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OVERLAPPING MULTIGRID METHODS AS AN EFFICIENT APPROACH FOR SOLVING THE BLACK-SCHOLES EQUATION

100%
Bernardelli M.
Metody Ilościowe w Badaniach Ekonomicznych
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2015
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vol. 16
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issue 1
25-36
EN
In this paper the modification of a two-level multigrid method by allowing an overlap between adjacent subdomains and its application to a one-dimensional Black-Scholes equation is described. The method is based on the finite-difference schema known as implicit Euler. Numerical experiments confirm the superiority of the proposed method in relation to the classic multigrid method in form of shortening computation time, memory savings and ease of parallelization. The comparison shows the advantages of overlapping grids vs method without them, mainly due to improved accuracy of the solution.
2
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The procedure of business cycle turning points identification based on hidden Markov models

100%
Bernardelli M.
Prace i Materiały Instytutu Rozwoju Gospodarczego SGH
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2015
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vol. 96: Analyzing and forecasting economic fluctuations
5-23
EN
In the paper the procedure, based on hidden Markov chains with conditional normal distributions and uses algorithms such as time series decompositions (STL), Baum-Welch algorithm, Viterbi algorithm and Monte Carlo simulations, is proposed to analyze data out of the business tendency survey conducted by the Research Institute for Economic Development, Warsaw School of Economics. There are considered three types of models, namely, with two-state, three-state and four-state Markov chains. Results of the procedure could be treated as an approximation of business cycle turning points. The performed analysis speaks in favor of multistate models. Due to, an increasing with the number of states, numerical instability, it is not obvious which model should be considered as the best one. For this purpose various optimization criteria are taken into consideration: information criteria (AIC, BIC) and the maximum-likelihood, but also frequency of obtaining a given set of parameters in the Monte Carlo simulations. The results are confronted with the turning points dated by OECD. The tested models were compared in terms of their effectiveness in detecting of turning points. The procedure is a step into automation of business cycle analysis based on results of business tendency surveys. Though this automation covers only some models from millions of possibilities, the procedure turns out to be extremely accurate in business cycle turning points identification, and the approach seems to be an excellent alternative for classical methods.
3
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The Viterbi paths in an analysis of business cycle synchronization

63%
Bernardelli M., Dędys M.
Prace i Materiały Instytutu Rozwoju Gospodarczego SGH
|
2015
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vol. 97: Economic cycles and uncertainty
9-23
EN
In the paper we investigate possibility of using the Viterbi paths to analyze two-dimensional macroeconomic time series. We build a two-dimensional Gaussian Markov-switching model with a four-state hidden Markov chain. The model is tested with two pairs of monthly indexes of industrial production for: Poland vs. France, and Poland vs. Germany. The most likely sequence of states of the hidden Markov chain is found for each pair. We compare that sequence with analogous sequences determined for a one-dimensional model with a two-state hidden Markov chain. The results of the comparison suggests the four state Viterbi path provides more valuable information about business cycle synchronization between the two economies than two separate two-state Viterbi paths.
4
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The Viterbi Path of hidden Markov models in an analysis of business tendency surveys

63%
Bernardelli M., Dędys M.
Prace i Materiały Instytutu Rozwoju Gospodarczego SGH
|
2015
|
vol. 96: Analyzing and forecasting economic fluctuations
25-47
EN
The aim of the paper is to show that turning points detection can be treated as a problem of pattern recognition. In the paper there are presented the results of applying normal hidden Markov models to a number of survey balances. Beyond a classical two-scale assessment of business activity a slightly more fuzzy classification of states is considered. To determine periods of unclear or difficult to evaluate situation unobservable Markov chains with three and four states are introduced. The outputs of the Viterbi algorithm, i.e. the most likely paths of unobservable states of Markov chains, are a basis of the proposed classification. The comparison of these paths with the business cycle turning points dated by OECD is described. The results obtained for three- and four-state Markov chains are close to those established in the references time series and seem to improve the speed with which, especially downshifts, are signaled. Furthermore, these results are more favorable than outcomes provided by conventional two-state models. The method proposed in this paper seems to be a very effective tool to analyze results of business tendency surveys, in particular, when multistate Markov chains are considered. Moreover, proposed decompositions allow an easy comparison of two time series as far as turning point are concerned. In the paper survey balances are compared with ‘hard’ economic data such as sold manufacturing production. The results confirm the accuracy of assessment provided by survey respondents.
5
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IMPROVING GLOBAL ELASTICITY OF BONUS-MALUS SYSTEM

63%
Topolewski M., Bernardelli M.
Metody Ilościowe w Badaniach Ekonomicznych
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2017
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vol. 18
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issue 1
120-133
EN
We optimize transition rules of bonus-malus system to achieve possibly best premium elasticity as defined by Loimaranta [1972] and later generalized as global elasticity by De Pril (1978). We use premium scale given by Norberg (1976). This issue constitutes a nonlinear nonconvex discrete optimization problem. To solve this problem, we apply improved greedy optimization algorithm, similar to one proposed by Morlock (1984). We analyse systems of different size for portfolios characterized by inverse Gaussian risk structure function with various parameters. We also propose alternative measures of global elasticity.
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