Stock analysis and forecast using chaos theory

Question

Yesterday, I was going through an article in which the user had mentioned that he has used chaos theory to predict stock prices and ended up with 30% + profit.(I am not intersted in the profits :P) After that I read a bit about chaos theory and found out that its basically finding patterns called fractals in the data available. After that I came to know it is used in various fields as weather prediction and stock prices determination.

I just wanted to know how if anyone has an idea on how we can do stock forecasting using chaos theory.

Like, it will be great if someone can provide me with some example w.r.t chaos theory.

Answer 1

First of all, I believe you meant fractals rather than fractules. No, sorry. First of all, do not fully rely upon advises of guys who made 30%+ profit until you have seen exactly that they did it. Even if the owner of Quantum Fund will tell you that he earned 30%+ profit, you can trust him - but don't still rely upon his advises.

Fractals are objects which seems to be very natural, however I do not know if they already were described formally. Say, Sierpinski carpet is known to be a fractal, but given some set $A$ you cannot say that it is a fractal.

The nature of the fractal is the following: each part of it has the same structure as the whole fractal (or there is a subset of each part, no matter how small is this part, which has the same structure as the whole fractal). This idea was applied by Benoit Mandelbrot to describe the structure of Elliot Waves - one of the most important tool in technical analysis of charts. Mandelbrot wrote about it in his paper. The main hypothesis: there are several factors which influence the price, some of them more, some - less, but the structure (not the amplitude) of these influences is the same.

The other application can be seen in the fractal Brownian motion. This stochastic process is used to model the price movements. You can also be interested in this case in this question.

Finally, I guess that there are methods to model the price movements with dynamical systems which have a chaotic behavior. E.g. Shiryaev in book "Essentials of stochastic finance" devoted one section to discuss the difference between time series based on chaotic (deterministic) processes and stochastic processes (see chapter "Dynamical Chaos Models").

That's what I know on the subject. I cannot provide a good example unless I put here a plot of some dynamical chaotic system to show that it looks like the price movements. If you make your question more specific, maybe I can write more.

P.S. You may also ask this question on http://quant.stackexchange.com

Answer 2

Here is a personal opinion: I don't believe in any mathematical models for stock prices or derivates or other financial products. But here is a possible connection of chaos theory to stock price modelling, I'll simply throw some buzz words at you:

If you have a dynamical system with a long time trend and short time "random noise influences", you can try to model this system with a stochastic differential equation. The famous Black-Scholes formula for option pricing is derived from a linear Ito stochastic differential equation. On a conceptual level, it combines the long time trend coming from fixed interest rates with the short time noise coming from day-to-day trading by many different agents, each having a small influence.

The expectation values of certain stochastic differential equations solve certain partial differential equations. The expectation value of an Ito process, for example, solves the Kolmogorov forward equation. This is also true for the equations that describe the classical flow of a fluid, the Navier-Stokes equations.

For a paper adressing this topic, see

• Peter Constantin, Gautam Iyer: "A stochastic Lagrangian representation of the 3-dimensional incompressible Navier-Stokes equations" (arXiv).

A first approximation of a weather forecast would need to solve the Navier-Stokes equations of the atmosphere of the Earth. This means that you could, in theory, use stochastic differential equations to do a weather forecast.

Both stochastic differential equations and partial differential equations describe systems with infinite many degrees of freedom. But it is possible to approximate these systems with systems that have a finite number of degrees of freedom, that is, by a system of ordinary differential equations. You can try to devise this approximation in such a way that the finite system captures some characteristic properties, like turbulence of solutions of the Navier-Stokes equations.

Here is a book devoted to this topic:

• Thomas Bohr, Mogens H. Jensen, Giovanni Paladin, Angelo Vulpiani: "Dynamical systems approach to turbulence".

This is somewhat related to the numerical approximation of PDE via spectral methods, but in this case the trick is to find the approximation that results in the best results concerning specific properties of the system that you are interested in.

So, in this sense chaos theory aka dynamical systems can be used as an approximation to PDE which describe the time evolution of the probability laws of stochastic processes, which are the most common modelling tool of quants, I guess (I'm not a quant).