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Mathematical Financial Econometrics: Non-normal Distributions and Risk Forecasting

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A special issue of Mathematics (ISSN 2227-7390). This special issue belongs to the section "Financial Mathematics".

Deadline for manuscript submissions: closed (30 September 2023) | Viewed by 26943

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Special Issue Editors

Prof. Dr. Javier Perote

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Department of Economics and Economic History and IME, Faculty of Economics and Business, University of Salamanca, Campus Miguel de Unamuno (Edif. F.E.S.), 37007 Salamanca, Spain
Interests: economics; econometrics; financial markets; risk management; mathematical social sciences; experimental science; computer science; energy business; environmental science

Prof. Dr. Andrés Mora-Valencia

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School of Management, Universidad de los Andes, Calle 21 No. 1‐20, Bogotá, Colombia
Interests: risk quantification; quantitative finance; risk management; tail risk; operational risk

Dr. Trino-Manuel Ñíguez

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School of Organizations, Economy and Society, Westminster Business School, University of Westminster, 35 Marylebone Road, London NW1 5LS, UK
Interests: economics; econometrics; quantitative finance; time series; risk forecasting

Special Issue Information

Dear Colleagues,

In the last several decades, financial markets have become more complex and volatile with the inception of new instruments for hedging portfolios, trading on future markets, tracking commodities, or even creating synthetic assets based on new technologies. All of this has highlighted the need for providing accurate econometric models capable of measuring and forecasting financial risks that help to stabilize financial markets and avoid crises emerging from the financial system.

For this purpose, financial econometrics has undergone a huge development with a wide variety of univariate and multivariate volatility models, statistical techniques and tools, and risk measures. The main focus has been placed on capturing the non-normality of financial return series, which exhibit leptokurtosis, skewness, volatility clustering, or long-memory, among other well-known features.

This Special Issue is devoted to recent developments in non-normal distributions and risk measures in financial econometrics, either from a theoretical or an empirical perspective. All contributions related to these topics are welcome, particularly those related to risk forecasting, new distributions, volatility models, estimation and testing methodologies, non-parametric and semi-nonparametric modeling, and their applications in financial, commodity, or energy markets.

Prof. Dr. Javier Perote
Prof. Dr. Andrés Mora-Valencia
Dr. Trino-Manuel Ñíguez
Guest Editors

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Keywords

non-normal distributions
risk forecasting
volatility modeling
nonparametric and semi-nonparametric models
tail dependence
high frequency
portfolio management
financial engineering
derivatives
statistical science
machine learning
financial and commodity markets
cryptocurrencies

Published Papers (14 papers)

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Research

25 pages, 593 KiB

Open AccessArticle

Expectation and Optimal Allocations in Existential Contests of Finite, Heavy-Tail-Distributed Outcomes

by Ralph Vince

Mathematics 2024, 12(1), 11; https://0-doi-org.brum.beds.ac.uk/10.3390/math12010011 - 20 Dec 2023

Viewed by 733

Abstract

Financial time series and other human-driven, non-natural processes are known to exhibit fat-tailed outcome distributions. That is, such processes demonstrate a greater tendency for extreme outcomes than the normal distribution or other natural distributional processes would predict. We examine the mathematical expectation, or simply “expectation”, traditionally the probability-weighted outcome, regarded since the seventeenth century as the mathematical definition of “expectation”. However, when considering the “expectation” of an individual confronted with a finite sequence of outcomes, particularly existential outcomes (e.g., a trader with a limited time to perform or lose his position in a trading operation), we find this individual “expects” the median terminal outcome over those finite trials, with the classical seventeenth-century definition being the asymptotic limit as trials increase. Since such finite-sequence “expectations” often differ in values from the classic one, so do the optimal allocations (e.g., growth-optimal). We examine these for fat-tailed distributions. The focus is on implementation, and the techniques described can be applied to all distributional forms. We make no assertion that the empirical data for any financial time series comports to the generalized hyperbolic distribution (GHD), which we will use as a proxy of any heavy-tailed distribution herein. Rather, we have selected the GHD to highlight the process for determining expectation and other important time-dependent metrics in existential contests, using the GHD as a generic proxy for the specific distributional form an implementor of the presented technique might ascribe to the empirical data. Full article

(This article belongs to the Special Issue Mathematical Financial Econometrics: Non-normal Distributions and Risk Forecasting)

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26 pages, 2438 KiB

Open AccessArticle

Financial Volatility Modeling with the GARCH-MIDAS-LSTM Approach: The Effects of Economic Expectations, Geopolitical Risks and Industrial Production during COVID-19

by Özgür Ömer Ersin and Melike Bildirici

Mathematics 2023, 11(8), 1785; https://0-doi-org.brum.beds.ac.uk/10.3390/math11081785 - 09 Apr 2023

Cited by 5 | Viewed by 3652

Abstract

Forecasting stock markets is an important challenge due to leptokurtic distributions with heavy tails due to uncertainties in markets, economies, and political fluctuations. To forecast the direction of stock markets, the inclusion of leading indicators to volatility models is highly important; however, such series are generally at different frequencies. The paper proposes the GARCH-MIDAS-LSTM model, a hybrid method that benefits from LSTM deep neural networks for forecast accuracy, and the GARCH-MIDAS model for the integration of effects of low-frequency variables in high-frequency stock market volatility modeling. The models are being tested for a forecast sample including the COVID-19 shut-down after the first official case period and the economic reopening period in in Borsa Istanbul stock market in Türkiye. For this sample, significant uncertainty existed regarding future economic expectations, and the period provided an interesting laboratory to test the forecast effectiveness of the proposed LSTM augmented model in addition to GARCH-MIDAS models, which included geopolitical risk, future economic expectations, trends, and cycle industrial production indices as low-frequency variables. The evidence suggests that stock market volatility is most effectively modeled with geopolitical risk, followed by industrial production, and a relatively lower performance is achieved by future economic expectations. These findings imply that increases in geopolitical risk enhance stock market volatility further, and that industrial production and future economic expectations work in the opposite direction. Most importantly, the forecast results suggest suitability of both the GARCH-MIDAS and GARCH-MIDAS-LSTM models, and with good forecasting capabilities. However, a comparison shows significant root mean squared error reduction with the novel GARCH-MIDAS-LSTM model over GARCH-MIDAS models. Percentage decline in root mean squared errors for forecasts are between 39% to 95% in LSTM augmented models depending on the type of economic indicator used. The proposed approach offers a key tool for investors and policymakers. Full article

(This article belongs to the Special Issue Mathematical Financial Econometrics: Non-normal Distributions and Risk Forecasting)

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15 pages, 2800 KiB

Open AccessArticle

Estimating Financial Fraud through Transaction-Level Features and Machine Learning

by Ayed Alwadain, Rao Faizan Ali and Amgad Muneer

Mathematics 2023, 11(5), 1184; https://0-doi-org.brum.beds.ac.uk/10.3390/math11051184 - 28 Feb 2023

Cited by 3 | Viewed by 2268

Abstract

In today’s world, financial institutions (FIs) play a pivotal role in any country’s economic growth and are vital for intermediation between the providers of investable funds, such as depositors, investors and users. FIs focus on developing effective policies for financial fraud risk mitigation however, timely prediction of financial fraud risk helps overcome it effectively and efficiently. Thus, herein, we propose a novel approach for predicting financial fraud using machine learning. We have used transaction-level features of 6,362,620 transactions from a synthetic dataset and have fed them to various machine-learning classifiers. The correlation of different features is also analysed. Furthermore, around 5000 more data samples were generated using a Conditional Generative Adversarial Network for Tabular Data (CTGAN). The evaluation of the proposed predictor showed higher accuracies which outperformed the previously existing machine-learning-based approaches. Among all 27 classifiers, XGBoost outperformed all other classifiers in terms of accuracy score with 0.999 accuracies, however, when evaluated through exhaustive repeated 10-fold cross-validation, the XGBoost still gave an average accuracy score of 0.998. The findings are particularly relevant to financial institutions and are important for regulators and policymakers who aim to develop new and effective policies for risk mitigation against financial fraud. Full article

(This article belongs to the Special Issue Mathematical Financial Econometrics: Non-normal Distributions and Risk Forecasting)

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15 pages, 5381 KiB

Open AccessArticle

Oil Price Shocks to Foreign Assets and Liabilities in Saudi Arabia under Pegged Exchange Rate

by Nahla Samargandi and Kazi Sohag

Mathematics 2022, 10(24), 4752; https://0-doi-org.brum.beds.ac.uk/10.3390/math10244752 - 14 Dec 2022

Cited by 3 | Viewed by 1178

Abstract

The Saudi economy ought to maintain a significant amount of foreign exchange reserves due to the pegged exchange rate regime. As a hydrocarbon economy, we measure the dynamic response of external assets and liabilities of banks to the international oil price in Saudi Arabia. In the presence of extreme observations, we apply sophisticated frameworks, including cross-quantilograms, quantile-on-quantile and TVP-VAR approaches, to analyze weekly time-series data from 1993 to 2021. Our results from the cross-quantilogram and quantile-on-quantile frameworks demonstrate that foreign assets and liabilities responded asymmetrically to the volatilities of international oil prices under the bullish and bearish states of the market over different memories. The TVP-VAR results indicate that, during the COVID-19 pandemic, the Saudi economy encountered negative net foreign assets, which occurred mainly as a significant plague of international oil prices. Our findings are robust under different estimators. Full article

(This article belongs to the Special Issue Mathematical Financial Econometrics: Non-normal Distributions and Risk Forecasting)

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17 pages, 1109 KiB

Open AccessFeature PaperArticle

Estimating Value-at-Risk and Expected Shortfall: Do Polynomial Expansions Outperform Parametric Densities?

by Brenda Castillo-Brais, Ángel León and Juan Mora

Mathematics 2022, 10(22), 4329; https://0-doi-org.brum.beds.ac.uk/10.3390/math10224329 - 18 Nov 2022

Cited by 2 | Viewed by 1566

Abstract

We assess Value-at-Risk (VaR) and Expected Shortfall (ES) estimates assuming different models for the standardized returns: distributions based on polynomial expansions such as Cornish-Fisher and Gram-Charlier, and well-known parametric densities such as normal, skewed-t and Johnson. This paper aims to analyze whether models based on polynomial expansions outperform the parametric ones. We carry out the model performance comparison in two stages: first, with a backtesting analysis of VaR and ES; and second, using loss functions. Our backtesting results show that all distributions, except for normal ones, perform quite well in VaR and ES estimations. Regarding the loss function analysis, we conclude that polynomial expansions (specifically, the Cornish-Fisher one) usually outperform parametric densities in VaR estimation, but the latter (specifically, the Johnson density) slightly outperform the former in ES estimation; however, the gains of using one approach or the other are modest. Full article

(This article belongs to the Special Issue Mathematical Financial Econometrics: Non-normal Distributions and Risk Forecasting)

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20 pages, 3208 KiB

Open AccessArticle

Portfolio Optimization Considering Behavioral Stocks with Return Scenario Generation

by Michael N. Young, TJ Troy N. Chuahay, Yen-Hsien Lee, John Francis T. Diaz, Yogi Tri Prasetyo, Satria Fadil Persada and Reny Nadilfatin

Mathematics 2022, 10(22), 4269; https://0-doi-org.brum.beds.ac.uk/10.3390/math10224269 - 15 Nov 2022

Cited by 1 | Viewed by 2106

Abstract

This study extends the application of behavioral portfolio optimization by estimating the return of behavioral stocks (B-stocks). With the cause-and-effect relationships of the respective irrational behaviors on the stock price movements and the unique information provided by B-stocks in terms of knowing with a calculated probability when (time duration) a specific effect (e.g., positive cumulative abnormal return) after a certain trigger point (cause of the irrational behavior) is spotted, regression analysis is applied on the information in the duration to have more accurate return estimates. To fit in the framework of behavioral portfolio optimization, the scenarios used for the optimization are generated utilizing regression analysis, based on which the safety-first scenario-based mixed-integer program is applied to obtain the optimal portfolios. This study also proposes two new types of B-stocks with corresponding operational definitions for herding and ostrich-effect, along with the previously identified over-reaction, under-reaction, and disposition-effect B-stocks. Back-test results show that the portfolios are profitable and can significantly outperform a benchmark and the market. Full article

(This article belongs to the Special Issue Mathematical Financial Econometrics: Non-normal Distributions and Risk Forecasting)

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23 pages, 2987 KiB

Open AccessArticle

A Study on Early Warnings of Financial Crisis of Chinese Listed Companies Based on DEA–SVM Model

by Zhishuo Zhang, Yao Xiao, Zitian Fu, Kaiyang Zhong and Huayong Niu

Mathematics 2022, 10(12), 2142; https://0-doi-org.brum.beds.ac.uk/10.3390/math10122142 - 20 Jun 2022

Cited by 12 | Viewed by 2003

Abstract

In the era of big data, investor sentiment will have an impact on personal decision making and asset pricing in the securities market. This paper uses the Easteconomy stock forum and Sina stock forum as the carrier of investor sentiment to measure the positive sentiment index based on stockholders’ comments and to construct an evaluation index system for the public opinion dimension. In addition, the evaluation index system is constructed from four dimensions, which include operation, innovation, finance and financing, to evaluate the overall condition of listed companies from multiple perspectives. In this paper, the SBM model in the data envelopment analysis method is used to measure the efficiency values of each dimension of the multidimensional efficiency evaluation index system, and the efficiency values of each dimension are the multidimensional efficiency indicators. Subsequently, two sets of input feature indicators of the SVM model were established: one set contains traditional financial indicators and multidimensional efficiency indicators, and another set has only traditional financial indicators. The early warning accuracy of the two sets of input feature indicators was empirically analyzed based on the support vector machine early warning model. The results show that the early warning model incorporating multidimensional efficiency indicators has improved the accuracy compared with the early warning model based on traditional financial indicators. Then, the model was optimized by the particle swarm intelligent optimization algorithm, and the robustness of the results was tested. Moreover, six mainstream machine learning methods, including Logistic Regression, GBDT, CatBoost, AdaBoost, Random Forest and Bagging, were used to compare with the early warning effect of the DEA–SVM model, and the empirical results show that DEA–SVM has high early warning accuracy, which proves the superiority of the proposed model. The findings of this study have a positive effect on further preventing and controlling the financial crisis risk of Chinese-listed companies and promoting as well as facilitating the healthy growth of Chinese-listed companies. Full article

(This article belongs to the Special Issue Mathematical Financial Econometrics: Non-normal Distributions and Risk Forecasting)

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32 pages, 12618 KiB

Open AccessArticle

Estimating Structural Shocks with the GVAR-DSGE Model: Pre- and Post-Pandemic

by Chunyeung Kwok

Mathematics 2022, 10(10), 1773; https://0-doi-org.brum.beds.ac.uk/10.3390/math10101773 - 23 May 2022

Cited by 1 | Viewed by 1818

Abstract

This paper investigates the possibility of using the global VAR (GVAR) model to estimate a simple New Keynesian DSGE-type multi-country model. The long-run forecasts from an estimated GVAR model were used to calculate the steady-states of macro variables as differences. The deviations from the long-run forecasts were taken as the deviation from the steady-states and were used to estimate a simple NK open economy model with an IS curve, Philips curve, Taylor rule, and an exchange rate equation. The shocks to these equations were taken as the demand shock, supply shock, monetary shock, and exchange rate shock, respectively. An alternative model was constructed to compare the results from GVAR long-run forecasts. The alternative model used a Hodrick–Prescott (HP) filter to derive deviations from the steady-states. The impulsive response functions from the shocks were then compared to results from other DSGE models in the literature. Both GVAR and HP estimates produced dissimilar results, although the GVAR managed to capture more from the data, given the explicit co-integration relationships. For the IRFs, both GVAR and HP estimated DSGE models appeared to be as expected before the pandemic; however, if we include the pandemic data, i.e., 2020, the IRFs are very different, due to the nature of the policy actions. In general, DSGE–GVAR models appear to be much more versatile, and are able to capture dynamics that HP filters are not. Full article

(This article belongs to the Special Issue Mathematical Financial Econometrics: Non-normal Distributions and Risk Forecasting)

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23 pages, 1064 KiB

Open AccessArticle

Non-Normal Market Losses and Spatial Dependence Using Uncertainty Indices

by Catalina Bolancé, Carlos Alberto Acuña and Salvador Torra

Mathematics 2022, 10(8), 1317; https://0-doi-org.brum.beds.ac.uk/10.3390/math10081317 - 15 Apr 2022

Cited by 1 | Viewed by 1075

Abstract

We analyse spatial dependence between the risks of stock markets. An alternative definition of neighbour is used and is based on a proposed exogenous criterion obtained with a dynamic Google Trends Uncertainty Index (GTUI) designed specifically for this analysis. We show the impact of systemic risk on spatial dependence related to the most significant financial crises from 2005: the Lehman Brothers bankruptcy, the sub-prime mortgage crisis, the European debt crisis, Brexit and the COVID-19 pandemic, which also affected the financial markets. The risks are measured using the monthly variance or volatility and the monthly Value-at-Risk (VaR) of the filtered losses associated with the analysed indices. Given that the analysed risk measures follow non-normal distributions and the number of neighbours changes over time, we carry out a simulation study to check how these characteristics affect the results of global and local inference using Moran’s I statistic. Lastly, we analyse the global spatial dependence between the risks of 46 stock markets and we study the local spatial dependence for 10 benchmark stock markets worldwide. Full article

(This article belongs to the Special Issue Mathematical Financial Econometrics: Non-normal Distributions and Risk Forecasting)

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18 pages, 1182 KiB

Open AccessArticle

Contributions to Risk Assessment with Edgeworth–Sargan Density Expansions (I): Stability Testing

by Ignacio Mauleón

Mathematics 2022, 10(7), 1074; https://0-doi-org.brum.beds.ac.uk/10.3390/math10071074 - 27 Mar 2022

Viewed by 1128

Abstract

This paper analytically derives a stability test for the probability distribution of a random variable that follows the Edgeworth–Sargan density, also called Gram–Charlier. The distribution of the test is a weighted sum of Chi-squared densities of increasing degrees of freedom, starting with the standard equivalent Chi-squared under the same conditions. The weights turn out to be linear combinations of the parameters of the distribution and the moments of a Gaussian density, and can be computed exactly. This is a convenient result, since then the probability intervals can be easily calculated from existing Chi-squared distribution tables. The test is applied to assess the weekly solar irradiance data stability for a twelve-year period. It shows that the density is acceptably stable overall, except for some eventual and localised dates. It is also shown that the usual probability intervals implemented in stability testing are larger than those of the equivalent Chi-squared distribution under comparable conditions. This implies that the common upper tail interval values for rejecting the null stability hypothesis are larger. Full article

(This article belongs to the Special Issue Mathematical Financial Econometrics: Non-normal Distributions and Risk Forecasting)

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33 pages, 427 KiB

Open AccessArticle

“A Bias Recognized Is a Bias Sterilized”: The Effects of a Bias in Forecast Evaluation

by Nicolas Hardy

Mathematics 2022, 10(2), 171; https://0-doi-org.brum.beds.ac.uk/10.3390/math10020171 - 06 Jan 2022

Cited by 1 | Viewed by 1137

Abstract

Are traditional tests of forecast evaluation well behaved when the competing (nested) model is biased? No, they are not. In this paper, we show analytically and via simulations that, under the null hypothesis of no encompassing, a bias in the nested model may severely distort the size properties of traditional out-of-sample tests in economic forecasting. Not surprisingly, these size distortions depend on the magnitude of the bias and the persistency of the additional predictors. We consider two different cases: (i) There is both in-sample and out-of-sample bias in the nested model. (ii) The bias is present exclusively out-of-sample. To address the former case, we propose a modified encompassing test (MENC-NEW) robust to a bias in the null model. Akin to the ENC-NEW statistic, the asymptotic distribution of our test is a functional of stochastic integrals of quadratic Brownian motions. While this distribution is not pivotal, we can easily estimate the nuisance parameters. To address the second case, we derive the new asymptotic distribution of the ENC-NEW, showing that critical values may differ remarkably. Our Monte Carlo simulations reveal that the MENC-NEW (and the ENC-NEW with adjusted critical values) is reasonably well-sized even when the ENC-NEW (with standard critical values) exhibits rejections rates three times higher than the nominal size. Full article

(This article belongs to the Special Issue Mathematical Financial Econometrics: Non-normal Distributions and Risk Forecasting)

19 pages, 1956 KiB

Open AccessArticle

Cross-Hedging Portfolios in Emerging Stock Markets: Evidence for the LATIBEX Index

by Pablo Urtubia, Alfonso Novales and Andrés Mora-Valencia

Mathematics 2021, 9(21), 2736; https://0-doi-org.brum.beds.ac.uk/10.3390/math9212736 - 28 Oct 2021

Cited by 1 | Viewed by 1860

Abstract

We consider alternative possibilities for hedging spot positions on the FTSE LATIBEX Index, the index of the only international market exclusively for Latin American firms that is denominated by the euro. Since there is not a futures market on the index, it is unclear whether a relatively successful hedge can be found. We explore the plausibility of employing futures on four stock market indices: EUROSTOXX 50, S&P500, BOVESPA, and IPC, and simulate the results that could be obtained by a hedge position based on either unconditional or conditional second order moments estimated from different asymmetric GARCH models. Several criteria for hedging effectiveness suggest that futures contracts on BOVESPA should be preferred, and that a salient reduction in risk can be achieved over the unhedged LATIBEX portfolio. The evidence in favor of a better performance of conditional moments is very clear, without significant differences among the alternative GARCH specifications. Full article

(This article belongs to the Special Issue Mathematical Financial Econometrics: Non-normal Distributions and Risk Forecasting)

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12 pages, 1546 KiB

Open AccessArticle

Risk Transfer in an Electricity Market

by David Esteban Rodriguez, Alfredo Trespalacios and David Galeano

Mathematics 2021, 9(21), 2661; https://0-doi-org.brum.beds.ac.uk/10.3390/math9212661 - 21 Oct 2021

Cited by 2 | Viewed by 2162

Abstract

Energy is traded using different products; long-term contracts or electricity forward contracts can assure the future transaction price. However, due to the difficulties in storing electrical energy for long periods and in large amounts, risks must be incorporated when defining contract prices through a Forward Risk Premia (FRP). This study analyzes the transfer of uncertainty from electricity market variables to the FRP in long-term contracts. We evaluate a type of econometric risk with the construction of Autoregressive Distributed Lag contagion models for the FRP using electricity demand, spot price, power generation via different technologies, and the Oceanic Niño Index. As a case study, we consider the Colombian electricity market. Our results show empirical models where the FRP has a short-term response with the following variables: hydropower generation, coal power generation, electricity demand, and Oceanic Niño Index, even though its transaction is reflected one or two years after the occurrence of the event. Full article

(This article belongs to the Special Issue Mathematical Financial Econometrics: Non-normal Distributions and Risk Forecasting)

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28 pages, 677 KiB

Open AccessArticle

“Go Wild for a While!”: A New Test for Forecast Evaluation in Nested Models

by Pablo Pincheira, Nicolás Hardy and Felipe Muñoz

Mathematics 2021, 9(18), 2254; https://0-doi-org.brum.beds.ac.uk/10.3390/math9182254 - 14 Sep 2021

Cited by 7 | Viewed by 2300

Abstract

In this paper, we present a new asymptotically normal test for out-of-sample evaluation in nested models. Our approach is a simple modification of a traditional encompassing test that is commonly known as Clark and West test (CW). The key point of our strategy is to introduce an independent random variable that prevents the traditional CW test from becoming degenerate under the null hypothesis of equal predictive ability. Using the approach developed by West (1996), we show that in our test, the impact of parameter estimation uncertainty vanishes asymptotically. Using a variety of Monte Carlo simulations in iterated multi-step-ahead forecasts, we evaluated our test and CW in terms of size and power. These simulations reveal that our approach is reasonably well-sized, even at long horizons when CW may present severe size distortions. In terms of power, results were mixed but CW has an edge over our approach. Finally, we illustrate the use of our test with an empirical application in the context of the commodity currencies literature. Full article

(This article belongs to the Special Issue Mathematical Financial Econometrics: Non-normal Distributions and Risk Forecasting)

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