The Split-SV model

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Abstract

A modification of one of the most popular stochastic model in describing financial indexes dynamics is introduced. For describing a nonlinear behavior of volatility, a threshold noise indicator in the autoregressive time series of stochastic volatility is used. Toward this end, the model named the Split-SV model is introduced and its basic stochastic properties are investigated. Furthermore, the Empirical Characteristic Function (ECF) method is used for obtaining the parameter estimations of the model and a numerical simulation of the obtained estimates is given as well. Finally, the Split-SV model is applied for fitting the empirical data: the daily returns of the exchange rates of GBP and USD per euro.

Introduction

The stochastic analysis of financial sequences is commonly based on the time series modeling which is able to describe the distribution and behavior of a real data set. It has been shown empirically that the most of financial series, denoted as (Xt), exhibit nonlinear changes in the dynamics. Obviously, this implies a nonlinearity of the corresponding stochastic models. One of the main problem in this approach is to find the way to formally describe the volatility as a measure of variability of a particular financial series.

In the contemporary literature, mainly two different approaches to the mathematical modeling of volatility are used. The first one is described by a class of models of conditional heteroscedasticity. In this concept the relation between volatility and previously observed values of the series itself is expressed by using a deterministic approach. In the second group of models, dependence is presented by two independent sources of uncertainty. This determines behavior of the volatility and so behavior of the financial series (Xt). The model of this type was introduced by Taylor (1986) and named stochastic volatility model, or in short SV model. Today, SV models represent a very important class of nonlinear stochastic models.

Nowadays, there are many modifications and generalizations of Taylor’s model. We point out the works of Harvey et al. (1994), Jacquier et al. (1994), Shephard (1996), Kim et al. (1998) and, most recently, Wang et al. (2011), Tsiotas (2012), Venter and de Jongh (2014) and Shirota et al. (2014). Here we give the specific modification of Taylor’s model also, which can be favorable in a practical application.

It is well-known fact that many of the empirical data sets indicate outstanding nonlinearity which can be manifested in the various manners. The most common are sudden changes and “jumps” of volatility caused by the great fluctuation of the financial series (Xt). The main goal of our model is to explain that kind of nonlinearity in behavior of (Xt), i.e. its volatility series (σt). For this purpose, we set the noise threshold indicator, similarly as it was done in the time series of autoregressive conditional heteroscedasticity (ARCH) type, described in Stojanović and Popović (2007), as well as in the stochastic permanent breaking (STOPBREAK) processes, described in Stojanović et al., 2011, Stojanović et al., 2014. The basic idea of introducing the noise indicator in SV model is to follow the changes of volatility sequence as a function of dynamics of the sequence (Xt). On the other hand, a threshold structure of the noise indicator allows for one of the two independent noise sequences has a property of “optionality”. More precisely, if our first noise has “small” preceding realized value, our model “in the present” depends only on the realizations of the second noise. Vice versa, in the case of large-scale realizations, the model introduces extended values of the “optional” noise, and it will shapely react on quick and unexpected “jumps” of volatility. Therefore, it represents, in some way, the discrete-time analogue of the famous continuous Lévy-driven Ornstein–Uhlenbeck (OU)-based processes, introduced by Barndorff-Nielsen and Shepard (2001a) and applied in the modeling of SV processes by Barndorff-Nielsen and Shepard (2001b), Barndorff-Nielsen and Shephard (2003), Raknerud and Skare (2012) and some other authors. For these reasons, the basic structure of our stochastic model is somewhat more complex than Taylor’s SV model. Moreover, the constructed model, named Split-SV model, differs from the other related models, such as the Threshold SV model introduced by So and Li (2002) or the SV mixture model, introduced by Durhan (2007).

The paper is organized as follows. Definition of the Split-SV model is given in Section  2. The basic stochastic properties of this model are described in Sections  3 Properties of the autoregressive sequence, 4 Properties of the sequence. A procedure of the parameters’ estimation, based on Empirical Characteristic Function (ECF) method, is considered in Section  5. Section  6 is devoted to the Monte Carlo simulations, as an illustration of practical usage of the ECF method. Finally, an application of the previous estimation procedure in fitting the daily returns of the exchange rates of GBP and USD per euro is presented in Section  7.

Section snippets

Definition of the model

Generally, we define the Noise-Indicator model of Stochastic Volatility, or the Split-SV model, as the next sequence (Xt): {Xt=σtεt,σt=σe12Δt,Δt=aΔt1+ξtqt1, where tZ. We assume that the parameter a satisfies the non-triviality and stationarity conditions 0<|a|<1, and σ>0. Further, (εt) is a sequence of independent identically distributed (i.i.d.) random variables with Gaussian N(0,1) distribution, (ξt) is the i.i.d. sequence of random variables with Gaussian N(0,δ2) distribution, and the

Properties of the autoregressive sequence (Δt)

Let us remark that, under the stationarity condition |a|<1, random variables Δt have the mean μE(Δt)=0 and the variance υ2D(Δt)=D(ξtqt1)1a2=δ2mc1a2. Therefore, for an arbitrary tZ and kN we can express Δt as a function of Δtj,j=1,,k, in the following way Δt=j=0k1ajξtjqtj1+akΔtk. When k and |a|<1, we have Δt=j=0ajξtjqtj1, where the sum converges to Δt in mean-square and almost sure. According to the previous relations, we can obtain the autocovariance function Cov(Δt,Δt+k)=E

Properties of the sequence (Xt)

In this section, we investigate some stochastic properties of the series (Xt). As the random variable σte12ξtqt1 is Ft1 adaptive, we have E(Xt|Ft1)=σte12ξtqt1E(εte12ξtqt1|Ft1)=0. Thus, the sequence (Xt) is the martingale difference and therefore it is a sequence of uncorrelated random variables, and it follows: E(Xt)=E[E(Xt|Ft1)]=0. However, calculating the variance of Xt, D(Xt)=E(Xt2)=σ2E(eΔt), is somewhat complicated. A problem that occurs here is determining the exponential moments

Estimation of parameters. ECF method

The procedure of estimating parameters of SV models, because of their specific structure, is much more complex than with the most similar non-linear stochastic models. The most commonly used and studied methods are the simulated maximum likelihood method, introduced by Daníelsson (1994) and quasi-likelihood method introduced by Ruiz (1994), as well as the various modifications of these two methods, as the Laplace accelerated sequential importance sampling by Kleppe and Skaug (2012).

When the

Numerical simulation

In this section, we describe in detail the estimation procedure of the parameter θ based on the ECF method. As we have already noted, the implementation of the ECF method requires minimization of the p-dimensional objective function ST(p)(θ), while of the most important problems is the choice of value pN. Knight and Yu (2002) discussed the optimal value of p to achieve asymptotic efficiency of the estimator specifically for some Gaussian time series. Since the series (Yt), defined by Eq. (26),

Application of the model

In this section we applied the estimation procedure, described in the previous sections, to a real data set. We observed the daily log-returns of the exchange rates of British pound (GBP) and US dollar (USD) against the euro, in the period of 2003–2013 (Fig. 8). The sample sizes of these series are T=3631 and T=3640, respectively.

Table 3 presents some summary statistics of both of these series, denoted as (Xt), along with their squares (Xt2) and logarithmic processes (Yt). A simple comparison

Acknowledgments

We would like to thank the reviewers and editors for their valuable suggestions and comments that have allowed to improve the paper.

The second author was supported by the Serbian Ministry of Education, Science and Technological Development(No. #OI 174007).

The third author was supported by the Serbian Ministry of Education, Science and Technological Development(No. #OI 174015).

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