Recently, there are more and more literatures discuss on the issues of investment strategies of leveraged ETFs. In our works, we concentrate our issues on optimal leverage of ETF of S&P 500 index. Based on ARMA-GARCH model¡¦s assumption, we find out that the forecasting optimal leverage can be shown in a formula which contains return and characteristic function. In this paper, we use MA(1)-GARCH(1,1) to forecast volatility based on 1008 rolling window to forecast one day ahead¡¦s volatility; and our estimation time is start from 1954 to March 2011. In this paper, we present four dynamic leverage models (Normal, Student T, VG, and Best model¡¦s leverage) to find out the payoffs under these models. In our model, the forecasting accuracy is just about 55% which is slightly higher than SPX raise probability. But during long-term compound effect, the dynamic leverage models can out-perform than constant leverage. There may exist some important factors in these results, one of them is the crash forecasting ability. During 1980 to 2011 SPX has 14 big crashes and these models can effectively avoid 10 big crashes. In short-term investment horizon none of these five models are always outperform than others but in long-term investment horizon the strategy of best model¡¦s leverage can always earn money when investment horizon is 2400 days.

While the much more use of leverage could be effective in generating alpha o investment, the Kelly strategy is an attractive approach to capital creation and growth. It is originated from the Kelly criterion dubbed ¡§ fortunes formula ¡§ which maximizes the long run growth rate of wealth. There is a tradeoff of rate of return versus risk/volatility as a asymptotic function solution of leverage or position size determined by the application of EGARCH model in the different residual assumptions given by the Normal, Generalized Hyperbolic, and the Generalized Error distributions. No matter there is any timing ability in any strategy, risk management is much more important especially with many repeated trading. We present the performance and risk control of the leveraged ETFs tracked the S&P 500 index in the past ten years using optimal leverage strategy derived by the full Kelly and fraction Kelly criterion.

This study explores Bitcoin’s volatility characteristics using different extensions of the GARCH model. The volatility characteristics of bitcoin are compared with to a gold commodity and the S&P 500 index. The purpose is to identify which model fits best for the data and to see how the volatility changes during the time period of 1st February 2017 to 1stFebruary 2022. The dataset is divided into two time periods, one prior to the pandemic which is the low uncertainty period and the other after the pandemic being the high uncertainty period. The attention for cryptocurrencies and especially bitcoin, has risen expeditiously the last couple of years, this makes the analysis appropriate and current for the market. The result showed that bitcoin’s volatility is more effected by the volatility of gold than for S&P 500. The volatility shows that bitcoin was more similar to the behavior of the gold than the S&P 500 prior to the pandemic. Further is there still no clearer explanation and bitcoins behavior cannot be stated as a commodities or financial asset. The GARCH model results showed that bitcoin’s volatility is persistent over time and can therefore be an explanation that will apply well as for the next years. The high volatility time periods of bitcoin can be explained by optimism and overestimate bias. The bias connected the overly confident investment decisions to less accurate rents. Bitcoin is still new on the financial market which makes new knowledge extremely important in order to create safer investment portfolios.

This study examines the Chicago Board Option Exchange (CBOE) Volatility Index (VIX) which is the implied volatility calculated from short-term option prices on the Standards & Poor’s 500 stock index (S&P 500). Findings suggest VIX overestimates average volatility by approximately 3% but explains 55% of S&P 500’s proceeding month’s volatility. The implied volatility (IV) from options on the VIX add additional explanatory power for the S&P’s 500 proceeding kurtosis values (a measure of tail risk). The VIX option’s volatility smirks did not add additional explanatory power for explaining the S&P 500 volatility or kurtosis. A simple trading rule based on buying the S&P 500 whether the VIX, IV from the options on the VIX, and the VIX option’s volatility smirk decline over the preceding month results in an additional 0.96% return in the following month. However, this only occurs approximately 10% of the time and does not outperform a simple buy-and-hold strategy as the strategy has the investor out of the market the majority of the time.

Can the law of the natural distribution of random numbers expose malice in financial markets? This thesis aims to analyze the indices S&P 500 and STOXX 600, in an effort to identify days in which behavior in the market was the result of financial manipulation or non normal market movements. What was discovered by extending a previous study [10], was that we could accurately identify many days in which the market crashed or was affected by malpractice similar to the events in the 2007-2008 financial crisis.

This diploma thesis examines the possibilities of using the PE ratio in the creation of a stock portfolio on the Chinese and American stock market. The result of this work is the finding that low PE shares achieve higher risk-weighted returns over short and long investment horizons than shares with high PE values in both ex-amined markets. However, based on the detected volatility of the shares with the extreme values of PE, it is possible to recommend the use of this indicator for creation of the portfolio only to the most speculative investors.

Thesis advisor: Peter Ireland / It is not unreasonable to surmise that public sentiment views stock market behavior as an indicator of economic health. Historically, movements in the the stock market indeed correspond to business cycles, but this is not always the case, and the COVID-19 pandemic serves as a distinct case to highlight such an irregularity. The contrast between the behavior of the stock market and that of the economy during the pandemic compels an analysis of the pandemic's actual impact on the stock market, and this paper finds a negative and significant relationship between the interpolated daily closing prices of the S&P 500 and the daily number of COVID-19 cases. / Thesis (BA) — Boston College, 2023. / Submitted to: Boston College. College of Arts and Sciences. / Discipline: Departmental Honors. / Discipline: Economics.

本研究嘗試整合類神經網路與法則基礎（rule-based）系統技術，以建立S&P 500指數期貨的交易策略。本研究不同於先前研究之處有下列二方面：一、本研究採用法則基礎系統的方式提供神經網路的訓練範例；二、本研究以理解神經網路（Reasoning Neural Networks）取代後向傳導網路（Back propagation networks）以解決局部最小值與隱藏結點數未知的困境，而實證結果也顯示理解神經網路之表現優於後向傳導網路。首先，由期貨的日價格資料計算出十種技術分析指標值，用這些指標值來表示期貨市場內的各種可能狀況（case）。接著，我們提出FFM（Futures Forecast Model）與EFFM（Extended Futures Forecast Model）來處理市場的各種狀況，預測出隔日的期貨價格改變方向。以法則基礎方法所建立的FFM是用來處理明顯的狀況（obvious cases），並且提供類神經網路好的訓練範例。而EFFM包括四個理解神經網路系統與一個決策機置（voting mechanism），它被用來處理那些不明顯的狀況（non-obvious cases）。從實證模擬的結果顯示，在預測市場時FFM與EFFM有良好的合作 關係。因此，我們以FFM與EFFM為基礎建立一個整合的期貨交易系統（Integrated Futures Trading System，IFTS），並將它用於S&P 500 指數期貨市場作模擬交易，結果我們發現在1988到1993年的測試期間，IFTS 的投資報酬率高於買入持有投資策略。 / This research adopts a hybrid approach to implementing the trading strategies in the S&P 500 index futures market. The hybrid approach integrates both the rule-based systems technique and the neural networks technique. Our methodology is different from previous studies in two aspects. First, we employ Reasoning Neural Networks (RN) instead of back propagation networks to resolve the undesired predicaments of local minimum and the unknown of the number of hidden nodes. Second, the rule-based systems approach is applied to provide neural networks with good training examples. We, first, categorize the daily conditions of the futures market into a variety of cases through processing futures historical data. Then, the dual-forecast models, FFM (futures forecast model) and EFFM (extended futures forecast model), are proposed to predict the direction of daily price changes. The rule-based model, FFM, is designed to deal with the obvious cases and to provide the neural network-based model, EFFM, with good training examples. Meanwhile, EFFM, which consists of four RNs and a voting mechanism, is designed to handle the non-obvious cases. The simulation results show that the cooperation of FFM and EFFM does a good job in predicting the direction of daily price change of S&P 500 index futures. Based on FFM and EFFM, the integrated futures trading system (IFTS) is developed and employed to trade the S&P 500 index futures contracts. The results show that IFTS outperforms the passive buy-and-hold investment strategy over the six-year testing period from 1988 to 1993.

Sammanfattning Titel: Imorgon blir det börsfall! En studie om hur olika börser påverkar varandra i fördröjning. Seminariedatum: 28 maj, 2012 Ämne/kurs: FEK 61-90 Kandidatuppsats i Corporate Finance, 15 poäng  Författare: Stefan Koch, Malin Molin Handledare: Hans Mörner Examinator: Kent Sahlgren Nyckelord: Anomali, anomalier, veckodagseffekt, måndagseffekt, index, korrelation, S&P 500, OMXS 30, effektiva marknadshypotesen Syfte: Att undersöka ifall en eventuell anomali på det svenska OMXS 30 indexet eller det amerikanska S&P 500 ger en effekt på nästföljande dag på det motsatta indexet. Om en veckodagseffekt kan påvisas och den fördröjda korrelationen mellan indexen är tillräckligt stark kan metoden användas för att generera överavkastning.   Metod: Vi använder oss av en kvantitativ ansats för att med hjälp av statistiska metoder svara på vår problemformulering. De metoder vi har använt är hypotestestning av medelvärden med ett z test, beräknat korrelationskoefficienten mellan de två indexen och utfört en multipel regressionsanalys med dummyvariabler. Slutsats: Genom vår analys kom vi fram till att en veckodagseffekt inte kan påvisas på någon av de två undersökta indexen. En korrelation kunde finnas mellan de två indexen, däremot går det att ifrågasätta om korrelationens styrka är tillräckligt stark att handla utifrån. För att generera överavkastning krävs dessutom att den extra avkastning som genereras med hjälp utav vår metod med korta aktieaffärer överstiger den eventuella transaktionskostnaden som uppstår vid aktiehandel, något vi starkt betvivlar att den gör. / Abstract Title: Tomorrow the market falls! A study about how different stock markets affect each other in delay. Date of seminar: May, 28th, 2012 Course: FEK 61-90 Bachelor Thesis in Corporate Finance, 15 credits Authors: Stefan Koch, Malin Molin Advisor: Hans Mörner Examiner: Kent Sahlgren Key words: Anomaly, anomalies, day-of-the-week effect, weekend effect, index, correlation, S&P 500, OMXS 30, effective market hypothesis Objective: To examine whether a potential anomaly on the Swedish OMXS 30 index or the American S&P 500 has an effect on the next day on the opposite index. If a day-of-the-week effect can be proved and the delayed correlation between the indices is strong enough, our method could be used to generate excess returns. Methodology: We use a quantitative approach and statistic methods to answer our problem formulation. The methods we have been using are hypothesis testing of mean values with a z-test, calculations of correlation coefficients between the indices, and a multiple regression analysis with dummy variables. Conclusions: Through our analysis we found out that there was no day-of-the-week effect on any of the two examined indices. We could find a correlation between the two indices; however, we question whether the correlation is strong enough to trade on. To get excess returns it is required that the extra return that would be generated through our method with short trades exceed eventual transaction costs that occur through stock trading, something we strongly doubt that it would.

This Final thesis is focused on analysis of stock markets with more detailed view at S&P 500 index. The goal of market analysis is to reveal possible price bubble in relation to effect of quantitative easing, Federal Reserve Bank policy and behavioural factors. The content of practical part is to evaluate possible significant overvalue of S&P 500 index and possible price bubble of mentioned index.