# 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?

`SELECT * FROM wp_comments WHERE comment_post_ID = '2297' AND comment_approved = '1' ORDER BY comment_date`