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Do opisu powierzchni implikowanej zmienności potrzebne są przynajmniej dwie składowe. Wydaje się zatem, że dwufaktorowe modele stochastycznej zmienności, gdzie tylko jeden faktor odpowiada za stochastyczny charakter zmienności, będą niewystarczające do opisu ruchu powierzchni. Mogą one modelować przesunięcie powierzchni (w pionie), ale jej nachylenie już nie. (fragment tekstu)
W pracy przedstawiono pewne sposoby stochastycznego modelowania zmienności, gdzie ceny lub stopy zwrotu mogą być opisane jako procesy stochastyczne o czasie dyskretnym lub ciągłym. Omówione zostały właściwości wybranych modeli stochastycznych, takich jak: stochastyczne modele zmienności powrotu do średniej, model Hulla-White'a oraz model iloczynowy procesu zwrotów Taylora. (fragm. tekstu)
Celem artykułu jest porównanie oszacowań zmienności uzyskanych z modeli parametrycznych: GARCH i SV z oszacowaniem uzyskanym na podstawie zmienności zrealizowanej szacowanej w oparciu o dane różnej częstotliwości. W badaniu wzięto pod uwagę zwroty z wybranych instrumentów polskiego rynku finansowego: indeks WIG 20 oraz kurs walutowy EUR/PLN. Ujęta w badaniu próba objęła okres kryzysu finansowego, co stanowi istotne uzupełnienie wyników prezentowanych do tej pory w literaturze. (abstrakt oryginalny)
A. Kliber w pracy (...) przedstawia analizę na rynkach Polski, Czech, i Słowacji. Weryfikacji poddano hipotezę o zależności zmienności stopy procentowej w danym kraju od zmian w dynamice zmienności stóp procentowych w pozostałych państwach. Autorka próbuje też ustalić, jaką rolę w kształtowaniu i przenoszeniu zmienności między krajami odgrywa kurs walutowy. (fragm. tekstu)
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Content available remote Modelling and Forecasting WIG20 Daily Returns
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The purpose of this paper is to model daily returns of the WIG20 index. The idea is to consider a model that explicitly takes changes in the amplitude of the clusters of volatility into account. This variation is modelled by a positive-valued deterministic component. A novelty in specification of the model is that the deterministic component is specified before estimating the multiplicative conditional variance component. The resulting model is subjected to misspecification tests and its forecasting performance is compared with that of commonly applied models of conditional heteroskedasticity. (original abstract)
We combine machine learning tree-based algorithms with the usage of low and high prices and suggest a new approach to forecasting currency covariances. We apply three algorithms: Random Forest Regression, Gradient Boosting Regression Trees and Extreme Gradient Boosting with a tree learner. We conduct an empirical evaluation of this procedure on the three most heavily traded currency pairs in the Forex market: EUR/USD, USD/JPY and GBP/USD. The forecasts of covariances formulated on the three applied algorithms are predominantly more accurate than the Dynamic Conditional Correlation model based on closing prices. The results of the analyses indicate that the GBRT algorithm is the best performing method. (original abstract)
W celu oszczędnego modelowania zmienności cen dużej liczby aktywów finansowych J. Osiewalski i A. Pajor (2007, 2009) oraz J. Osiewalski (2009) wprowadzili wielowymiarowe hybrydowe modele MSV-MGARCH. Ich macierz warunkowych kowariancji jest iloczynem jednego procesu ukrytego i macierzy o prostej strukturze MGARCH (DCC Engle'a, SBEKK). Zaproponowane modele hybrydowe wydają się przydatne zarówno ze względu na ich dobre dopasowanie, jak i możliwość zastosowania do nawet 50 aktywów łącznie. Jednak jeden proces ukryty może nie wystarczać w przypadku niejednorodnego portfela inwestycyjnego. W tej pracy proponujemy ogólniejszą strukturę hybrydową, wykorzystującą dwa procesy ukryte. Przedstawiamy pełne wnioskowanie bayesowskie dla takiego modelu i sugerujemy strategię MCMC symulacji z rozkładu a posteriori. Podajemy dwa przykłady formalnego bayesowskiego porównywania modeli. Pokazują one korzyści ze stosowania dwóch procesów ukrytych. W szczególności nasze podejście jest wykorzystane w łącznym modelowaniu czterech szeregów czasowych: dwóch indeksów giełdowych oraz cen złota i srebra. Formalnie porównujemy model łączny i dwa odrębne modele (dla indeksów oraz dla cen metali). (abstrakt autorów)
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Content available remote Bayesian Analysis of a Regime Switching In-Mean Effect for the Polish Stock Market
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The study aims at a statistical verification of breaks in the risk-return relationship for shares of individual companies quoted at the Warsaw Stock Exchange. To this end a stochastic volatility model incorporating Markov switching in-mean effect (SV-MS-M) is employed. We argue that neglecting possible regime changes in the relation between expected return and volatility within an ordinary SV-M specification may lead to spurious insignificance of the risk premium parameter (as being 'averaged out' over the regimes). Therefore, we allow the volatility-in-mean effect to switch over different regimes according to a discrete homogeneous two- or three-state Markov chain. The model is handled within Bayesian framework, which allows to fully account for the uncertainty of model parameters, latent conditional variances and state variables. MCMC methods, including the Gibbs sampler, Metropolis-Hastings algorithm and the forward-filtering-backward-sampling scheme are suitably adopted to obtain posterior densities of interest as well as marginal data density. The latter allows for a formal model comparison in terms of the in-sample fit and, thereby, inference on the 'adequate' number of the risk premium regimes. (original abstract)
In the paper two particular Markov Switching Stochastic Volatility models (MSSV) are under consideration: one with a switching intercept in the log-volatility equation, and the other — with a regime-dependent autoregression parameter. While the former one is fairly common in the literature (as a tool of taking account for regimes of different mean volatility level), the latter has not been paid almost any attention so far. We note the fact, that state-varying mean volatility may arise from switches in the intercept or in the autoregression parameter. Hence, we aim to compare these two models in respect of goodness of fit to the data from the Polish financial market, employing Bayesian techniques of estimation and model comparison. Clear evidence of structural shifts in the volatility pattern is found. Two different regimes of the economy are characterized in terms of the mean volatility level and the variance of volatility. (original abstract)
This paper examines short-run relationships among the U.S., German and Greek bond markets in times of financial crises. Specifically, the connections among daily and weekly growth rates of the 10-year government bond yields of the U.S., Germany and Greece from July 13, 2006 to January 29, 2016 are considered and an empirical illustration of those, based on the vector autoregressive (VAR) model with stochastic volatility (SV) disturbances, is provided. Finally, sufficient weak and strong exogeneity conditions in the VAR-SV models are tested. Our results indicate that during the time period covered by the analysis, the weekly growth rates of the 10-year U.S. bond yields were not affected by the past growth rates of the 10-year German and Greek bond yields. Contagion effects were absent among all the 10-year bond markets considered. From October 2008 to April 2015 a 'flight to quality' effect be-tween Germany and Greece, as well as between the U.S. and Greece seems to have occurred. Since the strong exogeneity hypothesis of the 10-year US bond yields' weekly growth rates has not been rejected by the data, they can be predicted from the marginal model only, i.e. without taking the German and Greek bond yields into consideration.(original abstract)
In this study, we model realized volatility constructed from intra-day highfrequency data. We explore the possibility of confusing long memory and structural breaks in the realized volatility of the following spot exchange rates: EUR/USD, EUR/JPY, EUR/CHF, EUR/GBP, and EUR/AUD. The results show evidence for the presence of long memory in the exchange rates' realized volatility. FromtheBai-Perrontest,wefoundstructuralbreakpointsthatmatch significant events in financial markets. Furthermore, the findings provide strong evidence in favour of the presence of long memory. (original abstract)
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Content available remote Pomiar i modelowanie zmienności - przegląd literatury
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Celem niniejszego artykułu jest przedstawienie głównych koncepcji dotyczących pomiaru, prognozowania i modelowania zmienności cen instrumentów finansowych. (fragment tekstu)
J. Osiewalski and A. Pajor (2007, 2009) and J. Osiewalski (2009) introduced hybrid multivariate stochastic variance - GARCH (MSV-MGARCH) models, where the conditional covariance matrix is the product of a univariate latent process and a matrix with a simple MGARCH structure (Engle's DCC or scalar BEKK). The aim was to parsimoniously describe volatility of a large group of assets. The proposed hybrid specifications, similarly as other models from the MSV class, require the Bayesian approach equipped with MCMC simulation tools. In order to jointly describe volatility on two different markets (or of two different groups of assets), J. Osiewalski and K.Osiewalski (2011) consider more complicated hybrid models with two latent processes. These new specifications seem very promising due to their good fit and moderate computational requirements. This paper is devoted to hybrid specifications with three latent processes, even more complicated and located on the edge of possibilities of conducting exact Bayesian analysis. We present full Bayesian inference for such models and propose efficient MCMC simulation strategy. Our approach is used to jointly model volatility of six daily time series representing three different groups: two stock indices, prices of gold and silver, prices of oil and natural gas. We formally compare joint modelling to individual bivariate volatility modelling for each of three groups. (original abstract)
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Content available remote Online Testing of Switching Volatility
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Chapter 6 presents the methods for online detection of a change in the unconditional volatility. A test based on a moving sum is proposed and evaluated. The test has a controlled asymptotic size, i.e. the false alarm probability during an infinitely long monitoring period is fixed. The effects of autoregression and conditional heteroscedasticity have been also addressed and a test that allows for heteroscedasticity has been proposed. The testing procedures are exemplified with the Hang Seng Index to see whether a shift during the Asian crises period could have been detected online (fragment of text)
Transport plays a key role in inventory management since it affects logistic costs as well as environmental performance of the supply chain. Expected value and variability of supply lead time depend on the transportation means adopted, and influence the optimal values of order quantity, reorder level, and safety stock to be adopted. Fast transportation means allow reducing expected value of the lead time; they are characterized by the highest costs of externalities (i.e. air pollutant emission, noise, congestion, accidents). On the contrary, slow transportation means require high inventory level due to large order quantity; in this case costs of externalities tend to decrease. The Sustainable Order Quantity (SOQ) [1] allows identifying optimal order quantity, reorder level, safety stock as well as transportation means which minimize the sum of the logistic and environmental costs in case of stochastic variability of product demand. In this paper, the authors propose a new SOQ analytical model considering stochastic variability of supply lead time (LT). A solution procedure is suggested for solving the proposed model. The approach is applied to a real industrial case study in order to evaluate the benefits of applying it if compared with the traditional one. (original abstract)
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Content available remote On Sensitivity of Inference in Bayesian MSF-MGARCH Models
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Hybrid MSV-MGARCH models, in particular the MSF-SBEKK specification, proved useful in multivariate modelling of returns on financial and commodity markets. The initial MSF-MGARCH structure, called LNMSF-MGARCH here, is obtained by multiplying the MGARCH conditional covariance matrix $H_{t}$ by a scalar random variable $g_{t}$ such that ${ln g_{t}, t\in Z}$ is a Gaussian AR(1) latent process with auto-regression parameter $\phi$. Here we also consider an IG-MSF-MGARCH specification, which is a hybrid generalisation of conditionally Student t MGARCH models, since the latent process ${g_{t}}$ is no longer marginally log-normal (LN), but for $\phi$ = 0 it leads to an inverted gamma (IG) distribution for gt and to the t-MGARCH case. If $\phi$ 6= 0, the latent variables $g_{t}$ are dependent, so (in comparison to the t-MGARCH specification) we get an additional source of dependence and one more parameter. Due to the existence of latent processes, the Bayesian approach, equipped with MCMC simulation techniques, is a natural and feasible statistical tool to deal with MSF-MGARCH models. In this paper we show how the distributional assumptions for the latent process together with the specification of the prior density for its parameters affect posterior results, in particular the ones related to adequacy of the t-MGARCH model. Our empirical findings demonstrate sensitivity of inference on the latent process and its parameters, but, fortunately, neither on volatility of the returns nor on their conditional correlation. The new IG-MSF-MGARCH specification is based on a more volatile latent process than the older LN-MSF-MGARCH structure, so the new one may lead to lower values of $\phi$ - even so low that they can justify the popular t-MGARCH model. (original abstract)
The aim of this paper is to analyse the welfare consequences of the processes of liberalisation of trade between asymmetric states in terms of the various sizes and effectiveness of their economies and the type of international exchange. These characteristics ultimately define the distribution of benefits from the liberalisation of international trade. When it is inter-industry or vertical intra-industry and barriers in trade are smaller than the difference in the effectiveness of the economies, the trade liberalisation undoubtedly contributes to improved social welfare, regardless of the level of effectiveness and the size of the economy. In the situation, however, of horizontal intra-industry trade, changes in the welfares of asymmetric countries, caused by their progressing trade liberalisation, depend on the sizes and effectiveness of their economies. The welfare of society in either a very big and ineffective or in a small and very ineffective country could even decrease in such a situation. This is the case when the increase in consumers' surplus is not sufficient to compensate for the decreasing profits of firms. (original abstract)
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Content available remote Volatile ARMA Modelling of GARCH Squares
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This paper points out that the ARMA models followed by GARCH squares are volatile and gives explicit and general forms of their dependent and volatile innovations. The volatility function of the ARMA innovations is shown to be the square of the corresponding GARCH volatility function. The prediction of GARCH squares is facilitated by the ARMA structure and predictive intervals are considered. Further, the developments suggest families of volatile ARMA processes. (original abstract)
The s-period ahead Value-at-Risk (VaR) for a portfolio of dimension n is considered and its Bayesian analysis is discussed. The VaR assessment can be based either on the n-variate predictive distribution of future returns on individual assets, or on the univariate Bayesian model for the portfolio value (or the return on portfolio). In both cases Bayesian VaR takes into account parameter uncertainty and non-linear relationship between ordinary and logarithmic returns. In the case of a large portfolio, the applicability of the n-variate approach to Bayesian VaR depends on the form of the statistical model for asset prices. We use the n-variate type I MSF-SBEKK(1,1) volatility model proposed specially to cope with large n. We compare empirical results obtained using this multivariate approach and the much simpler univariate approach based on modelling volatility of the value of a given portfolio. (original abstract)
W artykule przedstawiono model zmienności stochastycznej, oparty na dekompozycji Choleskiego. Następnie model SV oraz podejście Bayesowskie zostało wykorzystane do modelowania zmienności dwuwymiarowych finansowych szeregów czasowych oraz budowy optymalnego portfela walutowego. Rozważono hipotetyczny portfel, w skład którego wchodzą złotówkowe kursy dwóch walut: dolara amerykańskiego i marki niemieckiej. W procesie optymalizacji portfela wykorzystano predyktywny rozkład stóp zwrotu oraz predyktywny rozkład macierzy warunkowych kowariancji, uzyskany w rozważanym modelu MSV za pomocą metod Monte Carlo (MCMC). (abstrakt oryginalny)
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