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).