Volatility Model And Information Flow Of Complex Systems

There are three main parts. Firstly, we study the numerical methods for solving problems in Black-Scholes equation, proposed the importance of stochastic volatility, and compared estimate values and accurate values. Secondly, we study the fluctuation analysis method in Beijing complex transportation systems, and proposed a new method to apply and gives the main results. Finally, according to the concept of entropy, we study the information flow and give the analysis results.In finance, one of the most popular and most useful formulas is the Black-Scholes equation for option pricing. There are various methods to approximate solutions to boundary values problems with this equation. However, the most popular one is numerical methods. In this paper, we will introduce three schemes, namely the ex-plicit, implicit and Crank-Nicolson finite difference method. The difference between traditional methods and our study is about volatility. Generally, Black-Scholes theory, it is assumed that the volatility is constant with respect to time and share price has no effect on it, but it is not necessarily true. This paper will discuss the approximation of solutions of Black-Scholes equation by using finite difference methods with a stochastic volatility model. In the stochastic volatility model, there is a brief discussion about the effect of parameters on values of volatility in the model. Moreover, the results will be compared with exact solutions.Traffic systems, especially urban traffic systems, are regulated by different kinds of interacting mechanisms which operate across multiple spatial and temporal scales. Traditional approaches fail to account for the multiple time scales inherent in time series, such as Empirical probability distribution function and Detrended fluctuation analysis which have lead to different results. The role of multiscale analytical method in traffic time series is a frontier area of investigation. In this paper, our main purpose is to introduce a new method-multiscale time irreversibility which is helpful to extract information from traffic time series we studied. In addition, to analyse the complexity of traffic volume time series of Beijing Ring 2,3,4 road between workdays and weekends which are from August 18,2012 to October 26,2012, we also compare the results by this new method and multiscale entropy method we have known well. The results show that the higher asymmetry index we get, the higher traffic congestion level will be, and accord with those which are obtained by multiscale entropy.Recently, an information theoretic inspired concept of transfer entropy has been introduced by Schreiber. It aims to quantify in a non-parametric and explicitly non-symmetric way the flow of information between two time series. This model-free based on Shannon entropies approach in principle allows us to detect statistical dependencies of all types, i.e. linear and nonlinear temporal correlations. However, we always analyze the transfer entropy based on the data, which is discretized into three partitions by some coarse graining. Naturally, we are interested in investigating the effect of the data discretization of the two series on the transfer entropy. In our paper, we analyze the results based on the data which are generated by the linear modeling and the ARFIMA modeling, as well as the dataset consists of seven indices during the period 1992-2002. The results show that the higher the degree of data discretization get, the larger the value of the transfer entropy will be, besides, the direction of the information flow is unchanged along with the degree of data discretization.