# Why is volume a totally independent variable from price?

In Martin Pring's book "Technical analysis explained", when talking about volume, he asserts that it is "a totally independent variable from price"

Why is this?

• Well, I think he should explain how he arrives at this statement. But normally people doing technical analysis are not very rigorous types (otherwise they wouldn't do technical analysis ;-) May 17 '15 at 19:00
• His statement appears only in the summary (or maybe I missed his explanation somewhere in the chapter). Personally, I dont think it's convincing because both volume and price share a common factor, that is crowd psychology. They should be quite dependent to each other. May 18 '15 at 1:42

## 2 Answers

Pring was (probably) simply referring to the fact that most indicators are function of price -- lots of different ways to twist and contort prices to define trends, reversal points, etc. Volume is another parameter entirely, as it doesn't depend on price; the market or share price can have an up day on average, high, or low volume, it can have a down day on average, high, or low volume, it can have a sideways day on average, high, or low volume. Whether price and volume are parameterized on a common factor, as you suggest, is certainly a valid question, and indeed what a lot of technical analysis is based on -- trying to divine crowd sentiment and intentions from the combination of price and volume.

More informed, certainly more thorough, insight was given by Mandelbrot. He noted that price change per market time (i.e. per trade, although some use volume, but I'm not sure how thoroughly tested this is) tends to follow a normal distribution (Gaussian, although it should never be assumed so), while the number of trades per clock time tends to follow fat-tailed distributions, the result being very non-Gaussian distribution of prices as is normally assumed for simplicity. This does suggest some coupling of price movement (but not direction) with number of trades, and possibly volume.

References:
Mandelbrot, The (Mis)Behaviour of Markets
Mandelbrot, Fractals and Scaling in Finance
Kobeissi, Multifractal Financial Markets

• Your first paragraph sounds reasonable to me, probably what he means is: volume is a price-free parameter. I'm more impressed and curious about the viewpoint of Mandelbrot that you mentioned, it seems very interesting (I've heard about Mandelbrot for long time but not actually read any of his books). May 18 '15 at 5:42
• I'd strongly recommend the first one I listed (Misbehaviour ...). VERY readable, and judging by the second book I listed, probably due to his co-author on the first one. Not very expensive, either. May 18 '15 at 5:48
• Thanks! Actually I've ordered them just now, will read it then. May 18 '15 at 5:56

So lets test if volume and price are independent. We can estimate this by using mutual information. When x and y are independent, their mutual information is 0 so measures of mutual information entropy should be zero.

Lets calculate this for the IBM case with 10 years of daily closing price and data volumen.

library(infotheo)

a<-c(186.76,186.22,183.9,182.21,185.81,186.81,184.71,187.9,186.67,188.25,195.04,192.62,189.83,190.45,192.49,194.5,193.55,192.69,191.77,194.52,193.29,196.64,195.68,195.19,197.77,197.02,196.4,190.01,192.27,192.15,191.73,190.22,189.63,193.14,195.11,196.47,193.53,191.44,191.26,190.03,189.3,188.91,190.08,192.57,192.19,188.72,186.46,187.06,187,184.89,186.39,185.68,185.94,184.78,183.08,183.76,184.36,185.69,184.37,184.51,185.98,186.37,186.22,184.29,182.25,181.22,182.56,182.35,182.26,183.6,182.82,181.55,182.14,180.88,180.72,180.37,181.71,181.27,186.35,188.39,188.53,188.04,187.22,188.42,187.7,188,189.86,188.49,192.36,192.49,192.5,190.85,194.09,193.63,195.24,194.4,195.78,194.57,194,191.67,189.15,189.64,187.1,185.97,184.3,186.63,187.47,187.34,187.95,187.88,187.38,189.36,190.07,190.1,191.23,190.41,191.16,192.99,192.25,192,192.3,191.56,191.95,190.68,191.2,190.14,189.99,191.54,191.72,191.28,191.81,192.96,192.8,193.75,194,193.11,191.62,192.31,189.01,190.06,189.64,189.83,187.17,186.91,188.67,189.04,185.71,189.36,186.42,185.93,183.52,183.8,181.75,179.84,182.05,169.1,163.23,161.79,162.18,162.08,161.87,163.6,163.46,164.35,164.4,164.36,162.65,161.82,161.46,162.07,163.49,163.3,161.92,162.79,164.16,164.16,161.89,161.43,160.64,160.92,162.15,161.76,161.95,162.17,161.54,162.67,164.52,164.05,163.27,161.86,162.99,160.51,161.07,155.38,153.06,151.41,151.93,157.68,158.51,161.44,162.24,161.82,162.34,160.51,160.05,160.44,162.06,159.51,156.07,155.05,158.42,159.11,156.44,156.81,155.8,154.57,157.14,156.95,152.09,155.39,155.87,156.36,153.67,151.55,155.48,153.31,154.66,158.47,156.96,157.91,156.72,155.75,158.56,158.2,158.52,160.4,160.96,162.19,163.89,163.65,162.91,164.83,162.81,160.87,161.94,160.48,161.03,159.42,161.18,158.5,160.77,157.81,156.8,157.98,154.28,157.08,156.96,159.81,159.81,162.88,164.63,163,159.2,160.59,160.4,162.67,160.5,159.18,160.45,162.04,162.07,161.85,162.34,162.86,162.38,162.3,164.13,163.13,160.67,166.16,164.26,165.36,170.24,169.78,170.73,173.92,174.4,171.29,173.67,173.97,173.08,170.05,170.99,172.68,171.12,170.55,172.28,174.05,173.26)
b<-c(4229458,4833043,4912569,5479729,4938122,3959058,4319687,5059611,12535240,5738487,10902080,6851710,6749697,5193721,8537265,5394135,4923642,4074541,6089856,6740548,5112029,5480087,8417865,4834989,5431064,5351281,8467967,11248880,5419311,3878576,4735761,4407111,3871007,5632139,4626270,4206079,3673954,3049853,2222041,3186237,3817452,2431943,2261646,3777411,2226577,5251499,4202156,3946049,3270670,3023761,2984907,2215309,2562907,4793716,3721398,2759903,4620386,3200540,2515006,2375817,2852248,3296692,2728403,4138348,4061486,4425086,2773423,3538718,2445357,3925158,3551005,10686760,3231689,3875358,2762820,3258470,4575380,4223820,6643078,5092954,2422363,2958707,3135115,3309559,3177816,2402540,4501073,4858779,7811181,8827453,8165628,4154446,4851325,3584165,3503667,3376430,3242107,3264124,3943779,4207339,5178022,2125035,3307945,3847041,2708590,2781486,2527208,1858635,1794941,1929422,2814815,2418687,2038470,2177642,2496475,1940334,1723557,3158767,2190185,1503582,2909422,2679529,1822765,2864944,2260202,2524233,2390391,2763994,2297955,2900953,2456411,2561528,3126159,2963346,8847306,3288454,3300715,3082583,4151355,2493862,2336317,2870283,3705432,2281749,3071521,2099525,2990954,2982451,2625359,5090174,3596738,3924621,6895843,5578469,4350238,23393800,20947830,11084830,7599154,6652126,4989112,7894973,4738734,3895869,5817961,4683808,4241264,4102438,4067487,3491874,4956409,3534400,3377542,3239343,4974605,4795042,5410056,3801179,4182186,4076193,6615896,4062338,3966022,2405512,4165523,3465647,6432090,3860288,3013455,2851383,3865355,4081495,3987625,8603486,6489279,6781567,5135799,7289467,8863708,4671176,4044120,1869977,1912278,3328597,2820875,4007568,5521275,4877949,6140885,4694891,4240432,4485123,4185540,4376654,4679570,4251472,5755866,8368458,11891690,6118450,4830128,7887907,5659328,4493190,8312100,6563442,4711986,5536804,3676106,5253628,3256440,2981444,4440579,3626629,3331904,3706909,2892929,4358252,3504269,3346955,2712218,4048109,4003088,7107766,4403866,5910972,3234006,3633335,3787426,4538724,5193874,4600260,5709280,4566291,6057170,3749638,3311792,5737794,3871119,9001842,5929727,4331090,5428222,4391017,3477801,4137986,4254808,3694519,4671478,3465581,3147566,2521583,2263480,2511035,3868071,2716418,3495848,3130813,4308880,9572126,9681557,4022236,8226707,3887859,3166588,5814978,4515143,5058348,3311743,4023794,3593415,3610295,2472627,3092479,2657123,2954387,2457170,2438669,2916087)

c<-data.frame(a,b)
d<-discretize( c, disc="equalfreq", nbins=NROW(c)^(1/3) )
mutinformation(d[,1],d[,2], method= "shrink")
0.09123699


You can see that the two variables are not totally independent in the case of IBM and I guess this is the same case for all securities.

You can calculate if there is causality of one time series causing the other and viceversa via granger causality, however this needs the data to be stationary.