First order differential equations of the form:

the rate of change of a variable + the original variable x a constant equals a constant times a function, or

dy/dt + p * y =  k1 * q(t)

has wide applicability in all physical settings. it's used to model the cooling and diffusion equations for example, as Arthur Mattuck in a brilliant and relatively easy to assimilate lecture shows.

For what variable in the market does its rate of change depend on its level and the movements of a second variable. The moves of stocks relative to bonds and currencies comes to mind. Is it predictive in certain cases and how do random perturbations affect the solution and its predictivity? Are there any methods used to solve these first order equations that are useful for markets without regard to stochastic, useless solutions?

Leo Jia writes: 

I once attempted to use it to model the market, but I did not proceed. The reason is that I realized the solution would be a function of two coefficients, i.e. K and K1 in this case, and so by varying the coefficients, one can fit the solution well onto the historical chart. The way to fit it wouldn't be very distinct from that of fitting a moving average onto a historical chart. So to me it seemed to fall into the same dilemma as trying to profit from a moving average model. Would anyone correct me?





Speak your mind

2 Comments so far

  1. isomorphismes on March 25, 2014 9:43 am

    Mattuck is an incredible lecturer. OCW all the way.

  2. isomorphismes on March 25, 2014 9:47 am

    Why do you say the stochastic approach is useless? As far as I can tell it’s a straight shot from ODE to PDE to SDE.

    All your basic famous ODE’s have solutions with a basis shaped like { C * exp( i * k * t ) }, with i = sqrt(-1) giving some kind of cyclicality and the k’s saying the time period / frequency of those (fixed, regular) cycles.

    Something else to think about, this is baby stuff to quants. Why on a broad level is there going to be an edge to 1st-order ODE’s which young engineers learn at age 19?


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