Fractal modeling of volatility dynamics in financial time series (Copyright Reserved)

Econophysics is an interdisciplinary research field, applying theories and methods originally developed by physicists in order to solve problems in economics, usually those including uncertainty or stochastic elements and nonlinear dynamics [1,2]. Its application to the study of financial markets has also been termed statistical finance referring to its roots in statistical physics [3].

A financial market is a mechanism that allows people to easily buy and sell (trade) financial securities (such as stocks and bonds), commodities (such as precious metals or agricultural goods), and other fungible items of value at low transaction costs and at prices that reflect the efficient market hypothesis [4-6]. In finance, financial markets facilitate:
(i) raising of capital in the capital markets,
(ii) transfer of risk in the derivatives markets,
(iii) international trade in the currency markets.

Long-range dependence (LRD) in financial time series has been widely studied [10-13]. It has been argued that the LRD may well be due to the time-varying trend. LRD can be defined in terms of the decay rates of long-lag autocorrelation C(t)~t--H ,as t®¥ for 0<> 0: step fi upwards,
fi <>> sx (5)
respectively. Meanwhile, for the case of long-range dependence (LRD) processes, one observes
with 0 < γ < 1 (6)

(7)

The fractional Brownian motion (FBM), BH(t) is one of the simplest stochastic model that can be used to model financial stocks or any time series with long-range memory (LRD) [21]. FBM is a Gaussian process defined in moving average representation as [16]
(8)
with the self-similar Hurst exponent 0 < H < 1 and B( ) is the Brownian motion. A general notation for the multiplicative constants VH and K, where VH = Г(2H+1) sin(πH) is the normalizing factor such that E[(BH,K (1) – BH,K (0))2] = K2, with K is the scale factor. The process is said to be a standard FBM, BH (t) when K = 1 and has zero mean with covariance
(9)

In addition, FBM is H-self-similar that is {BH(t)} and {a -H BH(at)} are equivalent in their joint distribution for all real a and 0 < H < 1. The discrete version of the increment processes or the fractional Gaussian noise WH(t) ≡ BH(t+1) – BH(t) is a stationary Gaussian process with zero mean and covariance
(10)
and as the lag k®¥,
(11)
where H = ½ which is refer to the Brownian motion. In general, for ½ < H < 1, WH(t) is said to be LRD or persistent. Meanwhile for 0 < H < 1, WH(t) is said to indicate short-range dependence (SRD) or antipersistent. These are the three different correlation behaviors.

Multifractional Brownian motion (MBM) is the generalization of fractional Brownian motion with the constant Hurst exponent H changed to Hurst function H(t) [13]. Using the time varying Hurst exponent one can model the stylized effects in time series based on the variation in their sample path irregularities.

Problems in economy and finance have attracted the interest of statistical physicists all over the world. Statistical physics concepts such as stochastic dynamics, short- and long-range correlations, self-similarity and scaling permit an understanding of the global behavior of economic systems without first having to work out a detailed microscopic description of the same system. Fundamental problems pertain to the existence or not of long-, medium- or/and short-range power-law correlations in various economic systems, to the presence of financial cycles and on economic considerations, including economic policy. A method like the detrended fluctuation analysis (DFA) is recalled emphasizing its value in sorting out correlation ranges, thereby leading to predictability at short horizon [7-8]. A well-known financial analysis technique, the so-called moving average, is shown to raise questions to physicists about fractional Brownian motion (FBM) properties [8].
In order to probe the extent of universality in the dynamic of complex behavior in financial markets and to provide a basic and appropriate framework for emerging economies, we focus in the international trade (foreign exchange) time series of certain emerging economies in East Asia namely Malaysia, Singapore, Indonesia, Thailand, and South Korea by studying and analyses the dynamical properties in the foreign currency exchange, such as volatility, scaling behavior [9]. These time series are then modeled using fractional Brownian Motion (FBM), fractional Gaussian noise (FGN), fractionally integrated ARFIMA (0,d,0). The focus parameters (LRD parameters) are the price, returns, absolute returns and volatility [10].

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