A constrained n-oscillator model in finance

This simple model comes out from a work I’m doing with a student on modelling the financial market.

We will divide the whole economy (comprising every possible asset) into some asset classes, e.g. housing, food, telecom, gold, oil, …

Denote this asset classes by A1, A2, …, An, and their prices (of the whole asset class) by P1, … , Pn respectively. Then the total economy is worth P = \sum P_i. Call Ri = Pi / P the relative price of the asset class P. Then \sum Ri = 1.

Each asset class Ai has a “fair value” Vi in the whole economy, and this “fair value” is a kind of adiabatic invariant (in changes very slowly). For example, computer prices drop fast because computers as a whole make up only some “fixed” % of the whole economy (people will only spend that much on computers), when the technology changes this % does not change much, and so prices drop.

We will denote by xi = Ri – Vi the “mispricing” of the asset class Ai. Then we have the constraint \sum xi = 0.

The “speculative energy” of the market is

E = 1/2 \sum ai xi^2 + 1/2 \sum bi \dot{xi}^2

With the above contsraint and energy function, we get a linear Hamiltonian system with n-1 degrees of freedom. Due to the fact that the Hamiltonian function is definite positive quadratic, the system is integrable with quadratic first integrals. Generically, we have n-1 normal modes of the system, which corresponds to n-1 different basic frequencies.  A general solution will be a linear combination of these normal modes, and so will be a quasi-periodic function with n-1 quasi-periods
Remark: this system is similar to, but  different from, the “coupled n-oscillator system” studied in physics textbooks.