給定聯合分配，可以容易地導出對應的條件分配。反之，給定條件分配的資訊，是否能導出對應的聯合分配呢？例如根據O. Paul et al.（1963，1968）對造成心血管疾病因素之追蹤研究，可得出咖啡量、吸菸量及是否有心血管疾病三者間的條件機率模型資料，是否能找到對應的聯合機率模型，以便可以更深入地研究三者之關係，是一個重要的議題。在選定參考點下，Chen（2010）提出以勝算比法找條件密度函數相容的充要條件，以及在相容性成立時，如何求得聯合分配。在二維中，當兩正值條件機率矩陣不相容時，郭俊佑（2013）以幾何平均法修正勝算比矩陣，並導出近似聯合分配，同時利用幾何平均法之特性，提出最佳參考點之選擇法則。本研究以二維的勝算比法為基礎，探討三維離散的相容性問題，獲得下列幾項結果：一、證明了三個三維條件機率矩陣相容的充要條件就是兩兩相容。二、當三維條件機率矩陣不相容時，利用幾何平均法導出近似聯合分配。三、利用兩兩相容的充要條件，導出三維條件機率矩陣相容的充要條件，並證明該充要條件與Chen的結果一致。四、在幾何平均法下，提出最少點法，有效率地找出最佳參考點，以產生總誤差最小的近似聯合分配。五、設計出程式檢驗三維條件機率矩陣是否相容，並找出最佳參考點，同時比較最少點法與窮舉法之間效率的差異。 / Given a joint distribution, we can easily derive the corresponding fully conditional distributions. Conversely, given fully conditional distributions, can we find out the corresponding joint distribution? For example, according to a longitudinal study of coronary heart disease risk factors by O. Paul et al. (1963, 1968), we obtain conditional probability model data among coffee intake, the number of cigarettes smoked and whether he/she has coronary heart disease or not. Whether we can find out the corresponding joint distribution is an important issue as the joint distribution may be used to do further analyses. Chen (2010) used odds ratio method to find a necessary and sufficient condition for their compatibility and also gave the corresponding joint distribution for compatible situations. When two positive discrete conditional distributions in two dimensions are incompatible, Kuo (2013) used a geometric mean method to modify odds ratio matrices and derived an approximate joint distribution. Kuo also provided a rule to find the best reference point when the geometric mean method is used. In this research, based on odds ratio method in two dimensions, we discuss their compatibility problems and obtain the following results on three-dimensional discrete cases. Firstly, we prove that a necessary and sufficient condition for the compatibility of three conditional probability matrices in three dimensions is pairwise compatible. Secondly, we extend Kuo’s method on two-dimensional cases to derive three-dimensional approximate joint distributions for incompatible situations. Thirdly, we derive a necessary and sufficient condition for the compatibility of three conditional probability matrices in three dimensions in terms of pairwise compatibility and also prove that this condition is consistent with Chen’s results. Fourthly, we provide a minimum-points method to efficiently find the best reference point and yield an approximate joint distribution such that total error is the smallest. Fifthly, we design a computer program to run three-dimensional discrete conditional probability matrices problems for compatibility and also compare the efficiency between minimum-points method and exhausting method.

違約機率校準檢定 - global test 由兩部分組成：第一部分為 level，探討真實的平均違約機率是否被高估；第二部分 shape，探討高低違約機率的表現情形。但 global test 與相關違約事件下的 level test 檢定尺度皆遠高於顯著水準 $\alpha$。本文先是針對相關違約事件，利用截斷分配使 level test 犯型一誤差機率更接近顯著水準，並提出虛無假設及對立假設為 $H_0: \theta \in \cup_{i=1}^2 \Theta_{i0}$ vs. $H_1: \theta \in \cap_{i=1}^2 \Theta_{i1}$ 的形式，引用交聯集檢定。更進一步透過 Liu \& Berger (1995, \textit{The Annals of Statistics}, 23, 1, 55-72) 建構齊一較強檢力檢定，改善檢定力。模擬結果顯示交聯集檢定與齊一較強檢力檢定的檢定尺度皆為 $\alpha$，且齊一較強檢力檢定的檢定力皆高於交聯集檢定。 / The calibration test of the PDs (probabilities of default) --- global test is twofold, the first part is the level test, which is about the mean of calibrated PDs. Second, the shape test is about whether a calibrated PD model differentiates correctly between low and high default probability events. In simulation results, we found that the type I error of global test is much greater than significant level $\alpha$, so is level test in correlation default events. In this study, firstly, we use the truncated level test to control previous error and suggest the hypothesis $H_0: \theta \in \cup_{i=1}^2 \Theta_{i0}$ vs. $H_1: \theta \in \cap_{i=1}^2 \Theta_{i1}$. Secondly, we introduce the intersection union test (IUT). Moreover, we construct an uniformly more powerful test (UMP test) by Liu \& Berger (1995, \textit{The Annals of Statistics}, 23, 1, 55-72). Simulation results show that the IUT and UMP test are size $\alpha$ tests, and the power of UMP test is greater than IUT.

在生態學、生物學、和比較文學的研究中，物種個數通常是評估生物多樣性的重要指標，單一群落物種數的估計已有非常豐富的相關研究。較為知名者包括Good (1953)提出未出現物種的機率，作為估計物種數的參考，往後Good的想法被大量延伸，推演出不少新的估計方法，像是Burnham and Overton (1978)的Jackknife估計法，Chao and Lee (1992)利用涵蓋機率的估計。相對而言，兩群落共有物種數的研究較少，現有研究中較為知名的有Chao et al. (2000)的估計式。 本研究延伸Good想法，探討Jackknife估計法在兩群落的應用，以出現一次的共有物種（一階Jackknife估計），推估未出現共有物種機率，並且仿造Burnham and Overton的想法，建立共有物種數的估計值及變異數。本文除了以電腦模擬，也使用實例（包括：金庸武俠小說、台灣野生水鳥、巴拿馬螃蟹和巴洛科羅拉多森林）檢驗本文的Jackknife估計法，利用涵蓋機率角度發現抽出某特定比例樣本時，估計值涵蓋母體共有物種數之機率值達到九成以上，且也與Chao提出的估計值比較。 / The number of species is frequently used to measure the biodiversity of a population in ecology, biology, and comparative literature. There are quite a lot of studies related to estimating the number of species. Among these studies, Good (1953) proposed a famous estimate (Turing’s estimate) for the probability of unseen species. Subsequently, many methods have been proposed for estimating the number of species based on Good’s idea. For example, the Jackknife estimator by Burnham and Overton (1978) and sample coverage probability by Chao and Lee (1992) are two famous estimates for the number of species. In contrast, there are not many studies for the number of shared species in two communities, and Chao et al. (2000) is probably the only one. This article extends Good’s idea and the Jackknife method to estimate the number of shared species in two communities. Similar to Burnham and Overton, we establish the estimate and its estimated variance, based on the number of species appearing exactly once. We also use computer simulation and real data sets (Jin-Yong martial arts novels, Taiwan wild birds, Panama crustacean, and Barro Colorado Island forest) to evaluate the proposed method. We found that the coverage probability for confidence interval covering the true number of shared species is more than 90%. In addition, we compare the proposed method with Chao’s method.

本論文研究如何由觀測的選擇權市場價格還原風險中立機率測度（等價平賭測度）。首先建構選擇權投資組合的套利模型，其中假設選擇權為單期，到期日時的狀態為離散點且個數有限，並且對應同一標的資產且不同履約價格。若市場不存在套利機會時，可使用拉格朗日乘數法則將選擇權套利模型導出拉格朗日乘子的可行性問題。將可行性問題作為限制式重新建構線性規劃模型以還原風險中立機率測度，並且利用此風險中立機率測度評價選擇權的公正價格。最後，我們以台指選擇權（TXO）為例，驗證此模型的評價能力。 / This thesis investigates how to recover the risk-neutral probability (equivalent martingale measure) from observed market prices of options. It starts with building an arbitrage model of options portfolio in which the options are assumed to be in one-period time, finite discrete-states, and corresponding to the same underlying asset with different strike prices. If there is no arbitrage opportunity in the market, we can use Lagrangian multiplier method to obtain a Lagrangian multiplier feasibility problem from the arbitrage model. We employ the feasibility problem as the constraints to construct a linear programming model to recover the risk-neutral probability, and utilize this risk-neutral probability to evaluate the fair price of options. Finally, we take TXO as an example to verify the pricing ability of this model.

This paper emphasis on the importance of default correlation, and also illustrate how the concept is connected with the Basel Ⅱ framework’s intention. Moreover, the paper brought out the different methodologies used by practitioners to arrive at the default correlation calculation, namely, the dispute between asset correlation and equity correlation. Furthermore, based on the model proposed by Hamerle, Liebig, and Scheule (2004), a panel logit model is set up to capture the relationship between the default events and the risk components endured by the specific industry. The model is therefore used to test the applicability of such model using Taiwan’s banking industry data. The result is consistent with our expectation about including the macroeconomic variables which will help to explain the default events happened within the banking industry. But, to my surprise, the proposition about the contemporary systematic random risk effect seems to be insignificant and a fixed effect is suggested to be assumed instead.

長久以來，信用風險一直是各銀行經營風險中最主要的來源，而就信用風險的衡量部份，巴塞爾委員會希望國際性銀行最低限度必須採用中等複雜程度的風險計算方法。也就是希望銀行能以新巴塞爾協定中信用風險的內部評等法為基本精神建置一套內部自有的信用風險模型來評估交易對手的信用風險。 同時，由於目前國內對於自有信用風險模型的建置和效力驗證的相關研究付之闕如，故本研究以新巴塞爾協定中信用風險的內部評等基礎法為基本精神，並且應用倒傳遞類神經網路方法，建構一套有效的信用風險模型並加以驗證以期能應用於銀行授信決策系統之中，也擬扮演一拋磚引玉的角色，以期未來有更多資源投入相關研究。 首先，本研究藉由文獻探討的方式，決定模型的輸入變數，接著利用ROE來做為評斷企業總體財務表現的指標，同時使用來對上市/櫃公司進行評分，根據評分的結果，再使用K-Means方法來針對所有ROE值為正的上市/櫃公司進行評等等級的切割，以計算所有上市/櫃公司各年度的評等。 研究結果發現： (1) 利用建模資料帶入模型，分別計算每一筆資料的違約機率，也就是該公司當年度的違約機率，再將每一個等級的所有資料的PD值求平均數，即可得到代表該等級的違約機率，而此估計出的違約機率也的確能隨著評等等級的遞增而增加。 因此，當我們要判斷一間公司的違約等級時，可利用本研究所建構出的信用評等模型，估計出該公司違約機率，以判斷該公司的違約等級，以為決策者提供重要的參考依據。 (2) 信用風險預警模型在預測公司下一年度違約與否的能力上，也有不錯的預測準確率；同時，本研究利用預測結果的型I誤差、型II誤差、模型區別率和模型預測率分析來分析預警模型的效度，經實證結果得知，預警模型在效度驗證方面也能有效滿足要求。 由以上的結果得知，本研究所自行發展的信用風險評等模型與信用預警模型相關建構流程、架構與方法論，可有效應用於銀行授信決策系統之中。

在現今關於公司現金決策的文獻中，多數文獻以預防性動機 (precautionary motive) 或是代理問題動機 (agency motive) 來解釋公司為何要存錢，但至今對於公司存錢的動機仍未有一致的定論。在本篇文章中，我們用籌資不易公司 (financially constrained firm) 在2000年網路泡沫以及2008年金融海嘯的表現來證明現金的預防性價值。我們發現事先有存錢的公司在上述事件期間仍能維持其原有的投資支出，反之事先沒存錢的公司在上述期間會擁有較低的股票報酬以及較高的違約風險。此外，本研究也發現曾在2000年網路泡沫期間遭遇資金問題的公司比較容易在2000年後開始存錢，這些預先存錢的公司在2008年金融海嘯來臨時也有相對較小的違約風險，顯示了公司在現金管理方面的學習行為及其好處。綜上所述，本文以外生事件突顯了現金的預防性價值，證明公司的未雨綢繆確實有用。 / This article examines the role of pre-saved cash in helping financially constrained firms during the 2000 dot-com crash and the 2008 financial crisis, both of which were exogenous shocks to industrial firms. The results show that constrained firms tended to increase capital investments during these severe economic downturns if they had pre-saved more cash. Constrained firms instead exhibited lower excess returns and incurred higher likelihoods of financial distress during the severe downturns if they had saved less cash prior to the events. Firms that experienced the 2000 dot-com crash and saved cash thereafter were less likely to default during the 2008 financial crisis, indicating the existence and benefit of learning effects. This study supports a precautionary motive for cash savings, showing that pre-saved cash helps financially constrained firms fund investment and reduces the likelihood of financial distress during severe market downturns. It demonstrates that saving for a rainy day really is valuable.

衡量投資組合的稀有事件時，即使稀有事件違約的機率極低，但是卻隱含著高額資產違約時所帶來的重大損失，所以我們必須要精準地評估稀有事件的信用風險。本研究係在估計信用損失分配的尾端機率，模擬的模型包含同質模型與異質模型；然而蒙地卡羅法雖然在風險管理的計算上相當實用，但是估計機率極小的尾端機率時模擬不夠穩定，因此為增進模擬的效率，我們利用Glasserman and Li (Management Science, 51(11),2005)提出的重點取樣法，以及根據Chiang et al. (Joural of Derivatives, 15(2),2007)重點取樣法為基礎做延伸的改良式重點取樣法，兩種方法來對不同的投資組合做模擬，更是將改良式重點取樣法推廣至異質模型做討論，本文亦透過變異數縮減效果來衡量兩種方法的模擬效率。數值結果顯示，比起傳統的蒙地卡羅法，此兩種方法皆能達到變異數縮減，其中在同質模型下的改良式重點取樣法有很好的表現，模擬時間相當省時，而異質模型下的重點取樣法也具有良好的估計效率及模擬的穩定性。 / When measuring portfolio credit risk of rare-event, even though its default probabilities are low, it causes significant losses resulting from a large number of default. Therefore, we have to measure portfolio credit risk of rare-event accurately. In particular, our goal is estimating the tail of loss distribution. Models we simulate are including homogeneous models and heterogeneous models. However, Monte Carlo simulation is useful and widely used computational tool in risk management, but it is unstable especially estimating small tail probabilities. Hence, in order to improve the efficiency of simulation, we use importance sampling proposed by Glasserman and Li (Management Science, 51(11),2005) and modified importance sampling based on importance sampling which proposed by Chiang et al. (2007 Joural of Derivatives, 15(2),). Simulate different portfolios by these two of simulations. On top of that, we extend and discuss the modified importance sampling simulation to heterogeneous model. In this article, we measure efficiency of two simulations by variance reduction. Numerical results show that proposed methods are better than Monte Carlo and achieve variance reduction. In homogeneous model, modified importance sampling has excellent efficiency of estimating and saves time. In heterogeneous model, importance sampling also has great efficiency of estimating and stability.

論文名稱：公司信用風險之衡量 校所組別：國立政治大學金融研究所 畢業時間：九十年度第二學期 提要別：碩士學位論文提要 研究生：林妙宜 指導教授：陳松男博士 論文提要及內容： 信用風險一直是整體金融環境非常重要的一環，銀行授信、商業交易、投資評估，都會對信用風險做仔細的研究與評估。本論文以台灣的公司為樣本，採用會計財務比率與股票價格，主要兩項反映公司體質的資訊，建構信用風險模型，期望能提供台灣公司信用風險衡量上，公正而有效的指標。 以財務比率為基礎的區別分析模型，選取變數為獲利能力指標的常續性EPS、現金流量指標的現金流量對負債、成長率指標的盈餘成長率、償債能力指標的負債比率，與經營能力指標的平均收帳天數，這五項財務比率涵蓋企業繼續經營與財務狀況的各個層面。區別分析模型在財務危機前一年可達正確分類率91．67%。 以股票市場價格為基礎的選擇權模型，可由每日之股票價格求算出預期違約機率，將市場對公司價值的衡量轉化為信用風險的程度，能即時掌握公司體質的變化，做出適當之因應。 關鍵字：信用風險、財務危機、會計資訊、財務比率、區別分析、股票價格、選擇權模型、預期違約機率 / Title of Thesis: Corporate Credit Risk Measurement Name of Institute: Graduate Institute of Money and Banking, NCCU Graduate Date: June, 2002 Name of Student: Lin, Miao-Yi Advisor: Dr. Chen, Son-Nan Abstract: Credit Risk has been the great concern in the financial market. Before the bank grants a loan or the company makes deals and investment, they first consider the credit risk of the conterparty. The empirical study tries to construct the credit risk models based on the public firms in Taiwan. Using financial ratios and stock prices, the two main sources of corporate financial information, we expect to provide a fair and efficient indicator to measure the corporate credit risk in Taiwan. In the discriminant analysis based on accounting data, the model chooses five financial ratios that cover the corporate operation and financial situation. They are earnings per share, operating cash flow to total debt, equity substantial growth rate, and average days to accounts receivable. The discrimanant analysis model can accurately classify 91.67% of the data as being default or solvency one year before the financial distress. In the option pricing model based on stock prices, the expected default probability can be solved by daily stock prices. In this model, how the market values the firm is turned into the level of credit risk, which can help us catch the changes of corporate soundness and make proper responses. Keywords: Credit Risk, Financial Distress, Accounting Data, Financial Ratio, Discrimanant Analysis, Stock Prices, Option Pricing Model, Expected Default Probability

Brigo與Mercurio提出了三種新的資產價格過程，分別是位移CEV過程、位移對數常態過程與混合對數常態過程。在這三種過程中，資產價格的波動度不再是一個固定的常數，而是時間與資產價格的明確函數。而由這三種過程所推導出來的歐式選擇權評價公式，將會導致隱含波動度曲線呈現傾斜曲線或是微笑曲線，且提供了參數讓我們能夠配適市場的波動度結構。本文利用台指買權來實證Brigo與Mercurio所提出的三種歐式選擇權評價公式，我們發現校準結果以混合對數常態過程優於位移CEV過程，而位移CEV過程則稍優於位移對數常態過程。因此，在實務校準時，我們建議以混合對數常態過程為台指買權的評價模型，以達到較佳的校準結果。 / Brigo and Mercurio proposed three types of asset-price dynamics which are shifted-CEV process, shifted-lognormal process and mixture-of-lognormals process respectively. In these three processes, the volatility of the asset price is no more a constant but a deterministic function of time and asset price. The European option pricing formulas derived from these three processes lead respectively to skew and smile in the term structure of implied volatilities. Also, the pricing formula provides several parameters for fitting the market volatility term structure. The thesis applies Taiwan’s call option to verifying these three pricing formulas proposed by Brigo and Mercurio. We find that the calibration result of mixture-of-lognormals process is better than the result of shifted-CEV process and the calibration result of shifted-CEV process is a little better than the result of shifted-lognormal process. Therefore, we recommend applying the pricing formula derived from mixture-of-lognormals process to getting a better calibration.