Parameter optimization for differential equations in asset price forecasting

A system of nonlinear asset flow differential equations (AFDE) gives rise to an inverse problem involving optimization of parameters that characterize an investor population. The optimization procedure is used in conjunction with daily market prices and net asset values to determine the parameters for which the AFDE yield the best fit for the previous n days. Using these optimal parameters the equations are computed and solved to render a forecast for market prices for the following days. For a number of closed-end funds, the results are statistically closer to the ensuing market prices than the default prediction of random walk. In particular, we perform this optimization by a nonlinear computational algorithm that combines a quasi-Newton weak line search with the BFGS formula. We develop a nonlinear least-square technique with an initial value problem (IVP) approach for arbitrary stream data by focusing on the market price variable P since any real data for the other three variables B, ζ1 and ζ2 in the dynamical system is not available explicitly. We minimize the sum of exponentially weighted squared differences F [K̃] between the true trading prices from day i to day i + n − 1 and the corresponding computed market prices obtained from the first row vector of the numerical solution U of the IVP with AFDE for ith optimal parameter vector where K̃ is an initial parameter vector. Here, the gradient (∇F (x)) is approximated by using the central difference formula and step length s is determined by the backtracking line search. One of the novel components of the proposed asset flow optimization forecast algorithm is a dynamic initial parameter pool which contains most recently used successful parameters, besides the various fixed parameters from a set of grid points in a hyper-box.