The Split-SV model
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 , 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 . 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 . The main goal of our model is to explain that kind of nonlinearity in behavior of , i.e. its volatility series . 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 . 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 : where . We assume that the parameter satisfies the non-triviality and stationarity conditions , and . Further, is a sequence of independent identically distributed (i.i.d.) random variables with Gaussian distribution, is the i.i.d. sequence of random variables with Gaussian distribution, and the
Properties of the autoregressive sequence
Let us remark that, under the stationarity condition , random variables have the mean and the variance Therefore, for an arbitrary and we can express as a function of , in the following way When and , we have where the sum converges to in mean-square and almost sure. According to the previous relations, we can obtain the autocovariance function
Properties of the sequence
In this section, we investigate some stochastic properties of the series . As the random variable is adaptive, we have Thus, the sequence is the martingale difference and therefore it is a sequence of uncorrelated random variables, and it follows: However, calculating the variance of , 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 based on the ECF method. As we have already noted, the implementation of the ECF method requires minimization of the -dimensional objective function , while of the most important problems is the choice of value . Knight and Yu (2002) discussed the optimal value of to achieve asymptotic efficiency of the estimator specifically for some Gaussian time series. Since the series , 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 and , respectively.
Table 3 presents some summary statistics of both of these series, denoted as , along with their squares and logarithmic processes . 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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