Volatility clustering is a stylized fact common in nance. Large changes in prices tend to cluster whereas small changes behave likewise. The higher the volatility of a market, the more risky it is said to be and vice versa . Below, we study volatility clustering using an agent-based model. This model looks at the reaction of agents as a result of the variation of asset prices. This is due to the irregular switching of agents between fundamentalist and chartist behaviors generating a time varying volatility. Switching depends on the performances of the various strategies. The expectations of the excess returns of the agents (fundamentalists and chartists) are heterogenous.

Efficient market exist such that financial market make the absence of arbitrage opportunity on intertemporal asset price, There are special existence due to volatility clustering effect provides that the conditional volatility predictor could control, applying on derivative such as option¡Bcurrency exchange¡Bswap¡Bexist possible arbitrage profits ,in this paper involve that forward market efficiency and how to prototype concrete, now we apply parity theory including covered interest parity and uncovered interest parity, then the study of both covered (CIP)and uncovered interest parity (UIP) plus FME are tested in the 30 and 90 forward markets for the NTD/USD exchange rate to examine market efficiency on using GARCH-M,EGARCH models , In the empirical tests, we find the NTS/USD dollar interest rate spread have I(o) property ,Results are provided for interest rate on stationarity indicating that interest differential is stationary ,the result also imply stationary relationship between Taiwan and USA on money policy, Using Taylor(1989) ¡As covered interest arbitrage models, The empirical results show lower positive profit opportunities on NTD or US returns, covered interest parity may hold because NTS/US exchange market after reopening becomes more efficient than market after reopening, the central bank money policy intervention is influential but we test market efficiency hypotheses on basis of Domowitz and Hakkio¡]1985¡^¡As ARCH-M model deeply employing GARCH-M¡BEGARCH models to estimate Risk Premium¡Athen employ Felmingham (2003.2) ¡As regression equation to test forward market efficiency , the empirical results shows that not only CIP¡BUIP theory fail but also Forward Market Efficiency hypotheses cannot hold ,then future spot rates could be predicted by forward rates are worthy of investigate., It may indicate that foreign securities are imperfect substitutes for domestic ones of equivalent maturity and that market participants, implying that there is arbitrage profit opportunity between Taiwan and the USA, there are many arguments to discuss whether forward rates as an unbiased predictor of future spot rate ,Forward Market efficiency give the presence of the time varying premium on different place, Ultimately, therefore, the unbiased nature of forward rates is an empirical, and not a theoretical, issue¡C

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Title: Volatility bursts and order book dynamics Author: Jana Plačková Department: Department of Probability and Mathematical Statistics Supervisor: Dr. Jan M. Swart Supervisor's e-mail address: swart@utia.cas.cz Abstract: The presented paper studies the dynamics of supply and demand through the electronic order book. We describe and define the basic rules of the order book and its dynamics. We also define limit and market orders and describe the differences between them and how they influenced the evolution of ask, bid price and spread. Next part of the paper is dedicated to the de- scription and definition of volatility and its basic models. The brief overview about volatility clustering and its modeling by economists and physicists can be found in the following part. In the last part we introduce a simple model of order book in which we observe ask, bid price and spread. Then we study the empirical distribution of spread and try to find its probability distribu- tion. The volatility clustering is then observed through the relative returns of spread. In the last part we introduce some possible improvement of the model. Keywords: volatility clustering, order book, limit orders, market orders 1

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The global economy has been severely impacted during the Covid-19 period. The U.S. stock market has also experienced greater volatility. Based on data from January 2020 to June 2021, this paper studies the volatility of daily returns on the stock market in the United States. The Standard and Poor's 500 (SPX) index and eight companies traded on major exchanges such as the New York Stock Exchange and the Nasdaq are used to calculate volatility. Combining the statistical analysis methods GARCH, GARCH-M, and TARCH, the time series of each security is modeled. It is demonstrated that the conditional heteroskedasticity of stock returns depends not only on the observed historical volatility (ARCH term) but also on the conditional heteroskedasticity of prior periods (GARCH term). As expected for financial markets, the COVID-19 outbreak increased the volatility of U.S. stock market returns. After the COVID-19 outbreak, the volatility of the U.S. stock market rose dramatically. It reached an extremely high level for the first quarter of 2020 and continued to move downwards in the following quarters. The significant heteroskedasticity in the return volatility indicates that external variables significantly affect the stock. Furthermore, this study combines the Capital Asset Pricing Model (CAPM) and the research of Engle et al. (1987), which provides a way to quantify the liquidity premium. However, with the results of the GARCH-M model, this study does not find a significant liquidity premium over time. Additionally, The TARCH model reveals a significant asymmetry in stock market returns during this epidemic, suggesting that negative news has a more substantial impact on U.S. financial markets. For investors and financial institutions, this research helps identify potential volatility in the face of similar risk events. It is helpful for investors to comprehensively consider various factors when investing in special periods or consider other investment portfolios to reduce investment risks in specific periods based on research results.

Thesis (MBA (Business Management))--Stellenbosch University, 2008. / ENGLISH ABSTRACT: This research report investigated whether evidence of volatility clustering exists on the FTSE/JSE Top 40 Index. The presence of volatility clustering has practical implications relating to market decisions as well as the accurate measurement and reliable forecasting of volatility. This research report was conducted as an in-depth analysis of volatility, measured over five different return interval sizes covering the sample in non-overlapping periods. Each of the return interval sizes' volatility were analysed to reveal the distributional characteristics and if it violated the normality assumption. The volatility was also analysed to identify in which way, if any, subsequent periods are correlated. For each of the interval sizes one-step-ahead volatility forecasting was conducted using Linear Regression, Exponential Smoothing, GARCH(1,1) and EGARCH(1,1) models. The results were analysed using appropriate criteria to determine which of the forecasting models were more powerful. The forecasting models range from very simple to very complex, the rationale for this was to determine if more complex models outperform simpler models. The analysis showed that there was sufficient evidence to conclude that there was volatility clustering on the FTSE/JSE Top 40 Index. It further showed that more complex models such as the GARCH(1,1) and EGARCH(1,1) only marginally outperformed less complex models, and does not offer any real benefit over simpler models such as Linear Regression. This can be ascribed to the mean reversion effect of volatility and gives further insight into the volatility structure over the sample period. / AFRIKAANSE OPSOMMING: Die navorsingsverslag ondersoek die FTSE/JSE Top 40 Indeks om te bepaal of daar genoegsame bewyse is dat volatiliteitsbondeling teenwoordig is. Die teenwoordigheid van volatiliteitsbondeling het praktiese implikasies vir besluite in finansiele markte en akkurate en betroubare volatiliteitsvooruitskattings. Die verslag doen 'n diepgaande ontleding van volatiliteit, gemeet oor vyf verskillende opbrengs interval groottes wat die die steekproef dek in nie-oorvleuelende periodes. Elk van die opbrengs interval groottes se volatiliteitsverdelings word ontleed om te bepaal of dit verskil van die normaalverdeling. Die volatiliteit van die intervalle word ook ondersoek om te bepaal tot watter mate, indien enige, opeenvolgende waarnemings gekorreleer is. Vir elk van die interval groottes word 'n een-stap-vooruit vooruitskatting gedoen van volatiliteit. Dit word gedoen deur middel van Lineêre Regressie, Eksponensiële Gladstryking, GARCH(1,1) en die EGARCH(1,1) modelle. Die resultate word ontleed deur middel van erkende kriteria om te bepaal watter model die beste vooruitskattings lewer. Die modelle strek van baie eenvoudig tot baie kompleks, die rasionaal is om te bepaal of meer komplekse modelle beter resultate lewer as eenvoudiger modelle. Die ontleding toon dat daar genoegsame bewyse is om tot die gevolgtrekking te kom dat daar volatiliteitsbondeling is op die FTSE/JSE Top 40 Indeks. Dit toon verder dat meer komplekse vooruitskattingsmodelle soos die GARCH(1,1) en die EGARCH(1,1) slegs marginaal beter presteer het as die eenvoudiger vooruitskattingsmodelle en nie enige werklike voordeel soos Lineêre Regressie bied nie. Dit kan toegeskryf word aan die neiging van volatiliteit am terug te keer tot die gemiddelde, wat verdere insig lewer oor volatiliteit gedurende die steekproef.

本研究利用GARCH動態過程的優點捕捉匯率報酬率之異質變異與波動度叢聚性質，並以GARCH動態過程為基礎，考慮跳躍風險服從Lévy過程，再利用特徵函數與快速傅立葉轉換方法推導出GARCH-Lévy動態過程下的歐式匯率選擇權解析解。以日圓兌換美元(JPY/USD)之歐式匯率選擇權為實證資料，比較基準GARCH選擇權評價模型與GARCH-Lévy選擇權評價模型對市場真實價格的配適效果與預測能力。實證結果顯示，考慮跳躍風險為無限活躍之Lévy過程，即GARCH-VG與GARCH-NIG匯率選擇權評價模型，不論是樣本內的評價誤差或是在樣本外的避險誤差皆勝於考慮跳躍風險為有限活躍Lévy過程的GARCH-MJ匯率選擇權評價模型。整體而言，本研究發現進行匯率選擇權之評價時，GARCH-NIG匯率選擇權評價模型有較小的樣本內及樣本外評價誤差。 / In this thesis, we make use of GARCH dynamic to capture volatility clustering and heteroskedasticity in exchange rate. We consider a jump risk which follows Lévy process based on GARCH model. Furthermore, we use characteristic function and fast fourier transform to derive the currency option pricing formula under GARCH-Lévy process. We collect the JPY/USD exchange rate data for our empirical analysis and then compare the goodness of fit and prediction performance between GARCH benchmark and GARCH-Lévy currency option pricing model. The empirical results show that either in-sample pricing error or out-of-sample hedging performance, the infinite-activity Lévy process, GARCH-VG and GARCH-NIG option pricing model is better than finite-activity Lévy process, GARCH-MJ option pricing model. Overall, we find using GARCH-NIG currency option pricing model can achieve the lower in-sample and out-of sample pricing error.

本研究以Lévy過程為模型基礎，考慮Merton Jump及跳躍強度服從Hawkes Process的Merton Jump兩種跳躍風險，利用Particle Filter方法及EM演算法估計出模型參數並計算出對數概似值、AIC及BIC。以S&P500指數為實證資料，比較隨機波動度模型、Variance Gamma模型及兩種不同跳躍風險對市場真實價格的配適效果。實證結果顯示，隨機波動度模型其配適效果勝於Variance Gamma模型，且加入跳躍風險後可使模型配適效果提升，尤其在模型中加入跳躍強度服從Hawkes Process的Merton Jump，其配適效果更勝於Merton Jump。整體而言，本研究發現，以S&P500指數為實證資料時，SVHJ模型有較好的配適效果。 / This paper, based on the Lévy process, considers two kinds of jump risk, Merton Jump and the Merton Jump whose jump intensity follows Hawkes Process. We use Particle Filter method and EM Algorithm to estimate the model parameters and calculate the log-likelihood value, AIC and BIC. We collect the S&P500 index for our empirical analysis and then compare the goodness of fit between the stochastic volatility model, the Variance Gamma model and two different jump risks. The empirical results show that the stochastic volatility model is better than the Variance Gamma model, and it is better to consider the jump risk in the model, especially the Merton Jump whose jump intensity follows Hawkes Process. The goodness of fit is better than Merton Jump. Overall, we find SVHJ model has better goodness of fit when S&P500 index was used as the empirical data.

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