لخّصلي

Forecasting of asset market prices is of interest from a practical and theoretical perspective. Methodology in forecasting can be divided broadly into two groups. One of these consists of purely statistical methods, e.g., time series, that strive to uncover a statistically significant pattern in the data. A second involves developing some understanding of the underlying processes and deriving, for example, differ- ential equations. In general, there are some parameters that need to be estimated in order for a prediction to be made. For example, in weather forecasting one could make forecasts of temperatures in a way that is strictly statistical based upon the available data for a particular city and its neighbors. The alternative, however, is to utilize physical laws and estimate some parameters statistically whereupon the differential equations can be used to make a forecast. An advantage of the latter is that it greatly reduces the degrees of freedom, which, in a practical sense means that there are many fewer coefficients to be estimated. In some cases, there is a conservation law (e.g., the conservation of mass for weather forecasting) that eliminates a wide spectrum of coefficients. The main disadvantages of the model- ing approach versus the purely statistical approach is that the former may not be easily possible until a deeper understanding is attained. A secondary problem is that unfamiliarity with modeling leads some to question whether the conclusions are built into the system that has been derived. On the other hand, in a purely statistical model, it appears to be a clear and fair test of the importance of different effects.
However, the issue of the origin of the model is less relevant when it is possible to perform out-of-sample forecast (as is the case for weather forecasting and stock price forecasting, for example) that can be tested statistically to determine the accuracy of the predictions. Related to this issue is the Akaike [1] criterion which is a measure of the balance between the number of parameters to be estimated and the deviation between the model and the actual data. In other words, dramatically increasing the number of parameters in order to provide a slight improvement in the error does not yield a better model, in general. In the case of out-of-sample forecasting of asset prices, however, the number of parameters to be estimated is purely a practical issue. If one can determine a set of parameters within the com- putational and time constraints to make an accurate forecast, then it is generally not desirable to reduce the number of parameters considerably in order to obtain a forecast that is almost as good. On the other hand the constraints on time may be sufficiently strong that one may be restricted in the number of parameters and the method of estimation. For example, if one is using the government data released during the trading day in order to optimize parameters and make a forecast, one may have a constraint of one minute. On the other hand, if one is using daily data to make forecasts for the next day, one has the overnight time interval from the close of trading to the open of the next day.
Our approach uses the asset flow differential equations (AFDE) that have been developed by Caginalp and collaborators since 1989 (see Caginalp and Balenovich [11] and references contained therein). This approach has several key ingredients as discussed in the next section. It utilizes a basic supply and demand adjustment equation, but also incorporates the finiteness of assets, rather than the assumption of infinite arbitrage capital of classical finance theory. Furthermore, the supply and demand are determined by a transition rate that is dependent on sentiment. Classically, sentiment should depend only on a discount or premium to valuation. However, recent decades of research work has documented a set of motivations beyond valuation. One of the most significant of these is the price trend, also called momentum. The equations can readily incorporate additional motivational aspects of trading as they are established. In fact, one way of implementing this is to modify the differential equations to difference equations, and then to use statistical methods to evaluate the coefficients. Thus, if one hypothesizes a price trend motivation, the confirmation is the determination of a coefficient that is positive and statistically significant. If the coefficient corresponding to a particular hypothesized behavioral motivation is not actually present, then the coefficient, by definition, will be within the standard error of zero.
The implementation of these differential equations for practical forecasts poses challenging mathematical tasks that are inverse problems. Once a set of parameters characterizing an investor population is specified, the differential equations can be solved for future times. However, the values of the parameters are not known a priori, so that parameters must be determined by optimization using the actual price history up to the present time. We perform this optimization by a nonlinear computational algorithm that combines a quasi-Newton weak line search with the BFGS formula. We use nonlinear least-square technique with initial value problem approach 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. Here, the gradient (∇F (x)) is approximated by using the central difference formula and step length s is determined by the backtracking line search among several choices in literature (see Nocedal and Wright [29]). We construct a pool of initial parameters Ki chosen via a set of grid points in a hyper-box. We select an initial parameter vector from the initial parameter pool because the optimization success of quasi-Newton method in the algorithm depends on the initial parameter. Besides the fixed part of various initial parameters, the dynamic part of the pool is updated by adding successful parameters so that we keep a pool of various and most recently used candidate parameters.
In this paper, we utilize a system of differential equations that incorporate the valuation and trend motivations similar to Caginalp and Balenovich [11]. The ques- tion of valuation of a particular asset is often difficult to assess unambiguously. Nevertheless, the methodology for developing a valuation model is well established in finance, and a model for valuation can be constructed for a particular asset class. We utilize a particular group of stocks for which the valuation is very clear namely the set of closed-end funds traded on the NYSE. Briefly, a closed-end fund (see Bodie et al. [6]) is formed as investors pool a sum of money for a particular investment, e.g., investing in common stocks in Japan. Shares are allocated to the shareholders, who may then trade their shares on the open market, just like any other stock on the NYSE, and with the same rules. For most closed-end funds, the net asset value per share (NAV) is computed daily or weekly, and reported to the shareholders. Unlike an open-end fund, however, the shares of a closed-end fund cannot be redeemed by the fund except under special circumstances. This, of course, introduces the possibility that the fund may trade (on the NYSE) lower than the NAV (called a discount) or higher (called a premium) than the NAV. The fact that these funds often trade at significant discounts (e.g. 5% to 25%) and sometimes at large premiums (e.g. 50%) has been a puzzle to classical finance, and many papers have addressed these issues (see Anderson and Born for a sum- mary [2]). A number of these papers have focussed on reconciling these discounts to the efficient market hypothesis (EMH) whose centerpiece is the concept that the market trading price should reflect all available public information and should therefore reflect the true value. For example, a fund may trade at a discount due to an inherent tax liability ([2], Chapter 6) due to profits that are unrealized for tax purposes.