The unimodality of a function seems a simple concept. But in the Euclidean space R^m, m=3,4,..., it is not easy to define. We have an easy tool to find the minimum point of a unimodal function. The goal of this project is to formalize and support distinctive strategies that typically guarantee convergence. Support is given both by analytic arguments and simulation study. Application is envisioned in low-dimensional but non-trivial problems. The convergence of the proposed iterated grid search algorithm is presented along with the results of particular application studies. It has been recognized that the derivative methods, such as the Newton-type method, are not entirely satisfactory, so a variety of other tools are being considered as alternatives. Many other tools have been rejected because of apparent manipulative difficulties. But in our current research, we focus on the simple algorithm and the guaranteed convergence for unimodal function to avoid the possible chaotic behavior of the function. Furthermore, in case the loss function to be optimized is not unimodal, we suggest a weaker condition: almost (noisy) unimodality, under which the iterated grid search finds an estimated optimum point. / Ph. D.

This thesis explores continuous-time reinforcement learning (RL) for stochastic control with two intimately related problems: mean-variance (MV) portfolio selection and linear-quadratic (LQ) control. For the former, we investigate markets where stock prices are diffusion processes driven by observable factors that are also diffusion processes yet the coefficients of these processes are unknown. Based on the recently developed RL theory for diffusion processes, we present data-driven algorithms that learns the pre-committed investment strategies directly without attempting to learn or estimate the market coefficients. For multi-stock Black–Scholes markets without factors, we develop a baseline algorithm and prove its performance guarantee by deriving a sublinear regret bound in terms of the Sharpe ratio. To optimize performance and facilitate real-world application, we further adapt the baseline algorithm into four variants. These enhancements incorporate techniques such as real-time online learning, offline pre-training, and mechanisms for managing leverage constraints and trading frequency. Following this, we perform a comprehensive empirical study to compare our RL algorithms against fifteen established portfolio allocation strategies based on S&P 500 constituents. The study employs multiple performance metrics, including annualized returns, variations of the Sharpe ratio, maximum drawdown, and recovery time. The results demonstrate that our continuous-time RL strategies are consistently among the best especially in a volatile bear market, and decisively outperform the model-based continuous-time counterparts by significant margins. We next study RL for a class of continuous-time LQ control problems for diffusions, where states are scalar-valued and running control rewards are absent but volatilities of the state processes depend on both state and control variables. We apply a model-free approach that relies neither on knowledge of model parameters nor on their estimations, and devise an actor--critic algorithm to learn the optimal policy parameter directly. Our main contributions include the introduction of an exploration schedule and a regret analysis of the proposed algorithm. We provide the convergence rate of the policy parameter to the optimal one, and prove that the algorithm achieves a regret bound of 𝑂(𝑁³/⁴) up to a logarithmic factor, where N is the number of learning episodes. We conduct a simulation study to validate the theoretical results and demonstrate the effectiveness and reliability of the proposed algorithm. We also perform numerical comparisons between our method and those of the recent model-based stochastic LQ RL studies adapted to the state- and control-dependent volatility setting, demonstrating a better performance of the former in terms of regret bounds. Along a different direction, we present a policy gradient-based actor–critic algorithm featuring adaptive exploration in both actor and critic. To wit, both the variance of the stochastic policy (actor) and the temperature parameter (critic) are decreasing in time according to certain schedules. In particular, endogenizing the temperature parameter reduces the need for manual tuning. Despite this added flexibility, the algorithm maintains the same sublinear regret bound of 𝑂(𝑁³/⁴) as achieved in the deterministic schedule. In the numerical experiments, we evaluate the convergence rate and regret bound of the proposed algorithm, with results aligning closely with our theoretical findings.

[pt] Consideramos produtos tortos sobre shifts de Bernoulli, cuja dinâmica fibrada é dada por difeomorfismos do intervalo. Estudamos o comportamento previsível e/ou histórico destes sistemas, referindo-nos à convergência e/ou não convergência, da média de Birkhoff, respectivamente. Utilizamos a derivada Schwarziana das fibras e a lei do arco-seno para identificar condições nas quais esses produtos tortos apresentam esses tipos de comportamento. Identificamos distintos tipos de comportamento em relação à derivada Schwarziana. Quando a derivada Schwarziana é negativa, o produto torto tem bacias entrelaçadas. Por outro lado, quando a derivada Schwarziana é positiva, o produto torto possui uma medida física. Finalmente, quando a derivada Schwarziana é nula, o produto torto tem comportamento histórico. No último cenário, estabelecemos uma conexão entre o comportamento histórico e a lei do arco-seno que nos permite obter resultados em outras configurações independentes do sinal da derivada Schwarziana. / [en] We consider skew products over Bernoulli shifts, whose fibred dynamics is given by diffeomorphisms of the interval. We study the predictable and/or historical behavior, referring to convergence and/or non-convergence, of the Birkhoff average, respectively. We employ the Schwarzian derivative of the fiber maps and the arc-sine law to identify conditions under which these skew products exhibit these types of behavior. We identify distinct types of behavior according to the Schwarzian derivative. When the Schwarzian derivative is negative, the skew product has intermingled basins. Conversely, when the Schwarzian derivative is positive, the skew product has a physical measure. Finally, when the Schwarzian derivative is zero, the skew product has historical behavior. In the latter scenario, we establish a connection between historical behavior and the arc-sine law that allows us to obtain results in other settings independent of the sign of the Schwarzian derivative.

Die vorliegende Dissertation ist in drei Themenblöcke unterteilt. Im Mittelpunkt des ersten Themenblocks steht die vergleichende Analyse der Fondsregulierung in den USA und Deutschland/der EU in Bezug auf Derivate und Verschuldung vor, während und nach der Finanzkrise 2007-2009. Ziel ist es, anhand der Darstellung der geltenden Regulierung und ihrer Anwendbarkeit auf die hypothetische Nutzung von Credit Default Swaps (CDS) durch Anleihefonds, aufzuzeigen, wie viel Flexibilität Fonds in beiden Ländern haben. Insgesamt ist der aus der Fondsregulierung erwachsende Spielraum im Hinblick auf den Einsatz von Derivaten und Verschuldung für Fonds in beiden Ländern hoch, so dass Fonds unbeobachtet ihre Zusammensetzung in Richtung risikoreicherer Kapitalanlagen lenken könnten. Der zweite Themenblock beschäftigt sich empirisch mit der Frage, in wie weit Fonds ihre Flexibilität tatsächlich ausnutzen. Als Erstes wird dabei untersucht, wie hoch das Verlustpotential der größten Fonds in den USA und Deutschland aus CDS ist. Ferner wird analysiert, ob Fondskommentare in Jahres- und Halbjahresberichten bezogen auf deren Nutzung von CDS mit den tatsächlichen CDS-Beständen konsistent sind. Basierend auf den Resultaten ist es zu empfehlen, nicht nur bestehende Regeln im Hinblick auf die spekulative Anwendung von Derivaten angemessen zu verschärfen, sondern auch die Publizitätspflichten in beiden Ländern weiter zu standardisieren. Unter Heranziehung umfangreicher Fondscharakteristika, insbesondere Managercharakteristika, untersucht der dritte Themenblock, was die Entscheidung der US Anleihefonds CDS zu benutzen, deren Nutzung zu erweitern sowie die Nutzungsweise beeinflusst. Ferner werden die exakten Typen von CDS, die von Anleihefonds gehalten werden, wie z. B. long oder short CDS, die sich auf Einzelnamen oder Gruppen von Titeln beziehen, aufgezeigt. / The thesis consists of three parts. The first part analyzes the regulation at the time surrounding the 2007-2009 financial crisis and after with respect to leverage and derivative holdings for mutual funds in the U.S. and Germany/the EU. After presenting a detailed overview of U.S. and German/European regulations, this study thoroughly compares the levels of flexibility funds have in both countries. All analyses reveal that under existing derivative and leverage regulation, funds in both countries are able to increase risk by using derivatives up to the point at which it is possible for them to default solely due to investments in derivatives. This makes the issue of regulation highly relevant for the public and regulators. The second part builds upon the first and empirically investigates the level of credit derivatives use by funds together with their communication toward investors. Firstly, the loss potential arising from investments into CDS for a sample of large U.S. and German mutual funds is analyzed. Secondly, it is investigated whether comments on CDS use contained in periodic fund reports are consistent with the disclosed CDS holdings. Based on the results, it seems advisable that regulators in both countries tighten rules restricting the speculative use of derivatives by funds to a reasonable level, as well as implement more standardized disclosure policies. The third part analyzes what determines whether U.S. corporate bond funds decide to use CDS in a particular period between mid-2004 to 2010, to which extent they use them and how, by relying on various fund characteristics including an extended set of manager variables. In addition, the types of various credit derivatives that funds use (e.g. long and short CDS on single-name or multi-name underlying positions) are presented. The results suggest that the characteristics of fund managers affect a fund’s risk taking via derivatives, in addition to fund fundamentals.

The agricultural sector plays an important role in the South African economy through job creation and earning foreign exchange. The role of agricultural co-operatives increased substantially over the last few decades. The research focuses firstly on the identification of derivative instruments in the market and their applicability to mitigate financial risks co-operatives experience. Secondly, research is conducted about the extent to which co-operatives use these derivatives to hedge financial risks. The research shows that most co-operatives are exposed to financial risks through different activities. It is, however, evident that although the derivative instruments are available, not all co-operatives make use of these instruments. Recommendations for further research include the development of a risk management framework and determining the different economic factors that have an influence on the use of derivatives by South African agricultural co-operatives. / Business Management / M.Comm.

by Ng Wing-Sze Cecilia. / Thesis (M.B.A.)--Chinese University of Hong Kong, 2003. / Includes bibliographical references (leaves 92-93). / ABSTRACT --- p.ii / TABLE OF CONTENT --- p.iii / CHAPTER / Chapter 1. --- COMPANY PROFILE --- p.1 / The Walt Disney Parks --- p.1 / Hong Kong Disneyland --- p.1 / Location --- p.1 / Park Developer & Operator --- p.2 / Financing --- p.2 / Infrastructure --- p.3 / Schedule of Operation --- p.4 / Chapter 2. --- HONG KONG DISNEYLAND BUSINESS MODEL --- p.6 / Revenue Model --- p.7 / Customer Base --- p.7 / Pricing Strategy --- p.8 / Financial Performance Variable --- p.9 / Risk Management Program --- p.10 / The Walt Disney Company Risk Management --- p.10 / HKDL Risk Management --- p.13 / Risk Management on Book Record --- p.13 / Chapter 3. --- PRECIPITATION RISK EXPOSURE --- p.15 / Introduction to Precipitation --- p.15 / Distinguish between Weather and Climate --- p.16 / Rainfall Risk Exposure --- p.16 / Precipitation in Hong Kong --- p.17 / Overview --- p.17 / Rainstorm Warning System --- p.18 / Practices on Rainy Days --- p.20 / Theme Park Industry --- p.20 / The Ocean Park --- p.21 / Rainfall Risk Mitigation --- p.21 / Chapter 4. --- WEATHER DERIVATIVES --- p.24 / Evolution --- p.24 / The Birth of Weather Derivatives --- p.24 / Weather Risk Management Association --- p.24 / Year 1999 --- p.25 / Year 2000 --- p.25 / Year 2001 --- p.26 / Year 2002 --- p.26 / Precipitation Derivatives --- p.27 / Market & Market Players --- p.28 / Types of Product --- p.30 / Index Derivatives --- p.30 / Event-Basis Derivatives --- p.32 / Chapter 5. --- Hedging Against Rainfall Risk with Weather Derivatives --- p.33 / Formation of Hedging Strategy --- p.34 / Hedging Objectives --- p.34 / Hedging Target --- p.35 / Dimension of Precipitation Impacts --- p.35 / Normal Revenue without Rainfall Risk --- p.40 / Revenue Forecasting for Year 1 --- p.41 / Specifications on the Contracts --- p.46 / Chapter 6. --- General Recommendations to HKDL for hedging with all kinds of Rainfall Derivatives --- p.49 / Choice of Market and Counter Parties --- p.49 / Index Model Design --- p.50 / Dimensions of Variables & Time Scale --- p.50 / Accumulated Rainfall Index --- p.51 / Methodologies of Rainfall Measurements --- p.54 / Location of Rainfall Measuring Stations --- p.54 / Measuring Instrument --- p.56 / Historical Data Consistency --- p.58 / Data Availability and Reliability --- p.59 / Choice of Strike Level --- p.59 / Tick Size and Maximum Payments --- p.62 / Pricing Approach --- p.63 / Chapter 7. --- Example of Rainfall Derivatives --- p.66 / Black/Red Rainstorm Signal Call --- p.66 / Specifications --- p.66 / Revenue model under Different Scenario --- p.68 / Chapter 8. --- Portfolio Management --- p.70 / Risk Management Information System --- p.70 / Issues on Book Keeping --- p.71 / Chapter 9. --- CONCULSION --- p.72

The agricultural sector plays an important role in the South African economy through job creation and earning foreign exchange. The role of agricultural co-operatives increased substantially over the last few decades. The research focuses firstly on the identification of derivative instruments in the market and their applicability to mitigate financial risks co-operatives experience. Secondly, research is conducted about the extent to which co-operatives use these derivatives to hedge financial risks. The research shows that most co-operatives are exposed to financial risks through different activities. It is, however, evident that although the derivative instruments are available, not all co-operatives make use of these instruments. Recommendations for further research include the development of a risk management framework and determining the different economic factors that have an influence on the use of derivatives by South African agricultural co-operatives. / Business Management / M.Comm.

This paper concentrates on the description of fractional order LTI SISO systems using generalized Laguerre functions. There are properties of generalized Laguerre functions described in the paper, and an orthogonal base of these functions is shown. Next the concept of fractional derivatives is explained. The last part of this paper deals with the representation of fractional order LTI SISO systems using generalized Laguerre functions. Several examples were solved to demonstrate the benefits of using these functions for the representation of LTI SISO systems.

Agriculture is of inestimable value to South Africa because it is a major source of job creation and plays a key role in earning foreign exchange. The most significant contribution of agriculture, and in particular maize, is its ability to provide food for the nation. For a number of decades government legislation determined prices, and as such the trade of grains on the futures exchange requires market participants to adapt to a volatile environment. The research focuses on the ability of market participants to effectively mitigate price volatility on the futures exchange through the use of derivative instruments, and the possibility of developing risk management strategies that will outperform the return offered by the market. The study shows that market participants are unable to use derivative instruments in such a way that price volatility is minimised. The findings of the study also indicate that the development of derivative risk management strategies could result in better returns than those offered by the market, mainly by exploiting trends on the futures market. / Financial Accounting / M. Comm. (Business Management)

企業為了避免公司價值的變動受到匯率及利率風險的影響，一般而言均會採用衍生性金融商品進行避險，而台灣企業在衍生性金融商品市場逐漸成熟的這段期間中，也逐漸加入這股使用浪潮中，但是，究竟什麼才是決定企業是否避險的重要因素呢？什麼才是企業在取捨不同衍生性金融商品作為避險工具時的考量呢？ 在此一研究領域中，存在許多國內外文獻進行探討，由於國內文獻大多著重於研究公司是否使用衍生性金融商品避險的重要決定因子，探討與何種理論相符，較少文獻針對公司在面臨外匯風險及利率風險後，如何選擇衍生性金融商品作為避險工具的議題加以研究，有鑑於此，本文著重於研究期貨、選擇權、遠期外匯、交換等衍生性金融商品的使用與公司特徵間的關係，探討是否與公司所面臨的匯率風險或利率風險相關？此外，2001年後海外可轉換公司債逐漸成為國內企業重要的融資工具，尤以上市電子公司最為重要，因此，本文的另一重點為探討「海外可轉換公司債」對企業避險行為及避險工具選擇之影響，並以台灣上市電子公司為研究樣本。最後，本文研究期間為1995年第1季∼2002年第2季，藉以觀察1997年下半年發生的東亞金融風暴及2001年中央銀行連續調降利率政策對企業避險行為及避險工具選擇之影響。 實證結果發現公司規模、長期負債比率、內部人持股比率、流動比率及外銷比率為台灣上市電子業公司是否使用衍生性金融商品避險的重要決定因素。在衍生性金融商品市場之交易成本具經濟規模型態及涉外風險多樣化的推論下，實證結果發現公司規模及外銷比率與選擇權、遠期外匯、交換衍生性金融商品的使用呈顯著正相關。此外，實證結果顯示台灣上市電子公司為規避發行海外可轉換公司債伴隨的外匯及利率風險，將傾向使用適合不確定性交易的選擇權及交換這兩種衍生性金融商品進行避險。由於長期負債對利率的敏感度較短期負債為高，所以長期負債的比率越高時，公司將會使用利率交換作為管理利率風險的工具。 東亞金融風暴後，台灣上市電子公司決定是否從事避險時，除考量公司規模外，更注重長期負債比率，以降低因東亞外匯市場大幅波動提高的財務危機成本。此外，外銷比率和公司避險決策呈十分顯著的正向關係，而且根據實證結果，避險工具含括選擇權、遠期外匯及交換，意味著國際化程度越高的公司在東亞金融風暴發生後意識到同時控管外匯風險及利率風險的重要。至於台灣上市電子公司規避外匯風險的工具，也因東亞金融風暴產生重要的衝擊，實證結果發現台灣上市電子公司在規避匯率風險時，將一部份從遠期外匯衍生性金融商品的使用轉為選擇權衍生性金融商品的使用，原因可能為在國內外情勢未明的情況下，國內上市電子公司貿然使用遠期外匯避險，反而可能使公司暴露於更大的匯率風險中，因此，公司將會使用較具彈性的選擇權作為避險工具。2001年中央銀行連續調降利率後，海外可轉換公司債和公司避險與否呈現顯著正相關，至於規避風險的種類，經由實證結果發現為利率風險。