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A realistic ARCH process is set up so as to duplicate, for all practical purposes, the properties of stock time series from 1 day to 1 year. The process includes heteroskedasticity with long memory, leverage, fat-tail innovations, relative return, price granularity, and holidays. Its adequacy to describe empirical data is controlled over a broad panel of statistics, including (robust L-statistics) skew, (robust) kurtosis, shape factor for the volatility distribution, and lagged correlations between combinations of return and volatility. These statistics are computed for returns and volatilities with characteristic time intervals ranging from 1 day to 1 year. This wide cross-check between stock time series and simulations ensures that the most important features of the data are correctly captured by the process up to 1 year. The by-products of the statistical analyses and estimations are (1) a positive skew, (2) a cross-sectional relation between kurtosis and heteroskedasticity, (3) a very similar cross-sectional distribution for the statistics evaluated over the empirical data set or for the process with one set of parameters and (4) the heteroskedasticity is very close to an integrated volatility process. [ABSTRACT FROM AUTHOR]

The canonical process used to describe financial time series is based on a logarithmic random walk. Adding a fat-tail distribution for the innovations in this framework creates fundamental inconsistencies, essentially related to a diverging integral. The problems are related to (1) the (infinite) values of statistical quantities for processes with fat-tail innovations, (2) the robustness of computations when dealing with large events (genuine or noise), and (3) the pricing of options with heteroskedasticity and fat tails. Instead of logarithmic processes, we advocate using geometric processes and relative returns so that these problems do not occur. The mathematical properties of geometric processes are explored for setups of increasing complexity. For empirical time series and for processes, the skews are evaluated using a robust estimator and are compared for both return definitions. Finally, the European option pricing framework is modified to use geometric processes for the underlying, allowing the incorporation of natural fat-tail innovations. [ABSTRACT FROM AUTHOR]

The present paper studies the non-parametric estimation of volatility in financial time series. Support Vector Regression (SVR) is applied and compared with alternative techniques for estimating a Conditional Heteroskedastic AutoRegressive Nonlinear (CHARN) model. A multiscale decomposition analysis decomposes a time series into several scales, and then several separated SVRs, equal to the number of scales, are fed for running the CHARN model by scaled decomposed series. A multiscale CHARN model, which is an incorporation of wavelet decomposition analysis with a CHARN model estimated by SVR, is proposed and investigated using real-world data sets. To exploit the locality advantage of wavelets, we use wavelet analysis and combine it with SVR. Wavelets are able to economically describe phenomena that are heterogeneous. Recent studies on heterogeneous agents in financial markets theoretically support advocated scale-based analysis of markets, and, of course, the multiscale CHARN model proposition, since it is believed that each scale corresponds to each heterogeneous agent who acts differently in the market. Our objective is to improve the accuracy of estimation by applying multiscale decomposition. The results of applying SVR to estimate the CHARN model in a single resolution on real-world time series indicate that it outperforms the technical benchmarks. Also, the multiscale CHARN model estimated by SVR is more accurate than a single scale model. [ABSTRACT FROM AUTHOR]