# Complex Variation, by Victor Niederhoffer

October 11, 2007 |

I've been studying complex variables lately because I find the imaginary very important these days, and I had to brush up on them for one of my daughters.

It led me to consider the imaginary part of the moves during a day or week, and the real part. Consider last week. O/H/L/C:

9/28 1538.20 1545.20 1519.00 1538.10
9/21 1491.80 1552.00 1485.20 1534.40

The real part of the move, from 1534.40 to 1538.10 was 3.70. The low of the week 1519 so there was a -15.40 point imaginary negative part, and the high was 1545.20 so the imaginary positive part was 10.80.

A similar calculation could be done for the day, looking at the amount below the previous close, the amount above the close, and the final move.

We can look at the two points on an Argand like diagram. I claim that the length and the angle between the two lines connecting the negative and positive imaginary could be useful as a predictor. Better yet, the two angles themselves and the real part. Similarities might be useful. Such angles should be quantified , classified, and subjected to prediction and falsification.

Another example. The week of August 17 showed a real move of -1.10 and a negative imaginary of -76.00 and a positive imaginary of 21.50.  A small real move but non-negligible imaginary moves.

I'd also be interested in trying volatility as the orthogonal parameter (it is to do with the imagination after all.)

## Michael Cook follows up:

I love complex variables - it is one of the most beautiful subjects in mathematics. Everything comes together and illuminates and integrates everything that's gone before in the traditional mathematics curriculum.

I don't understand how you are defining the imaginary part of price moves - can you clarify? I am intrigued!

## Alex Castaldo explains:

If I understand Vic correctly, he defines two complex numbers, the AboveMove and the BelowMove:

AboveMove = (c[t]-c[t-1]) + i (h[t]-c[t-1])
BelowMove = (c[t]-c[t-1]) + i (l[t]-c[t-1])

And plot these as two vectors on the Argand diagram. The real parts are the same, but the imaginary parts are different (and always of opposite sign). Next you can get the angles and the lengths.

Are these the complex components of the change simply because they exceed the bounds of the price at the start and end of the week? If so, why a week and not a day or a month? And perhaps more to the point, can the maths of complex numbers then be used to predict? Analyze the moves?

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1. Kevin Jacobs on October 11, 2007 6:52 pm

It is a very interesting point. Perhaps one could run such a test on a correlated European market (i.e. the DAX) and use the findings on the US markets.

2. Hirak Parikh on October 12, 2007 12:20 am

Intriguing and interesting concept! While the computer crunches the angles, a thought occurred to me. Why stop at 2-D? This can be be extended to multiple dimensions using values from weekly, monthly and yearly h/l data. Hyperplanar geometry. Now we are cooking!

3. Ken Webb on October 12, 2007 7:31 am

i think volatility consideration is inherent in the complex representation of the real and imaginary parts of a market move. the correct question may alternatively be “are these the imaginary components of the complex change over any given period T simply because they do not in reality exist at the end of the period T and have relevance only to the price at the beginning of period T insofar as they make one’s heart stop and jump?” i think the answer is yes, and the complex components relate to emotions in the marketplace. perhaps if we were able to treat each of these imaginary data points as a discrete sampling of a continuous signal, we could determine the frequency decomposition of that signal and then look for predominant frequency components at critical turning points in the marketplace. perhaps these “imaginary signals” do have relevance to predict future price movement. the “maths of complex numbers” include the manipulation of these real and imaginary representations of system signals composed of various frequency components, whether periodic or aperiodic, linear or non-linear, to a form which is more easily analyzed in alternative representations with simplified mathematics such as is the case with z-transforms / laplace transforms. to take this a step further, impulse response signals might be calculated from application of an additional signal (such as position limit trade in the S&P futures) at a critical market turning point identified by the above signal frequency decomposition analysis. electrical engineering program from university of michigan finally applied! thanks to all for above comments, questions, and input. victor, please contact me at 925-639-7375 if you have time to discuss.

setup of complex numbers and angles in this case:

@ = angle theta in radians
@up = angle for up move from prior close to high
@dn = angle for down move from prior close to low

REmv = c[t]-c[t-1] = real move from prior close to close
IMup = h[t]-c[t-1] = up move from prior close to high
IMdn = l[t]-c[t-1] = down move from prior close to low

and

Zup = REmv + i*IMup
Zdn = REmv + i*IMdn

then, graphically speaking, there would be two line segments in the imaginary Z-plane. x-dimension is REmv for both with y-dimension IMup for Zup and y-dimension IMdn for Zdn.

a complex number z is an ordered pair of real numbers x,y:
z = (x,y), where x is called the real part of z and y is the imaginary part of z. by pythagorean theorem, radius r = sqrt(x^2 + y^2), r being the distance from the origin to the point x,y in the imaginary plane and x = r*cos(@), y = r*sin(@) so:

z = x+iy = r*(cos(@)+i*sin(@))

let rUP = sqrt(RE^2+IMup^2) and rDN = sqrt(RE^2+IMdn^2), then:

Zup = rUP*(cos(@up)+i*sin(@up))
Zdn = rDN*(cos(@dn)+i*sin(@dn))

Euler’s formula, published in 1749, e^(i*@) = cos(@)+i*sin(@) so:

z = r*e^(i*@)

Zup = rUP*e(i*@up)
Zdn = rDN*e(i*@dn)

still looking to discover formula for ultimate barbeque sauce.

4. Anonymous on October 12, 2007 12:20 pm

Nonsense.

5. David Saphier on October 12, 2007 1:42 pm

I’m new here, so let me preface by saying there are a lot of smart people on this sight and I apologize if what I have to say is too simplistic. I don’t think imaginary numbers are required for this analysis, though it is an interesting metaphor. I would suggest looking at pivot (P) analysis and its resistance (Rx) and support (Sx) derivatives.
Here are the equations:
P = (H + L + C)/3
R1= 2*P – L
S1 = 2*P-H
R2= P + (R1-S1)
S2= P – (R1-S1)
This does not account for the distance from the previous close but that could be added to the P calculation. The derivative R and S values are based on the previous bars trading range and can be applied to any timeframe. An analysis can be run to determine the probabilities of violating any set of R and S values. Moreover, once a level is violated what is the probability of maintaining that level.

6. Mikhail on October 13, 2007 12:16 am

Not sure how one would test it, but perhaps the particular nature or form/shape of the distribution (price across time) of the imaginary movements may have some predictive value for the real part in the subsequent period…

7. Mikhail on October 13, 2007 2:52 am

Some additional random thoughts or perhaps obvious observations:

The degree of the angle can measure how efficient an imaginary movement was in generating a specific real outcome. Also the length of the line connecting the two opposing imaginary rays may be a proxy for the total energy expanded to generate a particular real movement…. E.g. if the line is quite long relative to the real movement, this would indicate that an enormous amount of energy was expanded in vain. Or it could suggest a major transfer of holdings among different market participants. Though both of these observations do not account for the liquidity in the market at the time of the movements, as total energy required to generate a certain real movement ought to be less in a relatively illiquid market than in a liquid one… Assuming that opposing resistance is held constant.

8. Anonymous on October 13, 2007 4:59 am

bollocks…complicating mathematically what has no price predictive behaviour

9. Gangineni Dhananjhay on October 13, 2007 5:54 am

I think the symetry in Complex Variables lends itself for stock market movements. z and zbar as in x + iy and x-iy are symmetrical and opposites.

10. Anonymous on October 14, 2007 7:25 am

The definition here boils down to some function of the price excursion beyond and below the closing price of the time interval chosen. Take away the fancy mathematical veil and its what every home trader has employed to unsuccessfully predict market behaviour.