have some sort of message that says they are available, e.g. "Ten thousand people per day see this billboard — shouldn't your message be here?"

Having been involved in marketing and sales most of my life, one thing I've learned is the last place you want to cut expenses is in your marketing and advertising budget. Good companies adhere to this rule but most companies don't. Those that cut their marketing budget end up being hurt because of that cut. Advertising and marketing is the equivalent of planting a crop so you can eat tomorrow. Sure, you can save money today by not spending the money needed to grow tomorrow's crop, but come tomorrow you're gonna be hungry.

Based on what I'm seeing as I drive down the highways of America, there are a lot companies cutting their advertising budgets. They will find that tomorrow they will not have the new sales to feed their company. Their company will end up hungry and weak, they will be forced to make further cuts, — and the downward spiral to starvation begins. Cutting advertising budgets is not a good sign for the economy.

Russ Sears comments:

My guess is that you are correct for individual businesses, it makes sense to advertise the most when things are bleakest. But like PE ratios (make sense for an individual stock, but not for the index) my guess is that this does not apply to the whole.

For example many businesses shut their doors during a recession, leaving much ad space. Also fewer start-ups, and most smart start-ups advertise heavily. Once they establish a client base they often let up.

Further, it could be that billboards go first out the budget, because they are hard to change to meet the changing environment. For example you don't see many billboards touting prices of cars, refunds, etc. but you do see some, in better times, with cars on them.

Riz Din writes:

With respect to the issue of marketing during recessions, the literature confidently supports Scott's experienced take:

Harvard Business School's Working Knowledge site says 'Maintain marketing spending. This is not the time to cut advertising. It is well documented that brands that increase advertising during a recession, when competitors are cutting back, can improve market share and return on investment at lower cost than during good economic times.'

mailto:knowledge@wharton comments 'Research shows that companies that consistently advertise even during recessions perform better in the long run. A McGraw-Hill Research study looking at 600 companies from 1980 to 1985 found that those businesses which chose to maintain or raise their level of advertising expenditures during the 1981 and 1982 recession had significantly higher sales after the economy recovered. Specifically, companies that advertised aggressively during the recession had sales 256% higher than those that did not continue to advertise.'

Regarding the question of whether it is better for a company to have been born in a time of hardship, in a short paper titled 'Entrepreneurs and Recessions: Do Downturns Matter?' Paul Kedrosky looks at 8,464 companies that have gone public between 1975 and 2006. Using the IPO as a yardstick of success, Kedrosky concludes:

'Knowing that a company was successful—at least as evidenced by having gone public—does not give us any information about whether that company was founded during a recessionary or non-recessionary period. At least in a general sense, that is suggestive in that, given smaller numbers of companies founded during recessionary periods, the implication is that companies founded in such times have a higher likelihood of turning out to be economically important'

(the study doesn't look at death rates of companies in different periods).



 There are four systems that limit one’s ability to run fast, these roughly break into the four classifications of runners: The sprinters, the middle distance men, the distance runners and the marathoners. Each of these runners  train and maximize their specialized capability. Key to developing a niche in running is learning to maximize the efficiency of the many systems working in unison. The sprinter works on explosiveness. The middle distance will work on his kick, or sprinting after running hard. The distance guy his endurance, while the marathoner will work on fuel efficiency with the extra long runs.

However, each of these runners has races where they must master the bordering system’s limits.

For example the sprinter power system, explosive power only lasts 15 seconds, but many sprinters run fastest in a 200 meter race and the best can extend into the 400 meter race where they race for about 45 seconds. The Middle distance man must have the explosiveness like the sprinter in the 400, but will need the stamina of a distance guy in the 1500 m or mile. The distance guy will need some of the kick of a middle distance guy in the mile, but will need the marathoner’s efficiency and economy of energy to delay and withstand “the wall” in the marathon. And the marathoner will need to pace their efforts, carefully tread the line of their own equilibrium breaking point, not to “blow-up” even before the wall, like the distance runner. And the starting point and finish lines are not always clear in life and investing.

More general estimates can help determine which system limits are most important with-in the current environment. And then prepare yourself for the point in the race you find yourself at. Lackey commented on discussing my running how similar the jargon is to BMX’ing when a race goes bad. There is the “blow-up” or “explosion”, “the gorilla jumped on your back” also the “crash and burn” are a few terms we share. Likewise the four types of runners talk of similar terms when they talk of pushing their systems limits.

Basically, what all these races seem to have in common, is that all the body’s systems are affected simultaneously when their specialized critical function breaks down. The champion in each of these races is the person that can hold it together longest while their specialized system has already been depleted.

This reminds me of last year blow-ups, when everything was converging to correlation 1.0 and the only ones making money were those short the markets, any market. Those that held off the down turn, were the winners in the endurance test. In a race the worst thing you can do often is to compensate for the pain. Like in the race it often comes down to who is able to maintain their form, balance and coordination the best under stress and break-down of their power generating systems. Easing up due to crossing the lactic acid threshold for example will slow the heart rate which dumps even more lactic acid in your system due to the lowered equilibrium from the slow heart rate. Changing stride for a blister will end in stressed or pulled muscles, etc. One must wonder how the financial systems will respond the the compensation the government(s) are giving to those systems most obviously over whelmed. Further, I will leave it up to those more skilled than me to catch those falling knifes, while everything's going down.

In contrast, after the race when things start to heal, it becomes much clearer which systems actually suffered most. It may be a specific area such as strained muscle, blisters, banged up knee. Or a system taxed dehydration, kidney problem, hypothermia, etc. After the race the correlations decouple, the healing efforts and attention shift to individual damaged parts.

Would the markets correlation’s regime changes respond similarly when placed under severe stress?

In other words are there three correlation regimes to the markets?

  1. The “efficient” regime when the systems limits are not stressed, the specialist clearly reins
  2. The stress regime when every system crashes together and
  3. The “healing” regime where capital goes to where the most recovery is to be expected?

(I'll leave it to the reader to come up with his own hypothesis to test of changing correlation predictabilty, concurrence or lagging corresponding to these regimes).

This article blames the current crisis on those unable to comprehend correlation’s regime change after years of "normal" or booming housing market. Those correlations became gospel and developed a "free lunch" top credit spread, which of course necessitated a change in which every correlation approaches close to 1.

One must wonder, after reading it, if the bears, who clearly see both "the cause and effect" and "the why and how" every correlation must go to 1 in times of stress and all system crash together, will likewise be able to see how and why things must decouple.

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