Stable distributions have a wide sphere of application: probability theory, physics, electronics, economics, sociology. Particularly important role they play in financial mathematics, since the classical models of financial market, which are based on the hypothesis of the normality, often become inadequate. However, the practical implementation of stable models is a nontrivial task, because the probability density functions of α‐stable distributions have no analytical representations (with a few exceptions). In this work we exploit the parallel computing technologies for acceleration of numerical solution of stable modelling problems. Specifically, we are solving the stable law parameters estimation problem by the maximum likelihood method. If we need to deal with a big number of long financial series, only the means of parallel technologies can allow us to get results in a adequate time. We have distinguished and defined several hierarchical levels of parallelism. We show that coarse‐grained Multi‐Sets parallelization is very efficient on computer clusters. Fine‐grained Maximum Likelihood level is very efficient on shared memory machines with Symmetric multiprocessing and Hyper‐threading technologies. Hybrid application, which is utilizing both of those levels, has shown superior performance compared to single level (MS) parallel application on cluster of Pentium 4 HT nodes.
In this paper, we develop efficient parallel algorithms for the statistical processing of large data sets. Namely, we parallelize the maximum likelihood method for the estimation of parameters of the mixed-stable model. This method is known to be very computationally demanding. Financial German DAX stock index data are used as empirical data in this work. Several hierarchical levels of parallelism were distinguished, analyzed and implemented using OpenMP and MPI library. Parallel performance tests were conducted on the IBM SP6 supercomputer. Obtained performance results show that implemented parallel algorithms are very efficient and scalable on distributed and shared memory systems. Speedups up to 800 times were obtained for 1024 parallel processes. Noticeably, our parallel application is able to efficiently utilize the Simultaneous multithreading (Intel Hyper-Threading) technology in modern processors. This research demonstrates that the application of modern parallel technologies allows a fast and accurate estimation of mixed-stable parameters even for large amounts of data and promotes a wider use of stable modelling for the statistical data processing.
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