Most papers in empirical finance implicitly or explicitly assume the same price of risk, for each priced systematic risk factor, across all risky assets within a given domestic market. In doing so, they rely on the assumption that markets are domestically integrated and, as such, that the price of risk is determined independently of individual investors attitude towards risk. This is true in frictionless markets where investors have complete information, homogenous beliefs, and hold the mean-variance efficient combination of the market portfolio and a risk-free asset. However, investors might not hold the market portfolio because of exogenous reasons. In fact, several recent papers have provided evidence that US investors do not, holding instead vastly undiversified portfolios. There are two main implications to the above. First, if one group of investors does not hold the market portfolio, then the remaining set of investors will also not be able to hold the market portfolio and will rationally expect to be compensated for bearing idiosyncratic risk. Therefore, idiosyncratic risk will be priced in expected returns. Second, the price of risk need not be the same across all assets in which case domestic markets are not integrated. In the first essay titled "Is Idiosyncratic Volatility Really priced?" I show that the positive relation between idiosyncratic volatility (IV) and returns found by Fu (2009) only exists for firms that are difficult to arbitrage. The relation between IV and returns is strong for small and illiquid stocks, but decreases with size and liquidity and becomes non-existent for the largest and most liquid firms. Furthermore, zero-cost portfolios based on IV and size do not yield positive returns when conservative trading costs are considered. This evidence is consistent with an efficient market, in which arbitragers exploit profitable investment opportunities and by doing so they prevent systematic mispricing in financial markets. In the second essay titled "Are the U.S. Equity Markets Domestically Integrated?" I investigate whether the three main U.S. equity markets are domestically integrated by comparing the price of commonly used risk factors across the NYSE, Amex, and Nasdaq. I find that the markets have significantly different prices of risks for several risk factors, indicating that the markets are segmented. The magnitude of the difference is both statistically and economically significant, and is not due to arbitrage constraints or model misspecification. Instead, I find evidence consistent with the investor-segmentation hypothesis, in which different investors choose to hold different subsets of firms and demand different prices of risk among the different groups of securities. I do not find that segmentation is restricted to a specific time period. On the contrary, it is present in all sub-periods. In contrast to the results regarding the pricing of idiosyncratic volatility, these results highlight the value of diversification and suggest that domestic equity markets are not fully efficient.

The diploma thesis deals with modelling of time series (stock and commodities) by using the models of volatility. The theoretical part focuses on the term of volatility and other terms connected to it. There is a theoretical description of the models as well. The practical part of the thesis focuses on the analysis of the time series and modelling of volatility using the program R.

The purpose of this paper is to select the best GARCH-type model for modelling the volatility of Bitcoin, Bitcoin Cash, Litecoin, Dogecoin and Ethereum. GARCH (1,1), IGARCH(1,1), EGARCH(1,1), TGARCH(1,1) and CGARCH(1,1) are used on the cryptocurrencies closing day return. We select the model with the highest Maximum Likelihood and run an OLS regression on the conditional volatility to measure the day-of-the-week effect. The findings show that EGARCH(1,1) model best suits Bitcoin, Litecoin, Dogecoin and Ethereum data and that the GARCH(1,1) model suits best Bitcoin data. The results show a significant presence of day-of-the-week effects on the conditional volatility of some days for Bitcoin, Bitcoin Cash and Ethereum. Wednesday has a significant negative effect on Bitcoin conditional volatility. Friday, Saturday and Sunday are found to be significant and positive on Bitcoin Cash conditional volatility. Finally, Saturday is found to be significant and positive on Ethereum conditional volatility.

The efficient market hypothesis has been around since 1962, the theory based on a simple rule that states the price of any asset must fully reflect all available information. Yet there is empirical evidence suggesting that markets are too volatile to be efficient. In essence, this evidence seems to suggest that the reaction of the market participants to the information or events that is the crucial factor, rather than the actual information. This highlights the need to include the behavioural finance theory in the pricing of assets. Essentially, the research aims to analyse the efficiency of six key sovereign debt markets during a period of changing volatility including the recent global financial and sovereign debt crises. We analyse the markets in the pre-crisis period and during the financial and sovereign debt crises to determine the impact of the crises on the efficiency of these financial markets. We use two GARCH-based variance bound tests to test the null hypothesis of the market being too volatile to be efficient. Proposing a GJR-GARCH variant of the variance bound test to account for variation in the asymmetrical effect. This leads to an analysis of the changing behaviour of price volatility to identify what makes the market efficient or inefficient. In general, our EMH tests resulted in mixed results, hinting at the acceptance of the null hypothesis of the market being too volatile to be efficient. However, interestingly a number of 2017 observations under both models seem to be hinting at the rejection of the null hypothesis. Furthermore, our proposed GJR-GARCH variant of the variance bound test seems to be more likely to accept the EMH than the GARCH variant of the test.

In the financial industry, it has been increasingly popular to measure risk. One of the most common quantitative measures for assessing risk is Value-at-Risk (VaR). VaR helps to measure extreme risks that an investor is exposed to. In addition to acquiring information of the expected loss, VaR was introduced in the regulatory frameworks of Basel I and II as a standardized measure of market risk. Due to necessity of measuring VaR accurately, this thesis aims to be a contribution to the research field of applying GARCH-models to financial time series in order to forecast the conditional variance and find accurate VaR-estimations. The findings in this thesis is that GARCH-models which incorporate the asymmetric effect of positive and negative returns perform better than a standard GARCH. Further on, leptokurtic distributions have been found to outperform normal distribution. In addition to various models and distributions, various rolling windows have been used to examine how the forecasts differ given window lengths.

Copulas functions are a flexible tool for modelling the dependence structure between variables. The joint and marginal distributions of Copulas are not constrained by the assumptions of normality. This study examines the dependence structure between the gold, platinum prices and the ZAR/U.S.D exchange rate using Copulas. The study found that marginal distributions of Copulas follows the ARMA (1, 1)-EGARCH (1, 1) and ARMA(1, 1)-APARCH (1, 1) models under different error terms including the normal, the student-t and the skew student-t error terms. It used the Normal, the Student-t, the Gumbel, the rotated Gumbel, the Clayton, the rotated Clayton, the Plackett, the Joe Clayton and the Normal time varying Copulas to analyse the dependence structure between returns prices of gold, platinum and ZAR/U.S.D exchange rate. The results showed evidence of a positive strong dependence between the returns prices of gold, platinum and returns on the Rand/U.S.D exchange rate for constant and time varying Copulas. The result also showed a co-movement of exchange rates and gold and platinum prices during a rise or declining prices of gold and platinum. The results imply that fluctuations in gold and platinum prices generate Rand/U.S.D exchange rate volatility. / Statistics / M. Sc. (Statistics)

La thèse comporte trois parties, dont l’objet d’étude est l’immobilier soit en s’intéressant directement au marché immobilier résidentiel parisien, soit indirectement en analysant les REITs dont l’actif sous-jacent est l’immobilier. La première partie porte sur l’examen des facteurs macroéconomiques et financiers qui ont influencé, au cours de la période 1996-2010, les prix résidentiels à Paris en appliquant un modèle VAR (vector autoregressive). Dans la deuxième partie, nous analysons, au cours de la période 2007-2012, l’interaction entre les rendements des REITS et les facteurs macroéconomiques/financiers dans dix pays développés (les Etats-Unis, le Canada, l’Australie, l’Europe, la Zone Euro, le Japon, Hong-Kong, la France, le Royaume-Uni et l’Allemagne) en appliquant un modèle VAR. Les REITs investissent sur l’immobilier, mais ils sont cotés sur un marché boursier. Par conséquent, ils héritent des caractéristiques à la fois de l’immobilier et des actions. Ce caractère hybride des REITs nous conduit à nous interroger sur leur risque et le rôle qu’ils peuvent jouer dans la gestion de portefeuille. C’est pourquoi dans la troisième partie nous examinons, au cours de la période 2001-2012, l’effet des deux composantes du risque (bêta et le risque idiosyncratique) et certains facteurs spécifiques aux REITs (taille, rapport Actif Net Comptable / valeur de marché et la mesure d’illiquidité) sur les rendements des REITs européens (la France, l’Allemagne, le Royaume-Uni, les Pays-Bas et l’Italie) en appliquant le modèle à trois facteurs de Fama et French et la méthodologie de Fama et MacBeth. / The thesis consists of three parts, whose purpose is the real estate market either being interested directly in the residential real estate market in Paris, or indirectly by analyzing REITs, whose underlying asset is the real estate. In the first part entitled “Which Macroeconomics and Financial Factors Affect Real Estate Prices in Paris”, we employ a vector autoregressive (VAR) model in order to examine, during the period 1996-2010, the relation between residential prices in Paris and several macroeconomic/financial factors. In the second part entitled “How Legislation, REIT System and Taxes Influence REITs Returns Sensitivity to Macroeconomic and Financial Factors? An International Perspective”, we analyze, by applying a VAR model, during the period 2007-2012, dynamic interactions among REITs returns and macroeconomic factors for ten developed countries (the United States, Canada, Australia, Europe, Eurozone, Japan, Hong Kong, France, Britain and Germany). REITs invest in real estate and they are publicly traded. Thus they inherit the characteristics of both real estate and stocks. This hybrid nature of REITs reveals the importance of their risk and the role they can play in portfolio management. Therefore, in the third part entitled “Cross-Sectional Expected European REITs Returns : does Volatility Matters ?”, we investigate, during the period 2001-2012, the effect of the two components of risk (beta and idiosyncratic risk) and some specific factors of REITs (size, Net Asset Value to Market Value and illiquidity measure) on European REITs returns (France, Germany, the UK, the Netherlands and Italy) by applying the Fama and French model and cross-sectional regressions.

摘要 本文是利用1998年底以前券商發行的15支認購權證為研究標的，試圖說明不同波動度的估計方法，會使得認購權證的理論價與市價產生不同的誤差，藉以提供券商在評價認購權證上作一參考。本文的實證結果發現：（1）在波動度的參數估計上，各模型均有波動度群集效果，但是訊息不對稱的效果各模型卻無一致性的結果；（2）在各模型的預測能力比較上，ARCH-M(1,1)模型都比ARCH(1,1)的預測能力佳。歷史波動度對於標的股的波動度小具有較佳的預測能力，而EGARCH-M(1,1)模型與GJR-GARCH-M(1,1)模型在預測波動度較大的標的股時具有較佳的預測結果；（3）以預測誤差百分比來比較各模型在預測認購權證上何者具有較小的誤差，結果發現：不論有無考慮交易成本及間斷性避險，預測能力最差的是歷史波動度，而預測能力最佳的則是隱含波動度模型，此乃因為台灣認購權證市場只有認購權證而無認售權證所致；（4）以市場溢價來比較那一支認購權證較值得投資者購買，結果發現：若權證處於價外，會使得市場溢價過高，而不利投資者購買；相反，若權證價格處於價內，則使得市場溢價較低，投資者購買較有利；（5）利用Delta法及Delta-Gamma法來計算大華01可發現：不同波動度的估計方法會影響該權證的涉險值，由於隱含波動度明顯高於其他方法所估算的值，故以隱含波動度計算的涉險值也就高於其他模型之涉險值。 目錄 謝辭 摘要 第一章 緒論 第一節 研究背景與動機 ………………………………………….1-1 第二節 研究問題與目的 ………………………………………….1-4 第三節 論文架構與流程 ………………………………………….1-5 第二章 文獻回顧 第一節 隨機波動度模型 ……………………………………….2-1 壹 Hull & White（1987）模型 …………………………..2-1 貳 Wiggins（1987）模型 ………………………………..2-3 參 Johnson & Shanno（1987）模型 …………………….2-4 肆 Scott（1987）模型 …………………………………...2-5 伍 Stein & Stein（1991）模型 …………………………..2-6 陸 Heston（1993）模型 …………………………………2-8 第二節 GARCH體系---波動度估計之方法 ……………………2-10 壹 GARCH模型 …………………………………………2-10 貳 EGARCH模型 ………………………………………..2-10 參 GJR-GARCH模型 ……………………………………2-11 肆 N-GARCH模型 ………………………………………2-12 伍 T-GARCH模型 ………………………………………2-12 第三章 研究方法 第一節 波動度之估計方法 ……………………………………….3-1 壹 歷史波動度 ……………………………………………3-1 貳 GARCH(1,1)模型 ……………………………………..3-2 參 EGARCH(1,1)模型 …………………………………..3-3 肆 GJR-GARCH（1,1）模型 …………………………...3-5 伍 ARCH-M（1,1）模型 ………………………………..3-7 陸 隱含波動度模型（Implied Volatility） ………………3-8 第二節 選擇權評價公式之探討 ………………………………….3-24 壹 Black & Scholes的選擇權評價模型 ………………...3-24 貳 考慮交易成本極間斷性避險下的選擇權評價模型 ...3-25 第四章 實證結果與分析 第一節 波動度的估計與預測能力 ………………………………4-1 第二節 選擇權評價理論的實證結果 …………………………4-18 第三節 認購權證涉險值（VAR）之衡量與應用 ………………4-53 第五章 結論與建議 ……………………………………………….5-1 附錄 …………………………………………………………附-1 參考文獻 …………………………………………………………Ⅰ

It is widely believed among South African agricultural market participants that the United States' corn price, as represented by the Chicago Board of Trade-listed corn contract, is causal to the price of white and yellow maize traded on the South African Futures Exchange. Although a strong correlation exists between these markets, the corn contract is far from causal to the South African maize price, as indicated by Auret and Schmitt (2008). Similarly, South African market participants believe that volatility generated in the United States corn market spills over to the South African market. Given the perceived volatility spill-over from the corn market to the maize market, market participants might inadvertently include a higher volatility component in an option price in the South African maize market than is necessary. This study sought to quantify the amount of volatility spill-over to the South African white and yellow maize market from the United States corn contract. This task was accomplished by applying an Exponential Generalised Auto Regressive Conditional Heteroscedasticity model, within an aggregate shock framework, to the data. The findings indicated that the volatility spill-over from the United States corn market to the South African maize market is not statistically significant. This result suggests that volatility in the South African market is locally driven; hence, it should not be necessary for a South African listed option contract to carry an international volatility component in its price. It was also found that the returns data of the South African maize market is asymmetrically skewed, indicating that bad news will have a greater effect on the price of maize compared with good news. / Thesis (MCom (Risk Management))--North-West University, Potchefstroom Campus, 2013.

It is widely believed among South African agricultural market participants that the United States' corn price, as represented by the Chicago Board of Trade-listed corn contract, is causal to the price of white and yellow maize traded on the South African Futures Exchange. Although a strong correlation exists between these markets, the corn contract is far from causal to the South African maize price, as indicated by Auret and Schmitt (2008). Similarly, South African market participants believe that volatility generated in the United States corn market spills over to the South African market. Given the perceived volatility spill-over from the corn market to the maize market, market participants might inadvertently include a higher volatility component in an option price in the South African maize market than is necessary. This study sought to quantify the amount of volatility spill-over to the South African white and yellow maize market from the United States corn contract. This task was accomplished by applying an Exponential Generalised Auto Regressive Conditional Heteroscedasticity model, within an aggregate shock framework, to the data. The findings indicated that the volatility spill-over from the United States corn market to the South African maize market is not statistically significant. This result suggests that volatility in the South African market is locally driven; hence, it should not be necessary for a South African listed option contract to carry an international volatility component in its price. It was also found that the returns data of the South African maize market is asymmetrically skewed, indicating that bad news will have a greater effect on the price of maize compared with good news. / Thesis (MCom (Risk Management))--North-West University, Potchefstroom Campus, 2013.