Parallelization of the Α‐stable Modelling Algorithms

Abstract

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.

Mixed-stable approach to the management of the portfolio using high-frequency financial data

“…Note that precise and fast calculation of ( , ) k p X Θ values in the optimization target function (2) is a non-trivial task. In this work, the improper integral of probability density function (3) was evaluated using the methods developed and approved in previous works [3,4].…”

“…of non-zero returns. As in previous works [3,4], the log-likelihood function (2) is optimized using the Nelder-Mead method [24].…”

“…This approach is computationally very demanding, especially for long financial series. However, the application of parallel computing makes it both precise and practical, as it has been shown in [3,4].…”

Modeling Baltic market benchmark index: a comparison of models

“…Visų modelių parametrų įverčiams gauti yra taikomas maksimalaus tikėtinumo (ML) metodas. Šis metodas duoda tiksliausius įverčius, tačiau reikalauja daug kompiuterinio lauko (jei nėra taikomi lygiagretieji skaičiavimai [3,4]).…”

Parallel Computing for Mixed-Stable Modelling of Large Data Sets

“…However, in this work we will demonstrate that the application of modern parallel technologies makes this approach both precise and practical. In this research, we develop and expand our parallel computing algorithms [2] to the mixed-stable modelling of highfrequency data. To our best knowledge, such an approach to statistical processing of big data was not used before.…”