Garch Models

This paper proposes and compares portfolio selection models under the assumption that the portfolios of returns follow a GARCH type process. We compute the price/return distribution at some future time approximating the GARCH process with a Markov chain. We consider either a GARCH(1,1) model or an asymmetric GARCH type model (E-GARCH, GJR-GARCH) . We present an expost comparison of portfolio selection strategies applied to some assets of the US Market. Since the optimization problems present more local optima, we implement an heuristic algorithm for the global optimum in order to overcome the intrinsic computational complexity of the models.

This paper studies time-varying market efficiency in twenty emerging and five developed equity markets for the period covering from January 2000 to June 2017 by using daily returns from MSCI indices and draws a comparison between emerging and developed market efficiency. I used four methods to test for efficiency in emerging market, GARCH-M (1,1) model (Generalised autoregressive conditional heteroscedasticity in mean) to test degree of volatility over the time, Autocorrelation test to test for correlation between two values, Runs test to test for randomness of values and Long memory by using rolling regression estimates. The results from these tests show that, most of the emerging markets are inefficient in weak form and developed markets are efficient in weak form. Some of the emerging markets like Brazil, India, China, Taiwan, Korea and Turkey show significant improvement in recent years. Comparison between emerging and developed market reveals that, information flows efficiently and symmetrically in developed markets compare to emerging markets. In addition, I observed that subprime crisis had a huge impact on emerging markets and created economic imbalance around the world.

The aim of this paper was to accurately and efficiently forecast from multivariate generalized autoregressive conditional heteroscedastic models. The Rotated Dynamic Conditional Correlation (RDCC) model with the Normal, Student’s-t and Multivariate Exponential Power distributions for errors were used to account for heavy tails commonly observed in financial time series data. The daily stock price data of Karachi, Bombay, Kuala Lumpur and Singapore stock exchanges from January 2008 to December 2017 were used. The predictive capability of RDCC models, with various error distributions, in forecasting one-day-ahead Value-at-Risk (VaR) was assessed by several back-testing procedures. The empirical results of the study revealed that the RDCC model with Student’s-t distribution produced more accurate and reliable risk forecasts than other competing models. To cite this article [Farid, S. & Iqbal, F. (2020). Forecasting Value-at-Risk of Asian Stock Markets Using the RDCC-GARCH Model Under D...

The estimation of inflation volatility is important to Central Banks as it guides their policy initiatives for achieving and maintaining price stability. This paper employs three models from the Generalized Autoregressive Conditional Heteroscedasticity (GARCH) family with a view to providing a parsimonious approximation to the dynamics of Nigeria’s inflation volatility between 1996 and 2011. Of the competing models, the asymmetric TGARCH (1,1) provides an appropriate paradigm for explaining the dynamics of headline and core CPI volatilities in Nigeria, while the symmetric GARCH (1,1) was found to be adequate for food CPI. The results are quite revealing. Firstly, model outcomes indicate high persistence parameters for the core and food CPI, implying that the impacts of inflation shocks on their volatilities die away very slowly. However, the impact of inflation shocks on headline volatility die out rather quickly. Secondly, substantial evidence of asymmetric effect was found for bot...

This brief focuses on risk management of a portfolio within the Crédit Agricole Group of Morocco.
The purpose of this project is to manage a portfolio of stocks based on micro-macro economic rations.
The loss of our portfolio is estimated by using the parametric VaR and Monte Carlo,that we have
implemented on VBA.
Then, we calculated the price of an option, using the Monte Carlo simulation and Black & Scholes
model,on the most dominant value in our portfolio. All with estimating volatility by GARCH(1.1 )
model .
Keywords : Value at risk, option, volatility, VBA, portfolio, GARCH (1.1), Monte Carlo, Black &
Scholes.

In European countries, the last decade has been characterized by a deregulation of power production and electricity became a commodity exchanged in proper markets. This resulted in an increasing interest of the scientific community on electricity exchanges for modeling both market activity and price process. This paper analyzes electricity spot-prices of the Italian Power Exchange (IPEX) and proposes three different methods to model prices time series: a discrete-time univariate econometric model (ARMA-GARCH) and two computational-intelligence techniques (Neural Network and Support Vector Machine). Price series exhibit a strong daily seasonality, addressed by analyzing separately a series for each of the 24 hours. One-day ahead forecasts of hourly prices have been considered so to compare the prediction performances of three different methods, with respect to the canonical benchmark model based on the random walk hypothesis. Results point out that Support Vector Machine methodology gives better forecasting accuracy for price time series, closely followed by the econometric technique.

The study aims to extend the GARCH type volatility models to their nonlinear TAR (Tong, 1990) and STAR-based (Terasvirta, 1994) counter parts where both the conditional mean and the conditional variance processes follow TAR and STAR nonlinearity. The paper further investigates the models under their fractional integration and asymmetric power variants. The STAR-based models are LSTARLST- GARCH, LSTAR-LST-FIGARCH, LSTAR-LST-FIPGARCH and LSTAR-LSTFIAPGARCH models, which may be easily applied to model and forecast various financial time series. In the empirical section, an application is provided to model the daily returns in WTI crude oil prices considering the regime shifts the crude oil prices were subject to during history. Models are evaluated in terms of their out-of-sample forecasting capabilities with equal forecast accuracy tests and also in terms of various error criteria. The results suggest that volatility clustering, asymmetry and nonlinearity characteristics are modeled m...

This study aims to highlight the importance of GARCH models in the volatility modeling and forecasting as a mechanism for crisis management and early warning. After presenting the theoretical background of the models have been applied at the level of nine Arab stock exchanges indicators, namely: Abu Dhabi, Bahrain, Dubai, Egypt, Kuwait, Morocco, Oman, Qatar and Saudi Arabia, using daily data between 2007 and 2012 (1304 daily observation). The study concluded there is the problem of Heteroskedasticity and continuity in shock in light of the crisis, which imposes the use of GARCH models.

The stock market is constantly changing with uncertainties. That's why the analysis of certain time series from the economic and financial world shows specific characteristics that are not theoretically taken
into account in ARIMA modeling Box Jenkins. The first characteristic concerned is the
dependence of the conditional variance of time, in other words, we are concerned by the heteroscedastic behavior of the variance. The second
characteristic concerns the probabilistic distribution of the series (the appearance of shocks is
not compatible with the normal law). Then for a more realistic modeling of these series,
ARCH / GARCH models are the most useful.
In this project my co-worker and I :

► We have studied the ARCH and GARCH models from a theoretical point ( properties and conditions
to be fulfilled by the models).

► We applied them to one of the indices (CAC40 ) using EVIEWS.

► We applied 3 different forecasting models to predict the conditional variance of CAC 40 stock indices.

► We finally compared the forecasts using multiple statistical tests.

It is crucial to model, quantify and understand the variables and dynamics that underlie the well-known extreme behaviour of spot electricity prices in wholesale markets. We explicitly model the conditional volatility and skewness of electricity prices. A GARCH-type model allowing for time-varying volatility and skewness, which is estimated assuming a Gram-Charlier expansion of the normal density function, is presented. This model is applied to data from Pennsylvania- New Jersey, NordPool and Victoria (Australia) markets. We document the existence of a rich structure in the conditional skewness of spot prices and we show the relationship between skewness and demand-supply related variables.

El presente Trabajo Fin de Master plantea como objetivo la aplicación práctica del modelo GARCH y distribuciones de colas anchas en el cálculo del riesgo de mercado, mediante metodología VeR Paramétrica, en una cartera de renta variable en el periodo de 1/1/2014 al 31/12/2018 para los principales títulos del sector bancario y financiero que cotizan en el mercado bursátil español y que pertenecen al IBEX-35. Se escogen los valores cotizados de Santander, BBVA, CaixaBank, Sabadell y Bankia. Se realiza metodología empírica mediante técnicas matemáticas, estadísticas y econométricas. Se comparan resultados obtenidos entre distribución normal y distribución T-Student (colas anchas), y según la forma de cálculo de la volatilidad (Estándar o GARCH). Finalmente se realiza el Backtesting en el periodo de 1/1/2019 al 30/6/2019 para los resultados de la cartera y se establecen las conclusiones finales del estudio.

This study is a comparative investigation in the field of GARCH estimation. Short and long term market risk components are estimated based on financial market indices of eight countries including members of the advanced G10 (United Nations, Japan, Germany, Canada) and members of the recently emerging (Brazil, India, China, Turkey) group respectively with the final aim to detect country specific patterns of volatility as well as differences between mature and emerging markets. Six models of conditional volatility including GARCH, ApARCH, T-GARCH, GJR-GARCH, E-GARCH and CGARCH as well as their semi-parametric extensions and the SPLINE-GARCH model are introduced and applied to the data respectively. Results provide supportive evidence that long term volatility is a time variant oscillatory evolving function and that the net impact on short term conditional volatility depends either on sign and size of shocks. Differences in exposure, magnitude of shocks and persistence of short term volatility are derived via comparison of the GARCH coefficients. While volatility appears to be more persistent in the mature market case, emerging market countries display higher systematic risk levels. The impact of the recent Global Financial Crisis on long and short term volatility components, in turn, is significantly higher for the mature market cases while emerging markets albeit display contagion effect but are more susceptible to region or country specific shocks. The results are in consent with the volatility co-movement hypothesis and constitute a starting point for extension to mutlivariate grounds

L’obiettivo del progetto di tesi è quello di mostrare gli aspetti teorici necessari alla realizzazione di una piattaforma web-based di supporto agli investimenti finanziari, attraverso la messa a disposizione di una serie di strumenti quantitativi complessi a mezzo un front-end intuitivo.
In particolare, l’elaborato è suddiviso in due parti principali:

1. capitoli teorici per introdurre i modelli esposti in piattaforma corredati da esempi;

2. capitoli tecnici e funzionali per descrivere le modalità d’implementazione della piattaforma e l’esperienza d’uso della stessa.

I capitoli teorici sono organizzati in modo da fornire in primo luogo un background su distribuzioni e processi stocastici degli oggetti principali dell’analisi quantitativa trattata: i return dei titoli negoziati sul mercato.
In secondo luogo, dopo una prima introduzione ai modelli autoregressivi AR e MA per la modellizzazione dei return, viene dedicato ampio spazio alle misure di rischio principali, al loro uso nella pianificazione del rischio di portafoglio e alle diverse modalità per la loro stima.
In fine, vengono riportati i principali modelli per la volatilità: ARCH, GARCH e GARCH multivariati.
I capitoli tecnici e funzionali, invece, in primo luogo dettagliano le modalità di sviluppo progettuale descrivendo l’architettura applicativa dal punto di vista logico e tecnologico, per poi descrivere l’esperienza d’uso della piattaforma in cui viene mostrato lo use case tipico di un trader in cerca di insights quantitativi.

Volatility in financial markets, particularly stock exchange markets, is an important issue that concerns theorists
and practitioners. Over the past 30 years, there has been a vast literature for modeling the temporal dependencies
in volatility of financial markets. Also, more recently researches have been examining the asymmetry and
non-linear properties in variance of financial assets, rather than the conditional mean. In this study, a
comprehensive empirical analysis of the mean return and conditional variance of Turkish Financial Markets is
performed by using various GARCH models. CGARCH and TGARCH appear to be superior for modeling the
volatility of financial instruments in Turkey during the years 2002–2014. It is also found that return series of all
markets include; leptokurtosis, asymmetry, volatility clustering, and long memory.
Keywords: asymmetric GARCH, volatility, financial markets, forecasting, BIST

The aim of this chapter is to provide a detailed empirical example of autoregressive conditional heteroskedasticity (ARCH) model and selected generalized ARCH models. Before the ARCH/GARCH models are estimated, several calculations and tests should be done. The mean model is determined using the autocorrelation function and partial autocorrelation function and also the unit root test. The existence of ARCH effect is tested using ARCH-LM test. After these steps are done, then ARCH/ GARCH models can be estimated. All these theoretical aspects are applied to Sofia Stock Indexes (SOFIX) using EViews 9 software package. The windows and output of EViews are presented. To show the output's academic writing format researchers' outputs are presented in a table.

The aim and objective of this study is to model and forecast the stock index volatility of Shanghai and Shenzhen index. The volatility is estimated by employing three GARCH-type models standard GARCH, GJR-GARCH and the EGARCH. Their performance are compared based on three statistical errors, the mean absolute error, root mean squared and mean absolute percentage and then by the Mincer-Zarnowitz regression. The EGARCH outperforms its counterparts in all three statistical errors and by the regression based analysis. These empirical results are of important significance to portfolio management and risk management processes.

This paper aims to measure the nature of volatility in the cryptocurrency
market before and during Covid-19 pandemic period. To achieve this goal, the Wald test, Granger Causality and Generalized Autoregressive Conditional Heteroskedasticity (1,1) have been applied considering the daily US dollar dominated closing prices of 15 leading cryptocurrencies and volatility index (VIX- CBOE) from 1 January, 2019 to 5 June, 2020. The presence of structural breaks in all the selected cryptocurrencies is observed which result in erroneous forecasting in cryptocurrency market. The small size of cryptocurrency market hinders the risk diversification. It is further noticed that cryptocurrencies are exposed to the systematic bubble risks and therefore it is very unpredictable. Inclusion of cryptocurrencies in the portfolio along with conventional
instruments like stocks, bonds, precious metals, commodities, and paper
currencies may gear up the overall return on investment and increase the possibility of risk diversification if necessary investment precautions are taken.

This paper investigates volatility spillovers in the stock market in Japan during the COVID-19 pandemic by using GARCH family models. The empirical analysis is focused on the dynamics of the NIKKEI 225 stock market index during the sample period from July 30, 1998, to January 24, 2022. In other words, the sample period covers both the period of the global financial crisis (GFC) and the COVID-19 pandemic. The econometrics includes GARCH (1,1), GJR (1,1), and EGARCH (1,1) models. By applying GARCH family models, this empirical study also examines the long-term behavior of the Japanese stock market. The Japanese stock market is much more stable and efficient than emerging or frontier markets characterized by higher volatility and lower liquidity. The paper establishes that NIKKEI 225 index dynamics is different in intensity in the case of the two most recent extreme events analyzed, namely the global financial crisis (GFC)of 2007-2008 and the COVID-19 pandemic. The findings confirmed the presence of the leverage effect during the sample period. Moreover, the empirical results identified the presence of high volatility in the sample returns of the selected stock market. Nevertheless, the econometric framework showed that the negative implications of the GFC were much more severe and caused more significant contractions compared to the COVID-19 pandemic for the Japanese stock market. This study contributes to the existing literature by providing additional empirical evidence on the long-term behavior of the stock market in Japan, especially in the context of extreme events.

Options are instruments which have the special property of limiting the downside risk, while not limiting the upside potential, thus their use in hedging. The share of the options market in the Indian capital market has increased to 64% in just over a decade. The trading turnover of options in the FY11 was Rs. 193,95,710 crore, and the trading volume generated by options market was almost two times that of the volume generated in the cash market and futures market put together. So trading and pricing of stock option have occupied an important place in the Indian derivatives market. Volatility is a critical factor influencing the option pricing; however, it is an extremely difficult factor to forecast. Hence the crucial problem lies with the accurate estimation of volatility. The estimated volatility can be used to determine future prices of the stock or the stock option. Empirical research has shown that using historical volatility in different option pricing models leads to pricing biases. The GARCH (1, 1) model can be a solution for this

This study introduces a state-of-the-art volatility forecasting method for container shipping freight rates. Over the last decade, the container shipping industry has become very unpredictable. The demolition of the shipping conferences system in 2008 for all trades calling at a port in the European Union (EU) and the global financial crisis in 2009 have affected the container shipping freight market adversely towards a depressive and non-stable market environment with heavily fluctuating freight rate movements. At the same time, the approaches of forecasting container freight rates using econometric and time series modelling have been rather limited. Therefore, in this paper, we discuss contemporary container freight rate dynamics in an attempt to forecast for the Far East to Northern Europe trade lane. Methodology-wise, we employ autoregressive integrated moving average (ARIMA) as well as the combination of ARIMA and autoregressive conditional eteroscedasticity (ARCH) model, which we call ARIMARCH. We observe that ARIMARCH model provides comparatively better results than the existing freight rate forecasting models while performing short-term forecasts on a weekly as well as monthly level. We also observe remarkable influence of recurrent general rate increases on the container freight rate volatility.

We proposed a strategic asset allocation framework in which forward-looking views on various fixed income asset-classes are developed on volatilities rather than returns. Such methodology rely on some powerful and practical tools such as asymmetric and non-linear univariate GARCH models linked by elliptic copulas. In such scheme, the portfolio construction process is fully driven by the new information brought by the data without any restriction on the calibration of the dynamic matrix of volatilities.

We think that such approach has the advantage of bringing flexibility to asset managers to strategically adjust their portfolio compositions in a timely fashion with an effective control on the tail risks. The robustness of the proposed methodology was controlled by Cholesky Monte-Carlo simulations and various VaR and CVaR computation. That said, the classic framework of Mean-Variance optimization could be very restrictive practically due to its well-known drawbacks, and other frameworks of optimization should be explored.

Key words: Dynamic Strategic Asset Allocation; GARCH; EGARCH; COGARCH; Copula; Value-at-Risk; tail risk; Portfolio construction; Quantitative strategies;

Purpose
The current paper aims to fill the gap in the literature by analyzing the nature of volatility of the Karachi Stock Exchange (KSE) -100 index and also develops an understanding as to which model proves to be the most suitable model for measuring volatility among the models used. The study contributes significantly to the literature as compared to the limited previous work done in Pakistan as it covers a longer time period from the introduction of the KSE-100 index on November 2, 1991 to December 31, 2013 for three types of data i.e., daily, weekly and monthly. In addition, to analyze the impact of global financial crises upon volatility, the data is divided into pre-crisis (1991-2007) and post-crisis (2008-2013) periods.
Design/Methodology/Approach
This study employs an advanced set of volatility models such as Autoregressive Conditional Heteroscedasticity [ARCH (1)]; Generalized Autoregressive Conditional Heteroscedasticity [GARCH (1, 1)]; GARCH in Mean [GARCH-M (1, 1)]; Exponential GARCH [E-GARCH (1, 1)]; Threshold GARCH [T-GARCH (1, 1)]; Power GARCH [P-GARCH (1, 1)] and also a simple Exponentially Weighted Moving Average (EWMA) model.
Findings
The results reveal that daily, weekly and monthly return series show non-normal distribution, stationarity, and volatility clustering. However, the heteroscedasticity is absent only in the monthly returns making only EWMA model to be used in monthly series to measure the volatility level. P-GARCH (1, 1) model proves to be a better model for modeling the volatility in case of daily returns; while regarding the weekly data GARCH (1, 1) model proves to be the most appropriate based on SIC and LL criterion. The study shows a high persistence of volatility, a mean reverting process, and absence of risk premium in the KSE market with an insignificant leverage effect only in case of weekly returns; however a significant leverage effect is reported regarding the daily series of the KSE-100 index. In addition, to analyze the impact of global financial crises upon volatility, the findings show that sub-periods demonstrated a slightly low volatility and global economic crisis did not cause a rise in volatility level.
Originality/Value
The literature about the volatility modeling in the Pakistan market depicts a limited literature focus on few models for relatively small sample size. The current thesis attempts to overcome these limitations and employs diverse models for three types of data series (daily, weekly and monthly). In addition, Pakistani economy shows turmoil throughout its history, which ranges from mild shocks to extreme shocks. This paper measures the impact of those shocks upon the volatility level of the KSE.
Applications
The research provides some insights for policy makers as well as investors who are concerned about the fluctuations of the KSE-100 index in Pakistan. For example, the significant ARCH effect may imply that the institutional investors who do not trade very frequently also hold dominance with respect to the movement of the price of stocks. As when they trade, it is in the large bulk and this heavy trade holds a significant impact on the price movements of the stock (Husain and Uppal, 1999). The skewness with the negative value in all return series imply that there are more chances of earning negative returns than the positive returns which refers towards the conservative attitude of investors towards investment in the KSE market (Mittal and Goyal, 2012; Kaluo and Friday, 2012).. The findings also reveal a high persistence of volatility and it contributes to the existence of the impact of shock, observed in present, for a longer time on future returns. The high volatility presence makes it possible to earn high profit but it also leads towards an inefficient market (Mittal and Goyal, 2012). A presence of an insignificant risk premium implies that investors are unable to earn returns above the average by taking the higher risk. The leverage parameter finds significance in case of daily returns which implies that negative shocks account for the greater volatility in the KSE market; however, the weekly data do not reveal significant asymmetrical or leverage effect.

Key Words: ARCH, GARCH, EWMA, Global Financial Crisis, Pakistan, Volatility

Financial series tend to be characterized by volatility and this characteristic affects both financial series of developed markets and emerging markets. Because of the emerging markets have provided major investment opportunities in last decades their volatility has been widely investigated in the literature. The most popular volatility models are the Autoregressive Conditional Heteroscedastic (ARCH) or Generalized Autoregressive Conditional Heteroscedastic (GARCH) models. This paper aims to investigate the volatility of Bucharest Stock Exchange, BET index as an emerging capital market and compare forecasting power for volatility of this index during 2000-2014. To do this, this paper use GARCH, TARCH, EGARCH and PARCH models against Generalized Error distribution. We estimate these models then we compare the forecasting power of these GARCH type models in sample period. The results show that the EGARCH is the best model by means of forecasting performance.

We study performance of Islamic and conventional indices of the Gulf Cooperation Council (GCC) countries in the wake of financial crisis of 2008 and test whether Islamic indices were less risky than conventional indices. We make use of data of the six GCC markets as well as the Dow Jones Islamic Market Index GCC. The mean and variance of each of the indices are analyzed based on augmented GARCH models. The results show that the financial crisis impacted on the mean returns of Bahrain, the other indices remained unaffected. The financial crisis, however, impacted volatility in three GCC markets (Kuwait, Bahrain, and the UAE), while the impact on the remaining markets (Saudi Arabia, Oman, and Qatar) and the Islamic index was insignificant. More interestingly, we show that the Islamic index did not exhibit lower volatility than its conventional counterparts.

This paper investigates the performance of various conditional volatility models to forecast the second moment of tanker freight rates. Justified by existing theoretical and empirical evidence, we focus on asymmetric Markov regime-switching models to study the major global routes for long-haul trade of crude oil during the sample period from June 2000 to May 2015. Moreover, in contrast to a number of existing studies, we examine seasonally adjusted freight rates. We find that regime-switching GARCH models outperform their single-regime complements in terms of in-sample fit and out-of-sample forecasting accuracy. In particular, the asymmetric MRS-EGARCH and MRS-APARCH exhibit superior in- and out-of-sample performance. To additionally examine the applicability in freight risk management, we compare Value-at-Risk and Expected Shortfall forecasts. Our results show that accounting for volatility regimes and asymmetry does not enhance the performance of one-day-ahead forecasts of either risk measure for both long and short trading positions.

Interest is in evaluating, by Markov chain Monte Carlo (MCMC) simulation, the expected value of a function with respect to a, possibly unnormalized, probability distribution. A general purpose variance reduction technique for the MCMC estimator, based on the zero-variance principle introduced in the physics literature, is proposed. Conditions for asymptotic unbiasedness of the zero-variance estimator are derived. A central limit theorem is also proved under regularity conditions. The potential of the idea is illustrated with real applications to probit, logit and GARCH Bayesian models. For all these models, a central limit theorem and unbiasedness for the zero-variance estimator are proved (see the supplementary material available on-line).

In recent years, the increasing use of digital currency has been featured in the banking
environment and payment market. The Bitcoin, as the main representative of digital currencies,
reached the $ 29 billion mark traded in the last 12 months within an average level of
capitalization at $ 5 billion. The Court of Justice of the European Union has already taken the
first steps towards the recognition of Bitcoin as a currency, which is not shared with economists,
who has the main criticism of high volatility in their negotiated prices. In this paper, we will
use the model I-GARCH to establish an indicator of volatility of Bitcoin and also for a portfolio
of traditional currencies that represent more than 70% of international trading volumes, to try
to determine how far Bitcoin is in relation to other currencies in terms of volatility at this time.