在統計學上我們經常會遭遇到如下的問題： minimze subject to 其中 和Ｃ都是已知。 上述非線性規劃（NONLINEAR PROGRAMING）問題的最佳解，是相當複雜的，以致於我 們無法用簡單的式子，將其解明確的表示出來。 RAO-GHANGURDE （１９７２）在“從有限母體抽樣的貝氏最佳解”這一篇文章中，對 這種非線性規劃問題，提出一個反覆演算的解法，來解決這類問題，由於，我們無法 看出其演算法的立論根據何在，收斂結果的精確性有多高，於是，本文在ｋ＝２及ｋ ＝３的情形下，由直覺的幾何觀點，提出了另一個求最佳解的方法，來驗證RAO-GHAN GURED 反覆演算法的類確性。 最後，本論文將上述非線性規劃問題的解法，應用到下面兩個例子上： （ａ）在 COCHRAN的“抽樣技巧”（ SAMPLING TECHNIQUES）這一本書裡，有關雙重 抽樣（DOUBLE SAMPLING ）的理論中，也遭遇到要解決這一類問題，但由他的公式， 所計算出來的解，並不一定會萬足所需要的限制條件。 （ｂ）在SMITH-SEDRASK （１９８２）的“推估魚群年齡成份的貝氏最佳解“和JINN -SMITH-SEDRASK（１９８７）的“推估魚群年齡成份的貝氏最佳雙重抽樣”這兩篇的 文章中，同樣的也遭遇到這一類的問題。

The US sub-prime mortgage crisis greatly affected not only the US economy but also other countries in the world. This thesis employs the extreme value theory and Value at Risk (VaR) analysis to assess the impact of the US sub-prime mortgage crisis on various stock markets of the MSCI indexes, including 10 countries and 7 areas. It is reasonable to guess that VaR value should increase after the crisis. The empirical analyses on these indexes conclude that (1) the American market indexes not only do not agree with the guess after the crisis but four American indexes are identical; (2) not all the Asia market indexes consist with the guess; (3) the European market indexes agree with the guess; (4) MSCI AC PACIFIC, NEW ZEALAND, and AUSTRALIA consist with the guess; (5) the behavior for the positive log returns is different from that for the negative returns in some MSCI indexes. Over speaking, the impacts of US sub-prime mortgage crisis on those countries are not the same.

本篇論文使用了LHP的近似方法去評價擔保債權憑證，並推導出漸進極值因子模型，又稱單因子copula模型，單因子copula模型被廣泛運用在CDO之風險管理與一些風險因子模擬之應用，但由於2008年之金融海嘯造成市場標準模型Gaussian copula model會有評價上的誤差，所以為了能在市場不穩定時能更精確的求算出分券價差，我們必須找到一個更簡單且快速捕捉到市場不穩定性的模型。在這篇論文中，我們引用了Anna Schloesser在2009年所提出以NIG copula model為基礎的兩個延伸，讓模型更穩健和且擁有良好的性質去進行模擬，NIG Regime-Switch 模型有兩大特色: (i)可以用一致的方法去評價不同到期日的分券，放寬了同一分券必須是相同到期日的假設，和(ii)有不同的相關係數狀態，對於金融風暴來說，狀態轉換可以有效地降低市場不穩定所帶來的評價誤差。本文也對不同模型下的CDO進行風險分析與避險，分券的期望損失廣泛被信評公司視為一項審定信用評等重要的風險衡量指標，但是並無法真實反映出擔保債權憑證分券之間相對風險之大小，因此本文採用期望損失率的觀念，利用期望損失佔本金的比例來比較各分券之相對風險，且本文也求算出CDO之避險參數，讓投資人了解對合成行擔保債權憑證分券避險時所需之避險部位，分券持有人也可依據所要規避的風險類型，選擇市場上現有的信用違約交換指數或是單一資產之信用違約交換(single-name credit default swap)來進行避險。 / This paper presents the Large Homogeneous Portfolio (LHP) approach to the pricing of CDOs and we derive the one-factor copula model. It is popular that the one-factor copula models are very useful for risk management and measurement applications involving the generation of scenarios for the complete universe of risk factors. However, since the financial crisis in 2008 induces some errors in the valuation by Gaussian copula model, which is originally adopted by credit rating firms, it is necessary to have a simple and fast model that can capture the market unstableness. In this paper we apply two extensions of the NIG copula model, which are first present by Anna Schloesser (2009), since they make the model well defined and powerful for scenario simulation. The NIG Regime-Switch copula model allows for two important features: (i) tranches with different maturities modeled in a consistent way, and (ii) different correlation regimes. The regime-switching component of the NIG copula model is especially important in view of the financial crisis. This paper also targets on different models to conduct risk analysis and hedging strategy. The expected loss of tranches is widely used by credit rating organizations as one of the important indicators for risk measurement. However, it can’t reflect the relative risk level between CDO’s tranches. Therefore, our research adopts the concept of expected loss rate, which use the proportion of expected loss to total principal amount to compare the relative risk of each tranche. Moreover, when we want to hedge the spread risk of synthetic CDO tranches, the holders of tranches can choose the existing CDS index or the single-name CDS based on different risks types to hedge. The employment of the NIG Regime-Switch copula model not only has more precise estimation for the spread of tranches but also possess more stable hedge ratio to hedge.

影像中的文字訊息，通常包含著與場景內容相關的重要資訊，如地點、名稱、指示、警告等，因此如何有效地在影像中擷取文字區塊，進而解讀其意義，成為近來電腦視覺領域中相當受矚目的議題。然而在眾多的場景文字偵測方法裡，絕大多數是以英文為偵測目標語言，中文方面的研究相當稀少，而且辨識率遠不及英文。因此，本論文提出以中文和英文為偵測目標語言的方法，分成以下四個主要程序：一、前處理，利用雙邊濾波器(Bilateral filter)使文字區域更加穩定；二、候選文字資訊擷取，考慮文字特徵，選用Canny 邊緣偵測和最大穩定極值區域(Maximally Stable Extremal Region)，分別提取文字邊緣和區域特徵，並結合兩者來優化擷取的資訊；三、文字連結，依中文字結構和直式、橫式兩種書寫方向，設置幾何條件連結候選文字字串；四、候選字串分類，以SVM加入影像中文字的特徵，分類文字字串和非文字字串。使得此方法可以偵測中文和英文兩種語言，並且達到不錯的辨識效果。 / Text messages in an image usually contain useful information related to the scene, such as location, name, direction and warning. As such, robust and efficient scene text detection has gained increasing attention in the area of computer vision recently. However, most existing scene text detection methods are devised to process Latin-based languages. For the few researches that reported the investigation of Chinese text, the detection rate was inferior to the result for English. In this thesis, we propose a multilingual scene text detection algorithm for both Chinese and English. The method comprises of four stages: 1. Preprocessing by bilateral filter to make the text region more stable. 2. Extracting candidate text edge and region using Canny edge detector and Maximally Stable Extremal Region (MSER) respectively. Then combine these two features to achieve more robust results. 3. Linking candidate characters: considering both horizontal and vertical direction, character candidates are clustered into text candidates by using geometrical constraints. 4. Classifying candidate texts using support vector machine (SVM), the text and non-text areas are separated. Experimental results show that the proposed method detects both Chinese and English texts, and achieve satisfactory performance compared to those approaches designed only for English detection.

距離期望值4至6倍標準差以外的罕見機率電流值，是當前積體電路設計品質的關鍵之一，但隨著精確度的標準提升，實務上以蒙地卡羅方法模擬電路資料，因曠日廢時愈發不可行，而過去透過參數模型外插估計或迴歸分析方法，也因變數蒐集不易、操作電壓減小使得電流值尾端估計產生偏差，上述原因使得尾端電流值估計困難。因此本文引進統計方法改善罕見機率電流值的估計：先以Box-Cox轉換觀察值為近似常態，改善尾端分配值的估計，再以加權迴歸方法估計罕見電流值，其中迴歸解釋變數為Log或Z分數轉換的經驗累積機率，而加權方法採用Down-weight加重極值樣本資訊的重要性，此外，本研究也考慮能蒐集完整變數的情況，改以電路資料作為解釋變數進行加權迴歸。另一方面，本研究也採用極值理論作為估計方法。 本文先以電腦模擬評估各方法的優劣，假設母體分配為常態、T分配、Gamma分配，以均方誤差作為衡量指標，模擬結果驗證了加權迴歸方法的可行性。而後參考模擬結果決定篩選樣本方式進行實證研究，資料來源為新竹某科技公司，實證結果顯示加權迴歸配合Box-Cox轉換能以十萬筆樣本數，準確估計左、右尾機率10^(-4) 、10^(-5)、10^(-6)、10^(-7)極端電流值。其中右尾部分的加權迴歸解釋變數採用對數轉換，而左尾部分的加權迴歸解釋變數採用Z分數轉換，估計結果較為準確，又若能蒐集電路資訊作為解釋變數，在左尾部份可以有最準確的估計結果；而篩選樣本尾端1%和整筆資料的方式對於不同方法的估計準確度各有利弊，皆可考慮。另外，1%門檻值比例的極值理論能穩定且中等程度的估計不同電壓下的電流值，且有短程估計最準的趨勢。 / To obtain the tail distribution of current beyond 4 to 6 sigma is nowadays a key issue in integrated circuit (IC) design and computer simulation is a popular tool to estimate the tail values. Since creating rare events via simulation is time-consuming, often the linear extrapolation methods (such as regression analysis) are applied to enhance efficiency. However, it is shown from past work that the tail values is likely to behave differently if the operating voltage is getting lower. In this study, a statistical method is introduced to deal with the lower voltage case. The data are evaluated via the Box-Cox (or power) transformation and see if they need to be transformed into normally distributed data, following by weighted regression to extrapolate the tail values. In specific, the independent variable is the empirical CDF with logarithm or z-score transformation, and the weight is down-weight in order to emphasize the information of extreme values observations. In addition to regression analysis, Extreme Value Theory (EVT) is also adopted in the research. The computer simulation and data sets from a famous IC manufacturer in Hsinchu are used to evaluate the proposed method, with respect to mean squared error. In computer simulation, the data are assumed to be generated from normal, student t, or Gamma distribution. For empirical data, there are 10^8 observations and tail values with probabilities 10^(-4),10^(-5),10^(-6),10^(-7) are set to be the study goal given that only 10^5 observations are available. Comparing to the traditional methods and EVT, the proposed method has the best performance in estimating the tail probabilities. If the IC current is produced from regression equation and the information of independent variables can be provided, using the weighted regression can reach the best estimation for the left-tailed rare events. Also, using EVT can also produce accurate estimates provided that the tail probabilities to be estimated and the observations available are on the similar scale, e.g., probabilities 10^(-5)~10^(-7) vs.10^5 observations.

本論文推導違約風險下四種新奇選擇權的評價模型及其避險比率，依序為數據選擇權、寬它選擇權、互換選擇權，極值選擇權。並比較無違約風險與違約風險下的評價模型之差異。假若違約風險不存在時，違約風險下各種類型選擇權的評價模型皆會縮減成為無違約風險下所對應的評價模型。避險比率亦為如此。數值範例則印證違約風險下選擇權的價值較無違約風險選擇權的價值低。本論文完成目前尚無任何學術研究於違約風險下四種新奇選擇權的評價模型及避險比率。這是一個重要貢獻。 關鍵詞：違約風險、新奇選擇權、數據選擇權、寬它選擇權、互換選擇權、極值選擇權。 / This paper presents the analytic pricing formula and the hedging ratio of four kinds of exotic options with correlated credit risk. They are Digital options, Quanto Options, Exchange Options and Extreme-value Options, respectively. Furthermore, compare the discrepancy of the models under the condition whether the default risk exists. Finding that if there is no default risk, all models that we derive will reduce to the corresponding models with no default risks, and so do the hedging ratio. Numerical examples certify that the value of the vulnerable options will be lower than that of the ordinary options. All above that finished has not been done by existing researches and it is a chief contribution in this paper. Keywords: Exotic Options, Credit Risk, Digital Options, Quanto Options, Exchange Options, Extreme-value Options, Default Risk.

蒙地卡羅模擬是在組合信用風險的管理上相當實用的計算工具。衡量組合信用風險時，必須以適當的模型描述資產間的相依性。常態關聯結構是目前最廣為使用的模型，但實證研究認為 t 關聯結構更適合用於配適金融市場的資料。在本文中，我們採用 Bassamboo et al. (2008) 提出的極值相依模型建立 t 關聯結構用以捕捉資產之間的相關性。同時，為增進蒙地卡羅法之收斂速度，我們以 Chiang et al. (2007) 的重要性取樣法為基礎，將其拓展到極值相依模型下，並提出兩階段的重要性取樣技巧確保使用此方法估計一籃子信用違約時，所有模擬路徑均會發生信用事件。數值結果顯示，所提出的演算法皆達變異數縮減。而在模型自由度較低或是資產池較大的情況下，兩階段的重要性取樣法將會有更佳的估計效率。我們也以同樣的思路，提出用以估計投資組合損失機率的演算法。雖然所提出的演算法經過重要性取樣的技巧後仍無法使得欲估計的事件在所有模擬路徑下都會發生，但數值結果仍顯示所提出的方法估計效率遠遠優於傳統蒙地卡羅法。 / Monte Carlo simulation is a useful tool on portfolio credit risk management. When measuring portfolio credit risk, one should choose an appropriate model to characterize the dependence among all assets. Normal copula is the most widely used mechanism to capture this dependence structure, however, some emperical studies suggest that $t$-copula provides a better fit to market data than normal copula does. In this article, we use extremal depence model proposed by Bassamboo et al. (2008) to construct $t$-copula. We also extend the importance sampling (IS) procedure proposed by Chiang et al. (2007) to evaluate basket credit default swaps (BDS) with extremal dependence and introduce a two-step IS algorithm which ensures credit events always take place for every simulation path. Numerical results show that the proposed methods achieve variance reduction. If the model has lower degree of freedom, or the portfolio size is larger, the two-step IS method is more efficient. Following the same idea, we also propose algorithms to estimate the probability of portfolio losses. Althought the desired events may not occur for some simulations, even if the IS technique is applied, numerical results still show that the proposed method is much better than crude Monte Carlo.