Graphical models for financial time series and portfolio selection

Criteria, Models and Strategies in Portfolio Selection.- A Model for Portfolio Selection with Order of Expected Returns.- A Compromise Solution to Mutual Funds Portfolio Selection with Transaction Costs.- Optimal Portfolio Selection of Assets with Transaction Costs and No Short Sales.- Portfolio Frontier with Different Interest Rates for Borrowing and Lending.- Multi-period Investment.- Mean-Variance-Skewness Model for Portfolio Selection with Transaction Costs.- Capital Asset Pricing: Theory and Methodologies.- Empirical Tests of CAPM for China's Stock Markets.

This thesis concerns graphical models, mainly with discrete variables, a class of probabilistic models that can be understood pictorially using an interaction graph. Some classical graphical models includes the naive Bayes classifier and the Ising model, where the interaction graph of the former has directed edges, and the latter has undirected edges. At the core of graphical models is the concept of conditional independence; X is conditionally independent of Y given variable Z if by knowing Z, Y is irrelevant for X. Having access to such knowledge typically simplifies the model and hence the interaction graph. For classification problems, the structure of the interaction graph may be of lesser importance, whereas the structure may be essential in problems where the goal is the understand the underlying phenomenon. A large number of different graphical models exists with special structures of the interaction; graphs with both directed and undirected edges, both discrete and continuous variables etc. Common to all these is that the graphical structure must be learned either from data, also referred to as structure learning, or specified through expert knowledge. Given an interaction graph, calculating posterior probabilities, also known as inference, is essential in graphical models. Although both structure learning and inference are, in general, NP-complete problems there exist efficient heuristics for both, and even exact methods for inference. Most notably is the junction tree algorithm.

This paper continues research within the framework of the scientific direction in econophysics at the Department of Information Systems and Mathematical Methods in Economics of the faculty of Economics of PSU. Modeling and prediction of financial time series is quite a perspective area of research, because it allows participants of financial processes to reduce risks and make effective decisions. For example, we could research financial processes with the help of fractal analysis. In the article there is studied and worked out in detail one of the methods of fractal analysis of financial time series – the box-counting method for assessment of the fractal dimension. This method is often used in studies conducted by domestic authors, but the authors do not delve into the characteristics and problems of using the box-counting method for analysis of time series, that means that the answers to the interested questions have not yet been given. The main problem is that, as a rule, the analyzed object in the tasks of applying the box-counting method to time series is a computer image of the plot of series. In the article there is proposed the procedure of adaptation of the box-counting method for assessment of the fractal dimension of time series, the procedure does not require the formation of a computer image of the plot. In the article there is considered following difficulties developed from this adaptation: 1) high sensitivity of the resulting estimation of the dimension to the input parameters of the method (the ratio of the sides of the covered by cells plane with the plot; the used range of lengths of the side of the cell; the number of partitions of the plane into cells); 2) the non-obviousness of choosing the optimal values ​​of these parameters. In the article there are analyzed approaches to the selection of these parameters that were proposed by other authors, and there are determined the most suitable approaches for the adapted box-counting method. Also there are developed unique methods for determining the ratio of the sides of the plane with the plot. In the paper there is written the computer program that implements the developed method, and this program is tested on the generated data. The study obtained the following results. The fact of sensitivity of the adapted box-counting method to input parameters is confirmed, that indicates the high importance of the correct choice of these parameters. According to the study, there is found out inability of the proposed methods of automatic determination the ratio of the sides of the plane in relation to artificial time series. There are obtained the most precise (in a statistical sense) estimates of fractal dimension, those found by means of the adapted box-counting method, with the fixed ratio of the sides 1:1. According to comparing the adapted box-counting method and R/S analysis, there are obtained the most precise estimates by the second method (R/S analysis). Finally in the paper there are formulated the possible directions for further research: 1) comparison of the accuracy of various methods for assessment of the fractal dimension on series of different lengths; 2) comparison of the methods of fractal analysis and p-adic analysis for modeling and prediction of financial time series; 3) determination of the conditions of applicability of various methods; 4) approbation of the developed methods for determining of the ratio of the sides of the plane with the plot on real economic data.

Effects of Price of Gold on Bombay Stock Exchange Sectoral Indices: New Evidence for Portfolio Risk Management

Mean-risk models have been widely used in solving portfolio selection problems in the last years. Since the mean-variance theory of Markowitz, an enormous amount of papers have been published extending or modifying the basic model in several directions like the simplification of the type and amount of input data, the introduction of alternative measures of risk, the incorporation of additional criteria or constraints. Recently, risk measures concerned with the left tails of distributions that evaluate the extremely unfavourable outcomes are used. The most used risk measure for such purposes is Value-at-Risk (VaR). In this paper we concentrate on the second direction of incorporating of a new risk measure in portfolio modeling. We define a new risk measure, which take into consideration the values exceeding a certain threshold in the extreme tail of the loss distribution, called Limited Value-at-Risk (LVaR). We study the properties of this risk measure. We build a new model for portfolio selection, named mean-LVaR model, in which risk is evaluated using LVaR risk measure. We study the properties of the new mean-risk model and compare it with the classical mean-VaR model. We derive the analytical form of LVaR risk measure in the case of normal distribution. We provide computational results and analyze the implications of using the mean-LVaR risk model in portfolio optimization problem.

Mean-variance portfolio optimization is widely used by financial professionals as a fundamental strategy for constructing portfolios that achieve the highest returns for a given degree of risk tolerance. This traditional approach suffers from computational instability issues, which are often solved by integrating regularization-based methods. However, regularized mean-variance models overlook sector considerations, emphasizing a need for market-informed portfolio optimization. In this paper, we propose an extended mean-variance portfolio selection framework that incorporates a new adaptive sparse-group least absolute shrinkage and selection operator (LASSO) regularization as a penalty parameter. In the proposed model, the sparse optimal portfolio is selected considering both sectors and assets simultaneously. Moreover, the inclusion of adaptive weights in the regularization, estimated based on chosen criteria, enables the use of more market data in a Markowitz portfolio model. We also develop an efficient alternating direction method of multipliers (ADMM) algorithm that guarantees convergence, for finding the optimal sparse portfolio. The effectiveness of the suggested portfolio selection model is validated by numerical results derived from the constituents of the Standard and Poor’s 500. These results demonstrate superior performance over our selected benchmark models across various evaluation measurements.

Modern Portfolio Theory (MPT) is based upon the classical Markowitz model which uses variance as a risk measure. A generalization of this approach leads to mean‐risk models, in which a return distribution is characterized by the expected value of return (desired to be large) and a “risk” value (desired to be kept small). Portfolio choice is made by solving an optimization problem, in which the portfolio risk is minimized and a desired level of expected return is specified as a constraint. The need to penalize different undesirable aspects of the return distribution led to the proposal of alternative risk measures, notably those penalizing only the downside part (adverse) and not the upside (potential). The downside risk considerations constitute the basis of the Post Modern Portfolio Theory (PMPT). Examples of such risk measures are lower partial moments, Value‐at‐Risk (VaR) and Conditional Value‐at‐Risk (CVaR). We revisit these risk measures and the resulting mean‐risk models. We discuss alternative models for portfolio selection, their choice criteria and the evolution of MPT to PMPT which incorporates: utility maximization and stochastic dominance. Copyright © 2009 Wilmott Magazine Ltd

A portfolio selection model with return as triangular intuitionistic fuzzy number is developed in this study to assess and select portfolios on China Stock Exchange. Although the portfolio selection has been widely investigated, and most studies have regarded return and risk as the main decisive criteria, there are many uncertainties in the financial market, such as social, political and human psychological factors, which makes it difficult for us to describe the risks in line with empirical evidence. To fill this gap, first, a literature review was conducted to clarify the current situation and shortcomings of portfolio selection. Second, the triangular intuitionistic fuzzy number was used to fuzzify the coefficients of the objective function and the constraints in the portfolio model. Third, the triangular intuitionistic fuzzy number model was transformed into a linear programming model by using the selected exact ranking function, and the model was solved by MATLAB. Finally, the historical monthly returns of 10 stocks in China’s Stock Exchange from December 2017 to November 2020, lasting 36 months, were selected to demonstrate the model. The results indicate that the portfolio selection model with triangular intuitionistic fuzzy number can better meet the uncertainty of the real securities market, and more reasonable investment decision guidance can be provided for investors. Besides, the limitations of this study are pointed out, and implications and directions for future research are discussed.

In this article, we conduct a statistical analysis of the autocorrelation functions (ACF) of multi-scale logarithmic returns computed over maximal monotonic uninterrupted trends (runs) in financial indices’ daily data. We analyze the Dow Jones Industrial Average (DJIA) and the Mexican IPC (Índice de Precios y Cotizaciones) over a period from 30 October 1978 to 19 May 2025. We examine how deterministic alternation of signs shapes the ACF of multi-scale returns, and we evaluate covariance stationarity via formal tests (e.g., Augmented Dickey–Fuller and Phillips–Perron). We conclude that, despite the persistent long-memory oscillations in the ACF, multi-scale return series pass the stationarity tests, an outcome with interesting implications for econometric modeling of financial time series.

The log‐returns of most financial assets exhibit bursts of outlying observations that are characteristic of heavy tails. As such, models formulated for financial time series must capture heavy‐tailedness. Regular variation is a natural and often used concept to describe and model heavy‐tailed phenomena. This article first provides some evidence for heavy tails in log‐returns and then describes the basics of regular variation. It turns out that the two most commonly used models for financial time series, the GARCH and heavy‐tailed stochastic volatility (SV) processes, have finite‐dimensional distributions that are regularly varying. Regular variation is also a powerful tool for deriving large sample properties of important statistics of financial time series, from maxima to sample autocorrelations of functions of the process. These techniques show that although GARCH and SV processes share many common features, there can be a substantial difference in the large sample properties of the maxima and sample autocorrelations.

An Automatic Procedure for the Estimation of the Tail Index

Statistical tests of stochastic process models used in the financial theory of insurance companies

Apparent Multifractality in Financial Time Series

This study adds evidence from the four emerging markets of Central Europe relevant to the econometric modelling of financial time series by modelling volatility in these markets. The sample has all the previously documented characteristics of the unconditional distribution of stock returns normally used to justify the use of the GARCH class of models of conditional volatility. Both univariate and multivariate models are considered. Strong GARCH effects are apparent in all series examined. The estimates of asymmetric models of conditional volatility show rather weak evidence of asymmetries in the markets. The results of the multivariate specifications of volatility have implication for understanding the pattern of information flow between the markets. The constant correlation specification indicates significant conditional correlations between two pairs of countries: Hungary and Poland, and Hungary and Czech Republic. The BEKK model of multivariate volatility shows evidence of return volatility spillovers from Hungary to Poland, but no volatility spillover effects are found in the opposite direction.

Modeling Time Varying Volatility and Asymmetry of Karachi Stock Exchange (KSE)