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.

Dans le monde économique, les contrats d'options sont très utilisés car ils permettent de se couvrir contre les aléas et les risques dus aux fluctuations des prix des actifs sous-jacents. La détermination du prix de ces contrats est d'une grande importance pour les investisseurs.Dans cette thèse, on s'intéresse aux problèmes d'évaluation des options, en particulier les options Européennes et Quanto sur un actif financier dont le prix est modélisé en multi dimensions par un modèle de diffusion-saut à volatilité stochastique avec sauts (1er cas considère la volatilité sans sauts, dans le 2ème cas les sauts sont pris en compte, finalement dans le 3ème cas, l'actif sous-jacent est sans saut et la volatilité suit un CEV modèle sans saut). Ce modèle permet de mieux prendre en compte certains phénomènes observés dans les marchés. Nous développons des méthodes numériques qui déterminent les valeurs des prix de ces options. On présentera d'abord le modèle qui s'écrit sous la forme d'un système d'équations intégro-différentielles stochastiques "EIDS", et on étudiera l'existence et l'unicité de la solution de ce modèle en fonction de ses coefficients, puis on établira le lien entre le calcul du prix de l'option et la résolution de l'équation Intégro-différentielle partielle (EIDP). Ce lien, qui est basé sur la notion des générateurs infinitésimaux, nous permet d'utiliser différentes méthodes numériques pour l'évaluation des options considérées. Nous introduisons alors l'équation variationnelle associée aux EIDP et démontrons qu'elle admet une unique solution dans un espace de Sobolev avec poids en s'inspirant des travaux de Zhang [106].Nous nous concentrons ensuite sur l'approximation numérique du prix de l'option en considérant le problème dans un domaine borné, et nous utilisons pour la résolution numérique la méthode des éléments finis de type (P1), et un schéma d'Euler-Maruyama, pour se servir, d'une part de la méthode de différences finies en temps, et d'autre part de la méthode de Monté Carlo et la méthode Quasi Monte Carlo. Pour cette dernière méthode nous avons utilisé les suites de Halton afin d'améliorer la vitesse de convergence.Nous présenterons une étude comparative des différents résultats numériques obtenus dans plusieurs cas différents afin d'étudier la performance et l'efficacité des méthodes utilisées. / In the modern economic world, the options contracts are used because they allow to hedge against the vagaries and risks refers to fluctuations in the prices of the underlying assets. The determination of the price of these contracts is of great importance for investors.We are interested in problems of options pricing, actually the European and Quanto options on a financial asset. The price of that asset is modeled by a multi-dimentional jump diffusion with stochastic volatility. Otherwise, the first model considers the volatility as a continuous process and the second model considers it as a jump process. Finally in the 3rd model, the underlying asset is without jump and volatility follows a model CEV without jump. This model allow better to take into account some phenomena observed in the markets. We develop numerical methods that determine the values of prices for these options. We first write the model as an integro-differential stochastic equations system "EIDS", of which we study existence and unicity of solutions. Then we relate the resolution of PIDE to the computation of the option value. This link, which is based on the notion of infinitesimal generators, allows us to use different numerical methods. We therefore introduce the variational equation associated with the PIDE, and drawing on the work of Zhang [106], we show that it admits a unique solution in a weights Sobolev space We focus on the numerical approximation of the price of the option, by treating the problem in a bounded domain. We use the finite elements method of type (P1), and the scheme of Euler-Maruyama, for this serve, on the one hand the finite differences method in time, and on the other hand the method of Monte Carlo and the Quasi Monte Carlo method. For this last method we use of Halton sequences to improve the speed of convergence.We present a comparative study of the different numerical results in many different cases in order to investigate the performance and effectiveness of the used methods.

<p>We investigate some simulation-based approaches for the valuing of the employee stock options. The mathematical models that deal with valuation of such options include the work of Jennergren and Naeslund [L.P Jennergren and B. Naeslund, A comment on valuation of executive stock options and the FASB proposal, Accounting Review 68 (1993) 179-183]. They used the Black and Scholes [F. Black and M. Scholes, The pricing of options and corporate liabilities, Journal of Political Economy 81(1973) 637-659] and extended partial differential equation for an option that includes the early exercise. Some other major relevant works to this mini thesis are Hemmer et al. [T Hemmer, S. Matsunaga and T Shevlin, The influence of risk diversification on the early exercise of employee stock options by executive officers, Journal of Accounting and Economics 21(1) (1996) 45-68] and Baril et al. [C. Baril, L. Betancourt, J. Briggs, Valuing employee stock options under SFAS 123 R using the Black-Scholes-Merton and lattice model approaches, Journal of Accounting Education 25 (1-2) (2007) 88-101]. The underlying assets are studied under the GARCH (generalized autoregressive conditional heteroskedasticity) effects. Particular emphasis is made on the American style employee stock options.</p>

我國勞工退休金新制自2005年7月1日開始實施，由過去的確定給付制改為具有確定提撥特色的「個人帳戶制」。對於勞工而言，確定提撥制和過去確定給付制不同之處在於承擔退休金投資風險的責任將由雇主轉由個人承擔。如何透過退休機制的設計以降低退休金的投資風險是近年來的重要議題，因此本文主要從個人偏好與風險的觀點探討保證投資收益率制度與制度轉換選擇權等兩個降低確定提撥制投資風險的重要配套措施。 在本論文的第一篇研究中發現，資產配置與國際投資對保證成本的影響頗大，在個人可選擇資產配置的情況下，高投資風險的資產選擇將造成政府未來龐大的或有負債。為了解決政府保證成本過高造成代內與代間的財富移轉，本文從使用者付費與個人效用的觀點探討保證投資收益率制度的設計，發現藉由設立保證投資收益上限可提升風險趨避者、損失趨避者與後悔趨避者等偏好下的預期效用，且能降低個人管理下方風險所需的提撥成本與退休計畫參加者所需繳交的保證費用，故建議政府可將投資收益率上限納入保證投資收益率制度，供退休計畫參加者選擇合於本身偏好的保證收益率上限。 在近來許多國家的公、民營退休體系由過去以確定給付制改為確定提撥制，為了降低在確定提撥制下的退休金投資風險，在美國的佛羅里達州之公務人員退休體系中，存在著可供個人選擇是否轉換到確定給付制的機制。在我國勞退新制中除了從過去的確定給付制改為確定提撥制外，亦輔以「年金保險制」供勞工選擇與轉換，若年金保險制具有確定給付制的特徵，則勞工等於是擁有一個從確定提撥轉換轉到確定給付制的選擇權，因此制度選擇權的探討對我國而言亦是相當地重要。在本論文的第二篇研究中發現，當風險趨避程度越高則轉換至確定給付制的機率越高，轉換到確定給付制的高峰期會出現在開始工作的初期與屆臨退休之際等兩段期間。隨著工作期間的延長，個人轉換到確定給付制的機率越低，但仍可有效地提升退休金的所得替代率與達到降低退休金下方風險的效果，在加入退休制度初期不得轉換的限制之後，會降低轉換到確定給付制的機率。 / The Labor Retirement Pension Act enacted in 2005 introduced defined contribution (DC) pension plan for substituting the traditional defined benefit (DB) pension plan. In the defined contribution pension plan, the investment risk is transferred to the participants. However, the design of rate of return guarantee makes the investment risk less severe for participants. In the first essay, we find that the asset allocation and foreign investment have large impact on the guarantee cost: the high risky investment may result in large potential liability of the government in the future if participants have the investment portfolio choice. This study develops a framework to analyze design of rate of return guarantee from the financial engineering and user paid principle view. We find that the cap of investment return guarantee not only increases the expected utility of risk aversion, loss aversion and regret aversion, but also decreases the contribution cost to participant associated with managing the downside risk. Around the world, the defined contribution (DC) plans have been the primary trend of pension reform in the both public and private sector. In an attempt to decrease the investment risk associated with DC plan, the public employees are provided with an option to buy back DB plan in the Florida State of U.S.A. In the second essay, we find that the higher level of risk aversion is, the higher probability to buy back DB plan is. During the employee’s early years of service and as the employees near retirement, the probability to exercise the option is the highest. The probability to exercise the option is decreasing with the years of service being increasing; the option also increase the pension replacement rate as well as decrease the downside risk of pension. The probability to exercise the option is lower, when the option to buy back the DB plan is prohibited during the employee’s early years of service.

<p>We investigate some simulation-based approaches for the valuing of the employee stock options. The mathematical models that deal with valuation of such options include the work of Jennergren and Naeslund [L.P Jennergren and B. Naeslund, A comment on valuation of executive stock options and the FASB proposal, Accounting Review 68 (1993) 179-183]. They used the Black and Scholes [F. Black and M. Scholes, The pricing of options and corporate liabilities, Journal of Political Economy 81(1973) 637-659] and extended partial differential equation for an option that includes the early exercise. Some other major relevant works to this mini thesis are Hemmer et al. [T Hemmer, S. Matsunaga and T Shevlin, The influence of risk diversification on the early exercise of employee stock options by executive officers, Journal of Accounting and Economics 21(1) (1996) 45-68] and Baril et al. [C. Baril, L. Betancourt, J. Briggs, Valuing employee stock options under SFAS 123 R using the Black-Scholes-Merton and lattice model approaches, Journal of Accounting Education 25 (1-2) (2007) 88-101]. The underlying assets are studied under the GARCH (generalized autoregressive conditional heteroskedasticity) effects. Particular emphasis is made on the American style employee stock options.</p>

Black-Scholes選擇權定價模型是各種選擇定價的開山始祖，無論在理論或實務上均獲致許多的便利及好評，美中不足的是，這種既定模型下結構化參數的估計問題，在真實體系的結構訊息未知或是不明朗時，或是模式錯誤，亦或政治結構或金融環境不知時，該模型在實證資料的評價上會面臨價格偏誤的窘境。是故，許多的數值演算法（numerical algorithms）便因應而生，這些方法一則源於對此基本模型的修正，一則是屬於逼近的數值解。 評價選擇權的方法雖不一而足，然所有的這些理論或模型可分為二大類即模型驅動的理論（model-drive approach）及資料驅動的理論（data-driven approach）。前者是建構在許多重要的假設，當這些假設成立時，則選擇權的價格可用如Black-Scholes偏微分方程來表示，而後再用數值解法求算出，許多的數值方法即屬於此類的範疇；而資料驅動的理論（data-driven approach），其理論的特色是它的有效性（validity）不像前者是依其假設，職是之故，他在處理現實世界的財務資料時更顯見其具有極大的彈性。這些以計算智慧（computation intelligence）為主的財務計量方法，如人工神經網路（ANNs），遺傳演算法（GAs），遺傳規劃（GP）已在財務工程（financial engineering）領域上萌芽，並有日趨蓬勃的態勢，而將機器學習技術（machine learning techniques）應用在衍生性商品的定價，應是目前財務應用上最複雜及困難，亦是最富挑戰性的問題。 本文除了對現有文獻的整理評析外，在人工神經網路方面，除用於S&P 500的實證外，並用於台灣剛推行不久的認購構證評價之實證研究；而遺傳規劃在計算智慧發展的領域中，算是較年輕的一員，但發展卻相當的快速，雖目前在經濟及財務上已有一些文獻，但就目前所知的二篇文獻選擇權定價理論的文獻中，仍是試圖學習Black-Scholes選擇權定價模型，而本文則提出修正模型，使之成為完全以資料驅動的模型，應用於S&P 500實證，亦證實可行。最後，本文結合計算智慧中的遺傳演算法（ genetic algorithms）及數學上的加權殘差法（weight-residual method）來建構一條除二項式定價模型，人工神經網路定價模型，遺傳規劃定價模型等資料驅動模型之外的另一種具適應性學習能力的選擇權定價模式。 / The option pricing development rapid in recent years. However, the recent rapid development of theory and the application can be traced to the pathbreaking paper by Fischer Black and Myron Scholes(1973). In that pioneer paper, they provided the first explicit general equilibrium solution to the option pricing problem for simple calls and puts and formed a basis for the contingent claim asset pricing and many subsequent academic studies. Although the Black-Scholes option pricing model has enjoyed tremendous success both in practice and research, Nevertheless, it produce biased price estimates. So, many numerical algorithms have advanced to modify the basic model. I classified these traditional numerical algorithms and computational intelligence methods into two categories. Namely, the model-driven approach and the data-driven approach. The model-driven approach is built on several major assumptions. When these assumption hold, the option price usually can be described as a partial differential equation such as the Black-Scholes formula and can be solved numerically. Several numerical methods can be regarded as a member of this category. There are the Galerkin method, finite-difference method, Monte-Carlo method, etc. Another is the data-driven approach. The validity of this approach does not rests on the assumptions usually made for the model-driven one, and hence has a great flexibility in handling real world financial data. Artificial neural networks, genetic algorithms and genetic programming are a member of this approach. In my dissertation, I take a literature review about option pricing. I use artificial neural networks in S & P 500 index option and Taiwan stock call warrant pricing empirical study. On the other hand, genetic programming development rapid in recent three years, I modified the past model and contruct a data-driven genetic programming model. andThen, I usd it to S & P 500 index option empirical study. In the last, I combined genetic algorithms and weight-residual method to develop a option pricing model.

本論文提出線性規劃的方法以還原隱藏於選擇權市場價格中的風險中立機率測度，並利用該機率測度計算選擇權的合理價格。模型中假設選擇權對應同一標的資產與到期日，資產價格於到期日的狀態為離散點且個數有限，當市場不具任何套利機會時，以極小化市場價格與合理價格之離差總和作為挑選風險中立機率測度的準則。最後，以臺指選擇權(TXO)的交易資料做為實證對象。實證中發現，加入平滑限制式與離差權重之線性規劃模型在評價歐式選擇權合理價格的效能最為優異。 / The thesis proposes a liner programming to recover the risk-neutral probability distribution of an underlying asset price from its associated market option prices, and we evaluate the fair prices of options via the resulting risk-neutral probability distribution. Assume that we face a series of European options with different exercise prices on the same maturity and underlying asset in this linear programming model. The criterion of choosing a risk-neutral probability distribution is minimizing the sum of total deviations subject to requiring that the fair prices of options are consistent with observed market option prices. Finally, we take the trading data of TXO as an empirical study. The empirical study indicates that the model with smooth constraints and weighted deviations has the best performance in pricing the rational price of European options.

This thesis is concerned with the investigation of numerical methods for the solution of the Hamilton-Jacobi-Bellman (HJB) equations arising in European and American option pricing with proportional transaction costs. We first consider the problem of computing reservation purchase and write prices of a European option in the model proposed by Davis, Panas and Zariphopoulou [19]. It has been shown [19] that computing the reservation purchase and write prices of a European option involves solving three different fully nonlinear HJB equations. In this thesis, we propose a penalty approach combined with a finite difference scheme to solve the HJB equations. We first approximate each of the HJB equations by a quasi-linear second order partial differential equation containing two linear penalty terms with penalty parameters. We then develop a numerical scheme based on the finite differencing in both space and time for solving the penalized equation. We prove that there exists a unique viscosity solution to the penalized equation and the viscosity solution to the penalized equation converges to that of the original HJB equation as the penalty parameters tend to infinity. We also prove that the solution of the finite difference scheme converges to the viscosity solution of the penalized equation. Numerical results are given to demonstrate the effectiveness of the proposed method. We extend the penalty approach combined with a finite difference scheme to the HJB equations in the American option pricing model proposed by Davis and Zarphopoulou [20]. Numerical experiments are presented to illustrate the theoretical findings.

The study investigates whether the main capital market theories and pricing models provide a reasonably accurate description of the working and efficiency of capital markets, of the pricing of shares and options and the effect the risk/return relationship has on investor behaviour. The capital market theories and pricing models included in the study are Portfolio Theory, the Efficient Market Hypothesis (EMH), the Capital Asset Pricing Model (CAPM), the Arbitrage Pricing Theory (APT), Options Theory and the BlackScholes (8-S) Option Pricing Model. The main conclusion of the study is that the main capital market theories and pricing models, as reviewed in the study, do provide a reasonably accurate description of reality, but a number of anomalies and controversial issues still need to be resolved. The main recommendation of the study is that research into these theories and models should continue unabated, while the specific recommendations in a South African context are the following: ( 1) the benefits of global diversification for South African investors should continue to be investigated; (2) the level and degree of efficiency of the JSE Securities Exchange SA (JSE) should continue to be monitored, and it should be established whether alternative theories to the EMH provide complementary or better descriptions of the efficiency of the South African market; (3) both the CAPM and the APT should continue to be tested, both individually and jointly, in order to better understand the pricing mechanism of, and risk/return relationship on the JSE; (4) much South African research still needs to be conducted on the efficiency of the relatively new options market and the application of the B-S Option Pricing Model under South African conditions. / Financial Accounting / M. Com. (Accounting)