Mathematics

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30 pages, 4023 KiB

Open AccessFeature PaperArticle

Modeling Implied Volatility Surface Using B-Splines with Time-Dependent Coefficients Predicted by Tree-Based Machine Learning Methods

by Zihao Chen, Yuyang Li and Cindy Long Yu

Mathematics 2024, 12(7), 1100; https://doi.org/10.3390/math12071100 - 06 Apr 2024

Viewed by 357

Abstract

Implied volatility is known to have a string structure (smile curve) for a given time to maturity and can be captured by the B-spline. The parameters characterizing the curves can change over time, which complicates the modeling of the implied volatility surface. Although [...] Read more.

Implied volatility is known to have a string structure (smile curve) for a given time to maturity and can be captured by the B-spline. The parameters characterizing the curves can change over time, which complicates the modeling of the implied volatility surface. Although machine learning models could improve the in-sample fitting, they ignore the structure in common over time and might have poor prediction power. In response to these challenges, we propose a two-step procedure to model the dynamic implied volatility surface (IVS). In the first step, we construct the bivariate tensor-product B-spline (BTPB) basis to approximate cross-sectional structures, under which the surface can be represented by a vector of coefficients. In the second step, we allow for the time-dependent coefficients and model the dynamic coefficients with the tree-based method to provide more flexibility. We show that our approach has better performance than the traditional linear models (parametric models) and the tree-based machine learning methods (nonparametric models). The simulation study confirms that the tensor-product B-spline is able to capture the classical parametric model for IVS given different sample sizes and signal-to-noise ratios. The empirical study shows that our two-step approach outperforms the traditional parametric benchmark, nonparametric benchmark, and parametric benchmark with time-varying coefficients in predicting IVS for the S&P 500 index options in the US market. Full article

(This article belongs to the Special Issue Recent Advances of Computational Statistics in Industry and Business III)

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11 pages, 1433 KiB

Open AccessArticle

Decision-Making Model of Performance Evaluation Matrix Based on Upper Confidence Limits

by Teng-Chiao Lin, Hsing-Hui Chen, Kuen-Suan Chen, Yen-Po Chen and Shao-Hsun Chang

Mathematics 2023, 11(16), 3499; https://doi.org/10.3390/math11163499 - 13 Aug 2023

Cited by 1 | Viewed by 977

Abstract

A performance evaluation matrix (PEM) is an evaluation tool for assessing customer satisfaction and the importance of service items across various services. In addition, inferences based on point estimates of sample data can increase the risk of misjudgment due to sampling errors. Thus, [...] Read more.

A performance evaluation matrix (PEM) is an evaluation tool for assessing customer satisfaction and the importance of service items across various services. In addition, inferences based on point estimates of sample data can increase the risk of misjudgment due to sampling errors. Thus, this paper creates a decision-making model for a performance evaluation matrix based on upper confidence limits to provide various service operating systems for performance evaluation and decision making. The concept is that through the gap between customer satisfaction and the level of importance of each service item, we are able to identify critical-to-quality (CTQ) service items requiring improvement. Many studies have indicated that customer satisfaction and the importance of service items follow a beta distribution, and based on the two parameters of this distribution, the proposed indices for customer satisfaction and the importance of service items represent standardization. The vertical axis of a PEM represents the importance index; the horizontal axis represents the satisfaction index. Since these two indices have unknown parameters, this paper uses the upper confidence limit of the satisfaction index to find out the CTQ service items and the upper confidence limit of the importance index to determine the order of improvement priority for each service item. This paper then establishes a decision-making model for a PEM based on the above-mentioned decision-making rules. Since all decision-making rules proposed in this paper are established through upper confidence limits, the risk of misjudgment caused by sampling errors can be reduced. Finally, this article uses a practical example to illustrate how to use a PEM to find CTQ service items and determine the order of improvement priority for these service items that need to be improved. Full article

(This article belongs to the Special Issue Recent Advances of Computational Statistics in Industry and Business III)

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16 pages, 407 KiB

Open AccessArticle

A Note on Weibull Parameter Estimation with Interval Censoring Using the EM Algorithm

by Chanseok Park

Mathematics 2023, 11(14), 3156; https://doi.org/10.3390/math11143156 - 18 Jul 2023

Viewed by 1113

Abstract

In many engineering applications, it is often the case that the observations are only available in interval form. In this note, by using the expectation-maximization (EM) algorithm, the parameter estimation of the Weibull distribution with interval-censored data is considered. The estimates obtained using [...] Read more.

In many engineering applications, it is often the case that the observations are only available in interval form. In this note, by using the expectation-maximization (EM) algorithm, the parameter estimation of the Weibull distribution with interval-censored data is considered. The estimates obtained using the EM algorithm are compared with those obtained using the conventional Newton-type methods, including the Davidon–Fletcher–Powell (DFP) and Berndt–Hall–Hall–Hausman (BHHH) methods. The results indicate that the estimates obtained using the proposed EM method demonstrate superior convergence properties compared to the conventional DFP and BHHH methods. Finally, a numerical study that illustrates the advantages of the proposed method is provided. Full article

(This article belongs to the Special Issue Recent Advances of Computational Statistics in Industry and Business III)

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15 pages, 3571 KiB

Open AccessArticle

Negentropy as a Measure to Evaluate the Resilience in Industrial Plants

by Orlando Durán, Gustavo Sáez and Paulo Durán

Mathematics 2023, 11(12), 2707; https://doi.org/10.3390/math11122707 - 15 Jun 2023

Cited by 1 | Viewed by 849

Abstract

Resilience is an essential quality of systems. This characteristic is based on the ability of a system to cope with disruptive events. To prevent decreases in system functionality and performance and to respond promptly to unexpected situations or shocks, systems must possess this [...] Read more.

Resilience is an essential quality of systems. This characteristic is based on the ability of a system to cope with disruptive events. To prevent decreases in system functionality and performance and to respond promptly to unexpected situations or shocks, systems must possess this capacity. One challenge lies in identifying and measuring resilience. Recently, various metrics have been proposed in the literature to represent the resilience of systems. Despite this, there is still no global resilience measure that can be used in any type of industrial system. This work investigated a series of moment statistics and explored the field of entropy in the search for a general resilience indicator. A set of 27 hypothetical cases were proposed to calculate the indices under evaluation. Then, a series of comparisons were made between these indices and two resilience indicators found in the literature. The main results of this work lead to the overall conclusion that it is possible to use some of these indicators as potential resilience indicators for engineering systems and production lines. Specifically, negentropy appears to be a good option for this purpose. Full article

(This article belongs to the Special Issue Recent Advances of Computational Statistics in Industry and Business III)

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28 pages, 752 KiB

Open AccessArticle

Yield Curve Models with Regime Changes: An Analysis for the Brazilian Interest Rate Market

by Renata Tavanielli and Márcio Laurini

Mathematics 2023, 11(11), 2549; https://doi.org/10.3390/math11112549 - 01 Jun 2023

Viewed by 1560

Abstract

This study examines the effectiveness of various specifications of the dynamic Nelson–Siegel term structure model in analyzing the term structure of Brazilian interbank deposits. A key contribution of our research is the incorporation of regime changes and other time-varying parameters in the model, [...] Read more.

This study examines the effectiveness of various specifications of the dynamic Nelson–Siegel term structure model in analyzing the term structure of Brazilian interbank deposits. A key contribution of our research is the incorporation of regime changes and other time-varying parameters in the model, both when relying solely on observed yields and when incorporating macroeconomic variables. By allowing parameters in the latent factors to adapt to changes in persistence patterns and the overall shape of the yield curve, these mechanisms enhance the model’s flexibility. To evaluate the performance of the models, we conducted assessments based on their in-sample fit and out-of-sample forecast accuracy. Our estimation approach involved Bayesian procedures utilizing Markov Chain Monte Carlo techniques. The results highlight that models incorporating macro factors and greater flexibility demonstrated superior in-sample fit compared to other models. However, when it came to out-of-sample forecasts, the performance of the models was influenced by the forecast horizon and maturity. Models incorporating regime switching exhibited better performance overall. Notably, for long maturities with a one-month ahead forecast horizon, the model incorporating regime changes in both the latent and macro factors emerged as the top performer. On the other hand, for a twelve-month horizon, the model incorporating regime switching solely in the macro factors demonstrated superior performance across most maturities. These findings have significant implications for the development of trading and hedging strategies in interest rate derivative instruments, particularly in emerging markets that are more prone to regime changes and structural breaks. Full article

(This article belongs to the Special Issue Recent Advances of Computational Statistics in Industry and Business III)

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28 pages, 3290 KiB

Open AccessArticle

The Application of Symbolic Regression on Identifying Implied Volatility Surface

by Jiayi Luo and Cindy Long Yu

Mathematics 2023, 11(9), 2108; https://doi.org/10.3390/math11092108 - 28 Apr 2023

Cited by 1 | Viewed by 1483

Abstract

One important parameter in the Black–Scholes option pricing model is the implied volatility. Implied volatility surface (IVS) is an important concept in finance that describes the variation of implied volatility across option strike price and time to maturity. Over the last few decades, [...] Read more.

One important parameter in the Black–Scholes option pricing model is the implied volatility. Implied volatility surface (IVS) is an important concept in finance that describes the variation of implied volatility across option strike price and time to maturity. Over the last few decades, economists and financialists have long tried to exploit the predictability in the IVS using various parametric models, which require deep understanding of financial practices in the area. In this paper, we explore how a data-driven machine learning method, symbolic regression, performs in identifying the implied volatility surface even without deep financial knowledge. Two different approaches of symbolic regression are explored through a simulation study and an empirical study using a large panel of option data in the United States options market. Full article

(This article belongs to the Special Issue Recent Advances of Computational Statistics in Industry and Business III)

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17 pages, 321 KiB

Open AccessArticle

Stress–Strength Inference on the Multicomponent Model Based on Generalized Exponential Distributions under Type-I Hybrid Censoring

by Tzong-Ru Tsai, Yuhlong Lio, Jyun-You Chiang and Ya-Wen Chang

Mathematics 2023, 11(5), 1249; https://doi.org/10.3390/math11051249 - 04 Mar 2023

Cited by 2 | Viewed by 1151

Abstract

The stress–strength analysis is investigated for a multicomponent system, where all strength variables of components follow a generalized exponential distribution and are subject to the generalized exponential distributed stress. The estimation methods of the maximum likelihood and Bayesian are utilized to infer the [...] Read more.

The stress–strength analysis is investigated for a multicomponent system, where all strength variables of components follow a generalized exponential distribution and are subject to the generalized exponential distributed stress. The estimation methods of the maximum likelihood and Bayesian are utilized to infer the system reliability. For the Bayesian estimation method, informative and non-informative priors combined with three loss functions are considered. Because the computational difficulty on working posteriors, the Markov chain Monte Carlo method is adopted to obtain the approximation of the reliability estimator posterior. In addition, the bootstrap method and highest probability density interval are used to obtain the reliability confidence intervals. The simulation study shows that the Bayes estimator with informative prior is superior to other competitors. Finally, two real examples are given to illustrate the proposed estimation methods. Full article

(This article belongs to the Special Issue Recent Advances of Computational Statistics in Industry and Business III)

18 pages, 956 KiB

Open AccessArticle

How Tourists’ Perceived Risk Affects Behavioral Intention through Crisis Communication in the Post-COVID-19 Era

by Shui-Lien Chen, Hsiang-Ting Hsu and Richard Chinomona

Mathematics 2023, 11(4), 860; https://doi.org/10.3390/math11040860 - 08 Feb 2023

Cited by 4 | Viewed by 1784

Abstract

In the post-COVID-19 era, with tourism activity beginning to revitalize, the behavioral intention of tourists has emerged as the focus of much research interest. While previous studies have suggested that tourists’ perceived risk affects behavioral intention, it has not been found that perceived [...] Read more.

In the post-COVID-19 era, with tourism activity beginning to revitalize, the behavioral intention of tourists has emerged as the focus of much research interest. While previous studies have suggested that tourists’ perceived risk affects behavioral intention, it has not been found that perceived risk is influenced by other factors that affect behavioral intention in the post-COVID-19 era. This study constructs a research model to understand how tourists’ perceived risk influences emotional attachment to destinations and tourists’ behavioral intention through crisis communication and NPI. Through face-to-face interviews, this study conducted a survey and collected data from 1047 tourists who visited Dadaocheng’s renowned Chinese herbal street in Taiwan and examined the causal relationships through structural equation modeling. The results indicated that an increase in perceived risk had a positive effect on crisis communication and NPI and affected tourists’ behavioral intentions through emotional attachment to the destination. This study provides an opportunity to establish an essential contribution to post-disaster crisis management, which may serve as a marketing reference for tourism operators in the post-COVID-19 era, as well as to address future pandemic challenges. Full article

(This article belongs to the Special Issue Recent Advances of Computational Statistics in Industry and Business III)

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25 pages, 1231 KiB

Open AccessArticle

Modeling Income Data via New Parametric Quantile Regressions: Formulation, Computational Statistics, and Application

by Helton Saulo, Roberto Vila, Giovanna V. Borges, Marcelo Bourguignon, Víctor Leiva and Carolina Marchant

Mathematics 2023, 11(2), 448; https://doi.org/10.3390/math11020448 - 14 Jan 2023

Cited by 3 | Viewed by 1564

Abstract

Income modeling is crucial in determining workers’ earnings and is an important research topic in labor economics. Traditional regressions based on normal distributions are statistical models widely applied. However, income data have an asymmetric behavior and are best modeled by non-normal distributions. The [...] Read more.

Income modeling is crucial in determining workers’ earnings and is an important research topic in labor economics. Traditional regressions based on normal distributions are statistical models widely applied. However, income data have an asymmetric behavior and are best modeled by non-normal distributions. The objective of this work is to propose parametric quantile regressions based on two asymmetric income distributions: Dagum and Singh–Maddala. The proposed quantile regression models are based on reparameterizations of the original distributions by inserting a quantile parameter. We present the reparameterizations, properties of the distributions, and the quantile regression models with their inferential aspects. We proceed with Monte Carlo simulation studies, considering the performance evaluation of the maximum likelihood estimation and an analysis of the empirical distribution of two types of residuals. The Monte Carlo results show that both models meet the expected outcomes. We apply the proposed quantile regression models to a household income data set provided by the National Institute of Statistics of Chile. We show that both proposed models have good performance in model fitting. Thus, we conclude that the obtained results favor the Singh–Maddala and Dagum quantile regression models for positive asymmetrically distributed data related to incomes. The economic implications of our investigation are discussed in the final section. Hence, our proposal can be a valuable addition to the tool-kit of applied statisticians and econometricians. Full article

(This article belongs to the Special Issue Recent Advances of Computational Statistics in Industry and Business III)

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17 pages, 349 KiB

Open AccessArticle

A New Process Performance Index for the Weibull Distribution with a Type-I Hybrid Censoring Scheme

by Tzong-Ru Tsai, Yuhlong Lio, Jyun-You Chiang and Yi-Jia Huang

Mathematics 2022, 10(21), 4090; https://doi.org/10.3390/math10214090 - 02 Nov 2022

Cited by 1 | Viewed by 1164

Abstract

A new life performance index is proposed for evaluating the quality of lifetime products. The maximum likelihood estimation method and the Bayesian approaches using informative and non-informative prior distributions are utilized to infer the parameters of the Weibull distribution and the proposed new [...] Read more.

A new life performance index is proposed for evaluating the quality of lifetime products. The maximum likelihood estimation method and the Bayesian approaches using informative and non-informative prior distributions are utilized to infer the parameters of the Weibull distribution and the proposed new life performance index under a Type-I hybrid censoring scheme. Monte Carlo simulation results show that two Bayesian approaches outperform the maximum likelihood estimation method in terms of the measures of relative bias, relative mean square error, and coverage probability for the point and confidence interval estimators, respectively. The Bayesian approach using a non-informative prior distribution is recommended if the knowledge of setting up the hyper-parameters in the informative prior distribution is not available. Two real data sets are provided for illustration. Full article

(This article belongs to the Special Issue Recent Advances of Computational Statistics in Industry and Business III)

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13 pages, 1786 KiB

Open AccessArticle

Estimation of the Six Sigma Quality Index

by Chun-Chieh Tseng, Kuo-Ching Chiou and Kuen-Suan Chen

Mathematics 2022, 10(19), 3458; https://doi.org/10.3390/math10193458 - 22 Sep 2022

Cited by 1 | Viewed by 1687

Abstract

The measurement of the process capability is a key part of quantitative quality control, and process capability indices are statistical measures of the process capability. Six Sigma level represents the maximum achievable process capability, and many enterprises have implemented Six Sigma improvement strategies. [...] Read more.

The measurement of the process capability is a key part of quantitative quality control, and process capability indices are statistical measures of the process capability. Six Sigma level represents the maximum achievable process capability, and many enterprises have implemented Six Sigma improvement strategies. In recent years, many studies have investigated Six Sigma quality indices, including

Q_{p k}

. However,

Q_{p k}

contains two unknown parameters, namely

δ

and

γ

, which are difficult to use in process control. Therefore, whether a process quality reaches the k sigma level must be statistically inferred. Moreover, the statistical method of sampling distribution is challenging for the upper confidence limits of

Q_{p k}

. We address these two difficulties in the present study and propose a methodology to solve them. Boole’s inequality, Demorgan’s theorem, and linear programming were integrated to derive the confidence intervals of

Q_{p k}

, and then the upper confidence limits were used to perform hypothesis testing. This study involved a case study of the semiconductor assembly process in order to verify the feasibility of the proposed method. Full article

(This article belongs to the Special Issue Recent Advances of Computational Statistics in Industry and Business III)

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