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Extreme Value Methods with Applications to Finance
Extreme value theory (EVT) provides tools for assessing risk of highly unusual developments, such as financial market crashes. This book presents a synthesis of recent research, with emphasis on dependent observations. It concentrates on modern topics, such as compound Poisson approximation, processes of exceedances, and nonparametric estimation methods, which have not been focused on in other books on extremes. Along with examples from finance and insurance that illustrate the methods, the book includes over 200 exercises, making it useful as a reference book, self-study tool, or comprehensive course text.
eBook
Future projections of Indian summer monsoon rainfall extremes over India with statistical downscaling and its consistency with observed characteristics
Indian summer monsoon rainfall extremes and their changing characteristics under global warming have remained a potential area of research and a topic of scientific debate over the last decade. This partially attributes to multiple definitions of extremes reported in the past studies and poor understanding of the changing processes associated with extremes. The later one results into poor simulation of extremes by coarse resolution General Circulation Models under increased greenhouse gas emission which further deteriorates due to inadequate representation of monsoon processes in the models. Here we use transfer function based statistical downscaling model with non-parametric kernel regression for the projection of extremes and find such conventional regional modeling fails to simulate rainfall extremes over India. In this conjuncture, we modify the downscaling algorithm by applying a robust regression to the gridded extreme rainfall events. We observe, inclusion of robust regression to the downscaling algorithm improves the historical simulation of rainfall extremes at a 0.25° spatial resolution, as evaluated based on classical extreme value theory methods, viz., block maxima and peak over threshold. The future projections of extremes during 2081–2100, obtained with the developed algorithm show no change to slight increase in the spatial mean of extremes with dominance of spatial heterogeneity. These changing characteristics in future are consistent with the observed recent changes in extremes over India. The proposed methodology will be useful for assessing the impacts of climate change on extremes; specifically while spatially mapping the risk to rainfall extremes over India.
Journal Article
Extreme events in finance : a handbook of extreme value theory and its applications
\"Extreme Events in Finance: A Handbook of Extreme Value Theory and its Applications features a combination of the theory, methods, and applications of extreme value theory (EVT) in finance as well as a practical understanding of market behavior including both ordinary and extraordinary conditions.\"
eBook
Analysis of Carbon Dioxide Value with Extreme Value Theory Using Generalized Extreme Value Distribution
Abstract-This paper applies the generalized extreme value (GEV) distribution using maximum likelihood estimates to analyze extreme carbon dioxide data collected by the Provincial Energy Office of Phitsanulok from 2010 to 2023. The study aims to model return levels for carbon dioxide emissions for the periods of 5, 25, 50, and 100 years, utilizing data from various fuels-Gasohol E85, Gasohol E20, Gasohol 91, Gasohol 95, ULG95, and LPG. By fitting the GEV distribution, this research not only categorizes the behavior of emissions data under different subclasses of the GEV distribution but also confirms the suitability of the GEV model for this dataset. The findings indicate a trend of increasing return levels, suggesting rising peaks in carbon dioxide emissions over time. This model provides a valuable tool for forecasting and managing environmental risks associated with high emission levels.
Journal Article
Probabilistic assessment of earthquake hazard in the Andaman–Nicobar–Sumatra region
The Andaman–Nicobar–Sumatra (ANS) region is a very hazardous area on the globe, which has witnessed a megathrust earthquake of Mw 9.2 on 26 December 2004 and several dozen large earthquakes in the past. We estimate earthquake hazard parameters (i.e. seismic a- and b-values, maximum expected earthquake magnitudes, mean return periods and probabilities of earthquakes) in 11 shallow and 4 intermediate to deep depth seismogenic zones of the ANS region using a uniform and comprehensive earthquake data for the duration 1906–2018. The earthquake hazard scenarios for all seismogenic zones are calculated using the Gutenberg–Richter frequency–magnitude relation and the Gumbel’s extreme value theory. The low b-values (< 1.0) for both types of zones in the entire region suggest that the region is very active, under high stress and capable to generate large to great earthquakes. The estimated maximum magnitudes in different time periods using the extreme value theory show that shallow–depth zones 7, 8 and 11 (west to the Sumatra) have capabilities to generate an earthquake of magnitude Mw ≥ 8.0 in the next 50 and 100 years, while all intermediate to deep zones can generate magnitude less than 8.0. The mean return periods of earthquakes of magnitude Mw 7.0 in shallow zones 4–9 and 11 (the Sumatra and Nicobar Islands) exhibit less than 25 years. It is less than 80 years in shallow zones 4–11 for magnitude Mw 7.5, while higher return periods have been observed in the intermediate to deep zones (except for zone 4). The high probabilities (> 0.90) for the earthquake of Mw 7.0 in the next 50 years and 100 years are observed in shallow zones 4–11 (the Sumatra and Nicobar Islands), while only intermediate to deep zone 4 (Sumatra) shows high probabilities. The low return periods (< 25 years) and high probabilities (> 0.90) for the earthquake of Mw 7.0 are observed in shallow zones 5–11 (the Nicobar Islands and Sumatra regions), which suggest high earthquake hazard in these zones. The spatial variations of earthquake hazard parameters from one zone to another suggest a large grade of crustal heterogeneity and seismotectonic complexity present in this area.
Journal Article
Generalized Pareto processes for simulating space-time extreme events: an application to precipitation reanalyses
To better manage the risks of destructive natural disasters, impact models can be fed with simulations of extreme scenarios to study the sensitivity to temporal and spatial variability. We propose a semi-parametric stochastic framework that enables simulations of realistic spatio-temporal extreme fields using a moderate number of observed extreme space-time episodes to generate an unlimited number of extreme scenarios of any magnitude. Our framework draws sound theoretical justification from extreme value theory, building on generalized Pareto limit processes arising as limits for event magnitudes exceeding a high threshold. Specifically, we exploit asymptotic stability properties by decomposing extreme event episodes into a scalar magnitude variable (that is resampled), and an empirical profile process representing space-time variability. For illustration on hourly gridded precipitation data in Mediterranean France, we calculate various risk measures using extreme event simulations for yet unobserved magnitudes, and we highlight contrasted behavior for different definitions of the magnitude variable.
Journal Article
Robust Portfolio Optimization Based on Semi-Parametric ARMA-TGARCH-EVT Model with Mixed Copula Using WCVaR
Portfolio returns generally follow multivariate distribution, whose effectiveness depends not only on the correct estimation of marginal distributions, but also on the accurate capture of the interdependent structure among them. To effectively estimate the marginal distribution and improve the accuracy, we present a hybrid ARMA-TGARCH-EVT model, which considers the leverage effect, thick tail and heteroscedasticity of the financial asset return series. This model utilizes the extreme value theory to process the tail data and uses the kernel regression estimation to process the intermediate data to make the marginal distribution smooth, natural and regular. Furthermore, a novel semi-parametric ARMA-TGARCH-EVT- Copula portfolio model is proposed to achieve the robustness of minimizing worst-case conditional value-at-risk (WCVaR). In the model, a mixed copula set is presented by t-copula and Archimedean copula to cover the wide joint dependence among logarithmic daily returns. To verify the effectiveness and practicality of our proposed model, a static numerical example and a dynamic portfolio based on the historical index data of two stock crash periods are given. The results show that the new model is superior in terms of daily average logarithmic return, cumulative logarithmic return and sharp ratio.
Journal Article
Tolerance Interval for the Mixture Normal Distribution Based on Generalized Extreme Value Theory
For a common type of mixture distribution, namely the mixture normal distribution, existing methods for constructing its tolerance interval are unsatisfactory for cases of small sample size and large content. In this study, we propose a method to construct a tolerance interval for the mixture normal distribution based on the generalized extreme value theory. The proposed method is implemented on simulated as well as real-life datasets and its performance is compared with the existing methods.
Journal Article
Understanding Changing Trends in Extreme Rainfall in Saudi Arabia: Trend Detection and Automated EVT-Based Threshold Estimation
The increasing occurrence of extreme rainfall events often leads to flash floods, infrastructure damage, loss of human life, and significant economic impacts. There is a pressing need for data-driven assessments and the application of robust analytical approaches to better understand these changes. Analyzing ground-level daily rainfall data from 1985 to 2023 from 26 monitoring stations, this study first employs the Mann–Kendall test using robust statistics including minimum, median, various quartiles, and maximum rainfall values for detecting long-term trends across Saudi Arabia. Next, the k-means clustering technique is applied to characterize the annual rainfall cycles across different regions of the country. Finally, the Peaks Over Threshold (POT) approach within Extreme Value Theory (EVT) is employed to identify site-specific thresholds for extreme rainfall using the Generalized Pareto Distribution (GPD). This automated, data-driven method offers a more objective alternative to the commonly used ad hoc percentile-based threshold selection, thereby enhancing the rigour and reproducibility of extreme rainfall analysis. Local specific thresholds were computed ranging from about 16 to 47 mm from Arar and Jazan, respectively. These thresholds were then used to calculate the frequency and intensity of extreme rainfall events. The fitted GPD parameters were further used to estimate return levels (RLs) for different return periods (2-, 5-, 10-, 20-, 50-, and 100-year) into the future. The results underscore considerable spatial variability in extreme rainfall behaviour across Saudi Arabia, with a higher likelihood of intense and infrequent precipitation events in the coming decades.
Journal Article