# Two Books on Difference Equations, from Victor Niederhoffer

June 28, 2011 |

An Introduction to Difference Equations by Saber Elaydi and Difference Equations by Paul Cull, Mary Flahive et al have me thinking of second order difference equations.

x(t+2) = a + b  x(t+1) + c  x(t) + u(t) where u ( t ) is a random variable.

The fibonacci equation has b and c = 1 and x (t) = x(t-1 ) + x (t-2) , x(0) = 1 , x( 1) = 2 and the series is 1,2,3,5,8,13,21,… the simple way to solve for the x(t) in such a series is to make it a binomial (x) (x) - x -1 = 0 and then a very beautiful set of roots described in Edspec in a math team problem comes about with a(( 1 + 5 to 1/2)/2)to the 1/2 and a (( 1 -5 to 1/2)/2) to 1/2 term as the multiples to apply in the solution.

Okay, I think the second order difference equations have many applications in many markets. I propose to simulate the best fit for a second order difference equation with say the last 10 sets of 10 observations, and then to predict where it will go. I believe that a simplicity based solution using only the constants 1, 2, and 3 and no higher powers than a square should be found.

I propose that this would be predictive and the divergences from the prediction would be tradable. The exercise violates our rule that the higher the mathematics the less the use in predictions. But it seems to me a useful reflection that is much more relevant than the essays on stochastic difference equations that one sees in the literature such as this and it's a useful diversion from cross road puzzles and pattern recognition I think.

Name

Email

Website

1. daniel on June 28, 2011 5:31 am

Maybe there is something that I don’t understand, but aren’t these essentially VAR models?

2. vic on June 28, 2011 6:12 pm

the fit based on simulation would come up with a second order linear difference equation based on a simplicity limitation. Yes, that is a second order garch model. vic

3. Paul Cromwell on June 28, 2011 10:02 pm

Mr. Neiderhoffer,
I’ll be quick as I know you have better ways to spend your time. Education of a Speculator changed my life. It made me want to become a speculator and gave me a road map that would serve me well even though a speculator I would not become.
I feel like prior to reading Education I saw the world in only black and white, while afterwards I saw things in technicolor.
I am starting a blog focused on becoming a more disciplined, self-aware trader. I will be mentioning Education of a speculator from time to time and the positive impact that it had on my life. I will also bring up what I gained from Nassim Taleb’s writings. I know that you to have had difference’s of opinion at least in the media.
I am writing my next blog entry focused on what I learned from Education of a Speculator. I plan to send it your way before I publish it I hope that you will take a quick look at it and hopefully approve it. I will take any feedback that you have seriously.

4. Andre Wallin on June 29, 2011 10:36 am

before market declined below 1300 5 weeks ago it didn’t break 1300. Now we break 1300 with ease.

5. vic on June 29, 2011 2:19 pm

Thanks MR. Cromwell for that kind remark. I’ll be happy to do what I can to assist you. vic

6. douglas roberts dimick on July 2, 2011 11:10 am

Quantitative Relativity of electronic exchanges indicates that such attempts (including prescribed applications such as Fibonacci)to formulate arbitrary (or that which is based on external market concepts) models function only relative to pattern recognition and corresponding psychologies.

If one formulates a given difference equation based on market indicators, the equation may be found; however, that occurrence (in)frequently or otherwise does not prove that consilience constitutes market systematics.

VARs models merely serve as a form of pattern recognition, often relied upon for statistical probability determination. The problem with this approach is when market actors fail to recognize lack of consilience with competing, usually unknown models so operating.

As I have cited before, Madoff’s head trader (Josh) explains that “paradoxical” orientation and alignment of market participants (or what he calls “the herd”), whereby models become standardized, hence pattern recognition and the illusion of mean reverting strategies.

To correlate or not to correlate… that is the question.

dr

7. Nim Chimpsky on July 3, 2011 6:59 am

To Mr Dimick:
“To be is to do,” Socrates
“To do is to be,” Sartre
“Scooby dooby do” Scooby Do
“Yabba dabba do” Fred Flintstone