Volatility Estimation Related to Principal Component Analysis By Steve Rogers, member, EE-Pub Published: December 5, 2005
Abstract In the financial industry model-based methods are popular for volatility estimation and processing. Volatility may be estimated in numerous model-based methods. We will explore both simple model-based ideas as well as the use of the Kalman filter. Once the volatility is estimated, it is then analyzed to extract useful information. In some cases, such as with the GARCH (generalized autoregressive conditionally heteroskedastic) model, a volatility model is presupposed. In other approaches, the model is fitted to the data. Such an approach is the principal component analysis (PCA). In this paper we will integrate volatility estimation with PCA analyses.
Article Information Field of Study—Signal Processing Keywords—volatility-estimation, GARCH, adaptive-filters, Kalman-filters
I. INTRODUCTION One of the most important innovations in modeling volatility changes is the GARCH (generalized autoregressive conditionally heteroskedastic) model. The GARCH (p,q) model [1,2] is shown in Figure 1. The conditional variance of a series of prediction errors is a function of lagged errors, time, parameters, and predetermined variables [1]. Note that with the system in the form we may now consider system identification.
Figure 1 GARCH model If input data is available at each desired update the coefficients may be solved adaptively. They may be solved by a moving window in a fixed filter least squares approach.
Adaptive update mechanisms for adaptive/recursive system identification will be discussed in the next section. Principal components analysis (PCA) is a variable-directed technique. It is based strictly on data and makes no assumptions about data models or groupings within the data. It is an unsupervised method, meaning that the analyses are based strictly on data and not a desired characteristic. As related to volatility estimation [3-4], it is used to extract eigenvalues and eigenvectors from volatility correlation matrices. Extracting this information at each data input would be of value in determining financial market direction. PCA may be used in calculating value at risk [4] (VaR) and direct volatility estimation [3]. The general approach to the use of PCA for volatility estimation is shown in Figure 2.
Figure 2 PCA equations The result of this analysis is that the eigenvalues are the volatility factors. Thus, if we can arrive at a good estimate for the matrix M, there are standard algorithms for reliably solving for the eigenvalues. The largest eigenvalue is the first principal component. The next largest eigenvalue is the second principal component and so forth. Generally, we only need to know the first few largest eigenvalues to describe the dominant parts of the volatility motion. The following sections discuss a few of the approaches to obtaining volatility estimates at each sample update.
II. SYSTEM IDENTIFICATION Numerous parameter estimation techniques have been developed over the years including fixed filters, adaptive/recursive filters and Kalman filters [5-7]. Fixed filters require a window of time series data between filter parameter updates. In the adaptive approaches the filter parameter updates occur at each time step or introduction of new data. This paper will emphasize the adaptive or recursive and Kalman filter approaches since we are generally dealing with slowly time-varying parameters to be estimated. As long as the parameters are ‘slowly’ changing with respect to the update rate, the adaptive filter approach will usually work satisfactorily. A diagram of a typical general adaptive filter is given in Figure 3.
Figure 3 General Adaptive Filter Adaptive filters are filters that automatically change their characteristics to attain the desired response at a given update point in time. Adaptive filters involve one or input signals and a desired response signal that may or may not be accessible. All adaptive filters contain three modules [5-7]: 1) filtering structure, 2) adaptation algorithm, and 3) performance evaluation. The filtering structure module forms the output of the filter using measurements of the input signal(s). The structure may be linear or nonlinear. The performance evaluation module assesses the quality of the output with respect to the application requirements. There is normally some balancing needed between what is acceptable performance and what is mathematically tractable. The adaptation algorithm module uses the performance evaluation criteria or some derivation from it to modify the filter parameters and continually improve its performance. Detailed design requires a significant understanding of the application, even though there is much commonality among most applications. The designer must be able to discriminate his application to select the adaptive filter features to be applied. Most adaptive filters are supervised as illustrated in Figure 4. From the equations at the bottom of the figure we find that the error is formed by subtracting the filter output from the desired signal. The adaptive algorithm causes a change in the filter parameters according to an increment designed based on performance criteria.
Figure 4 Supervised Adaptive Filter Numerous adaptive filter update strategies have been developed including the normalized least mean square algorithm, which is one of the more commonly used. The algorithm is summarized in Figure 5.
Figure 5 Summary of the Normalized LMS Algorithm
III. DISCRETE KALMAN FILTERS
The Kalman filter is a special case of the optimal linear filter algorithms. The development is given in numerous texts [5-7] and will not be repeated here. The algorithm consists of five basic equations [7] shown in Figure 6.
Figure 6 Basic equations of a Kalman filter A block diagram of the Kalman filter algorithm is shown in Figure 7. The Kalman filter can be used for system identification of the supervised adaptive filter shown in Figure 4. In the parameter estimation approach the estimated states are the parameters to be estimated which cause the convergence of the model to the input signal in the same way that the FIR adaptive filter described above does. The application to system identification is described in Figure 8.
Figure 7 Kalman Filter Block Diagram
Figure 8 Kalman Filter Application to GARCH(p,q) System Identification Once the covariance matrix is calculated PCA may be applied to extract the eigenvalues and eigenvectors. Most of the software packages, such as matlab, excel, etc., have the
capability to calculate and extract the eigenvalues and/or eigenvectors. This information can then be used for further analyses of the volatility.
IV. CONCLUSION Volatility estimation is of great importance to the financial industry. Two basic approaches toward volatility estimation in real-time have been presented: 1) adaptive filters based on the GARCH structure, and 2) use of Kalman filters for application to the GARCH structure identification and principal component analyses. Future research involves testing the approaches using actual data and extension and testing with portfolios.
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