Parameter optimization for differential equations in asset price forecasting

Abstract

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 (MPs) 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 MPs for the following days. For a number of closed-end funds, the results are statistically closer to the ensuing MPs than the default prediction of random walk (RW). In particular, we perform this optimization by a nonlinear computational algorithm that combines a quasi-Newton weak line search with the Broyden–Fletcher–Goldfarb–Shanno formula. We develop a nonlinear least-square technique with an initial value problem (IVP) approach for arbitrary stream data by focusing on the MP 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[[Ktilde]] between the true trading prices from Day i to Day i+n−1, and the corresponding computed MPs obtained from the first row vector of the numerical solution U of the IVP with AFDE for ith optimal parameter vector, where [Ktilde] 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 that contains most recently used successful parameters, besides the various fixed parameters from a set of grid points in a hyper-box.

Keywords:

• numerical nonlinear optimization
• inverse problem of parameter estimation
• asset flow differential equations
• financial market dynamics
• market return prediction algorithm
• data analysis in mathematical finance and economics

AMS Subject Classifications :

• 65L09
• 90C31
• 62P05
• 93C15
• 91B28
• 37N40
• 62M20
• 90C53
• 91B24
• 91B42
• 91B84

Acknowledgements

The authors are grateful for helpful comments by Professor Mihai Anitescu and two anonymous referees. The support of International Foundation for Research in Experimental Economics (IFREE) has been appreciated.