Modeling The Fluctuation Of Financial Data And Empirical Studies

Firstly, in this paper we define a subseries of a time series, namely, a local alternate high-low point sequence, which characterize the rise-fall fluctuation of the whole series and is two-dimensional. It can be derived into another two-dimensional point sequence on the high-low level and up-down duration. By the definition, the particular subseries determine the direction change, the size correct and the duration of the local trend of the whole time series. If the new two-dimensional point sequence can be effectively builded a statistical model, it is a very important for the theoretical and practical value of time series data, especially financial data. For example, it is used to study the efficiency of financial market, behavioral finance, the pricing of the various derivatives of financial securities, the investment of securities and financial risk management and so on.Secondly, about the new two-dimensional point sequence, we build up two types of regime-switching models:one is a trend-switching model depended on the local high-low point, the other is the local high-low point model based on the regime-switching of bull-bear market. The former characterizes that the local trend of the dynamic process is resisted by the local high point (high level, upward duration) when the process rises, and supported by the local low point (low level, downward duration) when the process falls, meanwhile, the alternate local high-low points are before-after dependent and clustering. In essence, its thresholds are not fixed, but time-varying. The variable thresholds (or points) are a new two-dimensional sequence (high-low level, upward-downward duration). For the sequence, we set up an auxiliary model that is two-dimensional vector autoregressive. By means of the auxiliary model, we predict both the change of the direction and the correction of the size of the local trend. Moreover, because the forecast of the current two-dimensional thresholds (local high-low turning point) incorporate the information on the past thresholds, the critical information that controls the direction and the size of local trend is medium-and long-term. Furthermore, in the main model, which is also called for local trend (mean) model, the size of the local trend absorbs not only local short-term information, but also medium-and long-term information from the auxiliary model, so the switching of the direction and the correct of the size of the current local trend are impacted by both local short-term and medium-and long-term factors. We propose the beautiful, symmetric structure that the local trend (main) model is integrated with an auxiliary model. The specification can be also called the trend-switching model depends on the local high-low point or on both level and duration (Trend-LD); In order closer to real financial markets, the latter, based on the former, models local high-low points in bull and bear market respectively, and characterizes the dynamic behavior that the rise-fall fluctuation of local trend is not symmetric. Obviously, it incorporates not only the local information of rise-fall fluctuations, but also the large background information of bull-bear trend, so it is a blend of trend theory and wave theory in technology analysis. We believe that this dynamic behavior is a direct reflection of investors’ herding on security prices. In addition, we proposed a definition of weak stationarity of regime. It is a generalization of the common weak stationarity of stochastic process.Thirdly, this paper researches that investors’ behavior in the financial market asymmetrically affected by the local high-low levels and rise-fall durations of fluctuation of stock index. The results show that the dynamic behavior of the current high-low level of fluctuation is more dependent on the past fluctuations (high-low levels and rise-fall durations) than that of the rise-fall duration. The dependence between the high-low levels is much larger than that between the rise-fall durations. Moreover, the impact of the past local high-low levels on the subsequent rise-fall duration is greater than that of the past rise-fall durations on the subsequent local high-low level. This may result in the short-term profit of investors is more directly related to the local high-low level of fluctuation than to the local rise-fall duration (cycle). Therefore, the impact of the both on investors’ behavior is different.Finally, the two types of models are applied to the representative (developed and undeveloped) stock market indices. The former is used to model the six stock indices, the empirical result show that the forecasts of the direction and size of the local trend under the former’s hypothesis are significantly better than those under random walk hypothesis. It also means that even well-developed stock markets (e.g. Europe and North America) are not efficient, and the conclusion differs from those of the past many empirical studies. The latter is used to model the sixteen stock indices. Considering the transaction cost, By means of the model, we implement the buy-low-and-sell-high strategy along trend to invest the stock indices in the non-developed market with large rise-fall fluctuations. The findings show the performances of the strategy both at the end of the period and in the entire interval of investment simulation are significantly better than those of the buy-and-hold strategy (i.e., beat the market); However, to invest the stock indices in the developed markets with small rise-fall fluctuation, the performances are differences:(1) Implemented to invest in the stock index with a large observation data in North America, the performances of the strategy both at the end of the period and in the entire interval of investment simulation are almost significantly better than those of the buy-and-hold strategy;(2) With a less observation data in Europe and Japan, some are not significant as good as those of the buy-and-hold strategy, other are not significant different from those of the buy-and-hold strategy. Clearly, the simulating investment strategy based on the model, mainly takes advantage of the information of local high-low points. Nevertheless, if the local rise-fall fluctuation of the stock index is too small, taken the frequent trading cost into consideration, the simulating strategy is difficult to beat the market. The empirical result further shows that the non-developed stock markets are not weak efficient and the model-based strategy is practical for the non-developed stock markets, as even for the well-developed stock market in long-term view, except for the long-term strong bull market. From a behavioral finance perspective, the empirical results suggest that the psychological behaviors of investors in a different degree of development of the stock market are different, especially, herding behavior is a large difference. The more mature and sophisticated financial markets are, the more weak investors’ herding behavior is, and vice versa.