信用风险模型综述.doc

第二章 信用风险模型综述Chapter Two Credit Risk Model ReviewOverview因本文的研究工作主要集中在可违约债券期限结构模型，故本章的重点在于可违约债券期限结构模型的综述。其他信用风险模型，因为有两类在本论文有所涉及，所以也将对其作简单综述，即信用评分模型和实业界的信用风险管理模型。This dissertation focuses on the default-able bond term structure model, therefore the overview of the default-able bond term structure model is the main point of this chapter, the other two credit risk models-credit scoring model and credit VaR model, which are mentioned in the dissertation, are also overviewed .第一节 第一节 可违约债券期限结构模型Section1 The Default-able Bond Term Structure Model可违约债券期限结构模型的研究对象是在资本市场交易的可违约债券，研究对象的可交易特性要求模型必须考虑两种风险：信用违约风险和信用利差风险。可违约债券期限结构模型可以分成两类：1) 结构模型(Structural Model)，或者也可以称为公司价值模型(Firm Value Model)。2)强度模型(Intensity Model)，或者也可以称为简约模型（Reduced Form Model）。本节将对这两个流派的模型作详细地介绍和分析。总的来说，无论那一类模型，都包括了违约(default)过程、 回收(recovery)过程和利率过程。The study object of the default-able bond term structure model is the default-able bond which is traded in the capital market, so the model should handle consider two risks: credit default risk and credit spread risk. The default-able bond term structure model can be divided into two categories: 1) Structural Model, i.e.or Firm value model; 2) Intensity Model, i.e.or reduced form model. This section will introduce and analyze the two model categories in detail. In summaryTo sum up, in any of the models, there are default process, recovery process and interest rate process.违约过程：对这个过程的认识，不同类型的模型差别较大。结构模型假定资产符合某种随机过程，当资产到达某个阀值时便认为违约发生，违约率是内生的；而强度模型假定违约过程是一个具有某个违约强度的跳变过程，违约率是外生的。当然，两者之间有着本质上的联系，有些文献就把违约强度看成是资产的函数，如王莉君等(2003)。The default process is quite different in different models. The structural model supposes that the asset value follows some stochastic process, when the asset value reaches a certain cutoff,threshold the default probability is triggered, so the default is endogenetic. The intensity model supposes that the default process is a jump process with a certain default intensity, so the default probability is ectogenic. Of course, there is an intrinsical relationship between the two categories, for example, some papers supposes the default intensity is a function of the asset value, like those written by Wang Lijun(2003).回收过程：一般违约发生后，债权人并不会损失所有的债权资产，而是还能收回部分债权资产，通常用回收率表示收回的程度。目前的模型对于回收过程的假定都不尽相同，可以归结为以下三类：1)以面值回收，即回收资产=；2)以无风险债券的市值回收，即回收资产=；3)以可违约债券的市值回收，即回收资产=。上述符号本文在不同的模型间尽量使用相同的符号表示，所以如果同一符号在不同的模型中含义相同，将不会重复交待。而且本文在构建模型时也将尽量使用与综述一致的符号表示。，为回收率；为债券面值；为在到期时刻T支付1元的无风险债券在时刻t的价格；为在到期时刻T支付1元的可违约债券在时刻t的价格（时刻t以及之前尚未发生违约）。The recovery process: When a default takes placeoccurs, the creditor will not lose all of the loanassets, but claim back some parts of the loanpart of the assets, which is presented withby recovery rate. The assumption of the recovery process is different in different models. There are and can be divided into three categories: 1)recovery with redeemed at face value, the recovered value=; 2)recovery with redeemed atmarket value, which is face value that is the market capitalization discounted with risk-free interest rate, the recovered value=; 3) recovery with redeemed at default-able bond market value, the recovered value=.HereThis dissertation will try use same symbols among the different models, so if the symbol has same meaning in the different models, the dissertation will not define it again. And the dissertation will try use the same symbols in constructing the new model., is recovery rate, is face value, is the price of the risk-free bond at time t, which matures at time T, is the price of the default-able bond at time t, which matures at time T and has not defaulted at and before the time t.Skinner and Diaz (2000) 研究发现时变的回收率对正确定价是有一定的帮助，不过对于一般的产品其帮助是非常的有限，而对于那些支付依靠违约状态的信用风险衍生产品却是相当的重要。但实际上，目前大部分模型的回收率都没有考虑时变，例外的如Das and Tufano (1996)假定回收率随机变动。利率过程：由于研究对象是信用风险，所以利率过程一般都是直接地利用现有文献的模型，如Vasicek(1977), CIR(1985), Ho and Lee(1986), Hull and White (1990) and Heath, Jarrow, and Morton(1992)。需要研究的是利率过程与违约过程、回收过程之间的关系。如上文所说，利率对信用风险影响很大，所以在构造可违约债券期限结构模型时，相关性的研究是非常重要的。Skinner and Diaz (2000) found that the recovery rate that change with time is helpful for pricing, the help is very limited for general instrument, but is very importantcritial to for the credit derivatives whose payoffpayment depends on default statestatus. Now most of models do not suppose the recovery rate that changinge with time, except for Das and Tufano (1996).The interest rate process: because the study object is credit risk, the interest rate models are usually used in the interest rate process directly, such as Vasicek(1977), CIR(1985), Ho and Lee(1986), Hull and White (1990) and Heath, Jarrow, and Morton(1992). But the study about the relationship among the interest rate, the default process and the recovery process are needed. As mentioned in the above, the credit risk is strongly influenced badly by interest rate, so the study about correlation is very important.一、结构模型结构模型的研究开始于Merton(1974)的工作，他的开创性的工作奠定了结构模型的整个基础。Merton的主要思想是：可违约债券(如公司债券)可以等效成一个无风险债券 本文的无风险债券仅仅是指无违约风险的债券(default-free bond)，如国债，实际上无风险债券包括了利率风险等市场风险。减掉一个标的为该可违约债券发行人的资产市值的欧式看跌期权。设可违约债券在约定到期日的给付为F元，而A表示债券发行人的资产市值，则可违约债券的损益（Payoff）：I. Structural modelThe study of the structural model began in 1974 by the work of Merton(1977), who found the whole foundation of the structural model. Mertons main idea is that the default-able bond is equal to a default-able bond minus an European option value, whose underlying is the asset market value of the bond issuer.Suppose F is the face value of the default-able bond, A is the asset market value of the bond issuer, then the payoff of the default-able bond is (21)assetpayoff新巴塞尔协议Figure2-1F如果到期日债券发行人的资产超过或者等于F，那么债券持有人将得到F元；但是如果到期日债券发行人的资产少于F，那么债券持有人理论上能得到的最大值就是发行人的资产，即式子（21）。债券持有人的损益情况如图21。而图21还可以看成是一个标的为债券发行人资产的欧式看跌期权的发行人的损益，这里期权发行人的角色是由债券持有人充当的。If the asset market value is bigger than or equal to the face value F at the maturity, the bondholder will get F, but if the asset market value is smaller than the face value F, the maximum value the bondholder can receive is the asset of the bond issuer. The formula (2-1) and figurechart (2-1) illustrate the payoff, which is same as the payoff of European put option, which is issued by the bondholder, and the underlying of the option is asset market value of the bond issuer.所以，可以这样的看待可违约债券的交易：债券发行人卖给债券持有人一个无风险债券，同时债券持有人回卖给债券发行人一个欧式看跌期权，即期权持有人（债券发行人）有权在到期时以价格F元把他的所有资产卖给期权发行人（债券持有人）。这样我们利用Black and Scholes (1973)的期权理论就可以得到可违约债券的定价了，唯一不同的是，这里假定公司资产的市值习惯上，公司的市值就是指公司股票在证券市场的市值，但是这里的公司资产的市值是指在市场上公开出售公司资产的价值，注意区别。而不是公司股票价格符合几何布朗运动。ThereforeSo, the trading of the default-able bond can be viewed like this: the bond issuer sells a default-able bond to the bondholder, while the bondholder sells back a European put option, that is to say, the option holder(the bond issuer) has the right to sell his asset to the option issuer(the bondholder) at the price F. Then the price of the default-able bond can be achievedget by using the Black and Scholes(1973)s option theory, the only difference is the asset market valueUsually, the market value of the company means the market value of the company equity. Here, the market value means the market value of the company asset., not equity value following the Geometric Brown Motion.模型显著的特点就是违约率和回收率都是内生的，违约率就是到期日资产A小于债务F的概率，而如果违约发生，回收值就是资产A。尽管Merton的构思很巧妙，但其缺点也是很明显的，模型的很多假定与实际有比较大的差距。The characteristic of the model is that default probability and recovery rate are endogenetic, and default probability is the probability that the asset value A at maturity is less than debt F, and if default happens, the recovery value is asset value A.Though the idea of Merton is ingenious, the shortcoming is obvious, since quite a few of the assumptions of the model are different from the real world.1)假定公司资产市值符合几何布朗运动，但是公司资产不可能像股票那样在市场上得到频繁交易，假定几何布朗运动有点勉为其难了。实际上，更多的文章认为公司资产市值是一个跳变的过程。而且还有一个问题是公司资产价值并没有即时的市值可以观察，那么就是几何布朗运动的假定合理，公司资产价值到底是多少仍是相对难以确定的。关于公司的价值问题虽有一整套的理论和研究方法，但它仍是研究者尚待突破的任务之一。1)The model supposes that the asset market value follows the Geometric Brown Motion, but the asset of the company assetiscan not traded as frequently as stock in the capital market. In fact, more articles hold the idea that the asset value follows a jump process. Another problem is that there is no instant market value for company asset value, so even if the assumption of Geometric Brown motion is right, the estimation of the asset value is difficultit is difficult to estimate the asset value. Although there are a whole set of theories and methodologies for handling the problem, it remains to be a hard nut for researchers to crack.2)假定违约发生在到期日，这一点与实际情况大相径庭，很多时候，违约发生在到期日之前。2)Tthe model supposes that default happens at maturity, which is very different from the real world, more often than not, default happens before maturity.3)债务结构的问题。一般公司债券结构比较复杂，模型至少需要考虑所有比该债券优先的债务。这给问题的研究带来很大的复杂性。而在Merton(1974)的文中是假定了公司只发行一个简单的无息票债券。3)the problemIssue of the debt structure. Usually the debt structure of a company is complex, the model at least needs to consider all debt prior to the default-able bond. The model of Merton(1974) supposes that the company issues only a simple zero-coupon bond.4)违约触发问题。模型中假定违约的触发点就是F，这也比较的武断，实际上，有很多资不抵债的企业存在着，也有一些企业还没有等到资产减到F就不得不宣布破产。4)The trigger of the default. The model supposes that the trigger point of the default is F, which is different from the real world. In fact, some companies with negative equity are still aliveexists, while others with positive equity have to go bankrupt.在Merton其后，有着许多的模型拓展，如：Geske(1977)和Geske and Jonson(1984)得到了带息票的可违约债券的解析解；Ho and Singer(1982,1984)分析了各种契约保护(例如债券发行人的融资限制、债务的优先级、债务归还计划)对模型的影响；Chance(1990)考察了可违约债券的久期(duration)问题；Shimko, Tejima, and Deventer(1993)加入了Vasicek(1977)的利率模型，得到可违约债券的解析解；Wang(1999)加入了独立于公司资产波动的CIR(1985)利率模型；Szatzschneider(2000)改进了Wang(1999)的独立假定。After Merton, there are many expanded models. For example, Geske(1977)and Geske and Jonson(1984) got the closed solution of the default-able bond with coupon, Ho and Singer(1982,1984) studied the infection of all kinds of contracts protection(for example finance limit to bond issue), Chance(1990) studied the duration of the default-able bond, Shimko, Tejima, and Deventer(1993) got the closed solution based on the Vasicek(1977) interest rate model, Wang(1999) studied the model based on the CIR(1985) interest rate model which is independent with fluctuate of the company asset, Szatzschneider(2000) mended Wang(1999)s independent assumption.以上的结构模型由于都需要用到Black and Scholes (1973)的期权理论，所以被称为期权理论模型(Option Theoretic Models)。All of the above structure models use the Black and Scholes (1973) option theory, so they are called Option Theoretic Models.改进Merton的模型的另一个方向集中在上述缺点2和4，即放弃了违约必须在到期日的假定，而假定公司资产满足一定的随机过程，一旦到达某一标准，即被认为违约发生；同时对违约触发点加以修正。这实际上变成了一个随机过程的首过时问题，所以许多学者把这些模型单独归类，称为首过时模型（First Passage Time Models）。但是改进的模型保留了上述的1和3的缺点，而且带来新的问题：回收率不再是内生的了。Another direction to mend Mertons model focuses on shortcoming 2 and 4, that is to say, giving up the assumption that default must happen at maturity, the models suppose the company asset follows some stochastic process, then default can be triggered any time based ononce meet certainsome criterion, and the models mend the trigger point of the default. Many experts classify these models as First Passage Time Models. But the mended models still contain shortcoming 1 and 3, plus bringing and a new shortcoming arises,that is, recovery rate is not endogenic.Longstaff and Schwartz(1995)的模型假定资产价值A(t)符合几何布朗运动，利率r(t)符合Vasicek过程，而且利率和资产价值之间还有相关性Longstaff and Schwartz(1995) suppose the asset value A(t) follows the Geometric Brown Motion, interest rate r(t) follows Vasicek model, and there is correlation between the interest rate and the asset value 一旦资产价值A(t)首次击中违约阈值K，模型就认为违约发生，并按面值回收，回收率为(1-w)。这样，模型可以得到 Once the value of asset A(t) decreases to the trigger point of default hit the default threshold K for the first time, the model consider default occursthe default is triggered andwith the bond is recovered at face value. The recovery rate is (1-w). So, we can conclude that:(22)其中是风险债券的价格，是无风险债券的价格，可以认为是违约的概率(是在风险中性情况下的概率，不然式子（22）应该加上风险补偿)，也就是在时间段t, T之间A(t)到达违约阈值K的概率，Longstaff and Schwartz把它等效成在时间段t, T之间为负的概率， 并通过风险中性情况下的偏微分方程得到了该概率的表达式为：是当时的极限，其中表达式比较繁长，具体参见原文。Here, is the price of the risk bond, is the price of risk-free bond, can be considered as the probability of default (the probability is the one in risk-neutral condition, otherwise the risk premium risk should be added to formula 22). In other words, Longstaff and Schwartz believe, the probability of A(t) reaching valve K is equal to being negative during the time t, T. Through partial differential equation in risk-neutral condition,they get the probability expression: is the limit of when , , here, the expression of is complicated, the detail can be referred from the original paper.其他主要模型有，Black and Cox (1976)模型，利率非随机，违约阀值为，其中K和都是常数参数，违约一旦发生，回收值为资产市值A。Kim, Ramaswamy, and Sundaresan(1993)模型，利率符合CIR(1985)模型，违约阈值为，其中c表示债券息票率，表示公司流出现金流率。如果在到期日之前发生违约，则回收值为，其中为回收率，与时间有关；如果到期日发生违约，则回收值为。 Nielsen, Saa-Requejo, and Santa-Clara(1993)模型，利率符合Vasicek(1977)模型，违约阈值为，回收值为。 Briys and de Varenne(1997)模型，利率符合Vasicek (1977)模型，违约阀值为。如果在到期日之前发生违约，则回收值为；如果到期日发生违约，则回收值为。There are other important models. Another important model is In Black and Cox (1976)s model , with non-random interest rate is not random and the trigger valve of default threshold is . Here, K and are both constants. Once default happens, the recovery is the market value of the asset. In Kim, Ramaswamy, and Sundaresan(1993)s model, interest rate follows CIR(1985)s model. The trigger valve of default threshold is , c standing for coupon rate and y standing for cash outflow rate of the corporate. If default happens before maturity, the recovery is . Here, is recovery rate, which changes by with time. If default happens at maturity, the recovery is . In Nielsen, Saa-Requejo, and Santa-Clara(1993)s model, interest rate follows Vasicek(1977)s model. The default valve threshold is and the recovery is . In Briys and de Varenne(1997)s model, interest rate follows Vasicek(1977)s model, the default valvethreshold is . If default happens before maturity, the recovery is , otherwise, the recovery is .二、强度模型 Intensity Model强度模型，顾名思义，强度是该类模型的重点，一般直接假定违约强度。这类模型的共同特点是违约率和回收率都是外生变量，这是与结构模型最大的区别。从经济解释的角度讲，结构模型在解释违约机理方面，比强度模型相对突兀的假定违约强度要恰当些，当然这实际上是观察同一个问题的不同角度，而且两个角度是应有深刻联系的，例如下文的Madan and Unal(2000)从公司财务讨论违约机理，然后得到违约强度；另外，从金融实用的角度讲，哪个能更准确的定价，哪个就更好。本文以三篇代表性文章为主线介绍强度模型：Jarrow and Turnbull(1995)，Duffie and Singleton(1999), Madan and Unal(2000)。Intensity Model, from the definition, we can conclude that intensity is the focus of this type of these models. In these models, default intensity usually is directly supposed. The character of intensity models is that default probability and recovery rate are both exogenous, which differs from structural model. From economic view, structural model is better than intensity model in explanation to default mechanism. Of course, they are just different point of view to the same question in fact and the two views have deep strong relationship. For example, Madan and Unal(2000) discuss default mechanism from corporate finance then they get default intensity. In addition, from practicality, the more precise the model is, the better one it is. In this thesis, we introduce intensity model with three examples: Jarrow and Turnbull(1995), Duffie and Singleton(1999), Madan and Unal(2000).虽然与违约强度有关的模型最早可以追溯到Pye(1974)，不过一般认为强度模型的工作开始于Jarrow and Turnbull (1995) 模型（Bielecki and Rutkowski (2002)）。Jarrow and Turnbull (1995)模型假定：Although the default intensity model can cast back track to Pye(1974)s, it is usually accepted that the research about default intensity model starts from by Jarrow and Turnbull (1995)s model（Bielecki and Rutkowski (2002). Jarrow and Turnbull (1995) model supposes:1)违约过程是违约强度为常数l的泊松过程 (Poisson Process)。实际上，泊松过程是一个允许事件多次发生的计数过程，但是在Jarrow and Turnbull模型的应用中，只是应用第一个事件，违约一旦发生后，对该金融产品而言就不会再有违约发生。2)对于每个信用等级的债券i，回收率di 为常数。常数违约强度和常数回收率的假定暗含了一个问题，即模型不考虑信用利差风险。3)利率过程为一维的HJM(1992)模型。4)假定利率过程与违约过程、回收过程独立。这是一个比较大的妥协，正如上文提到的信用风险与市场风险的相关性非常显著，不容忽视。Jarrow and Turnbull (1995)在文中也承认这两者的相关性是很显著的。5)如果违约在到期日之前发生，那么回收值按照无风险债券的市值回收，即债权人得到，其中为到期日为T的无风险债券在t时刻的市值。6)违约发生后，可违约债券退化为无风险债券。这个假定换一个角度可能比较容易理解：在违约发生后，债权人把得到的金钱都投资到到期日为T的无风险债券。(1) Default process is a Poisson Process whose default intensity is a constant l. In fact, Poisson Process is a counting process which permittings an event happens again and again. But, Jarrow and Turnbulls model can only be used to the first event happened. Once default happens, there is no default happened to the financial product.(2) For bond i in each credit rating, the recovery di is a constant. The constant default intensity and the constant recovery suggest the model dose not take credit spread risk into consideration.(3) Interest rate process follows one dimension HJM(1992) model.(4) Interest rate process is independent to default process and recovery process. This is a huge compromise. As referred above, the correlation between credit risk and market risk is significant. It should not be ignored. Jarrow and Turnbull (1995) also admits the significance of the correlation.(5) If default happens before maturity, the recovery value is related with the market value of risk-free bond. In other words, the creditor gets . Here, is the market value of risk-free bond at time t, which matures at time T.(6) After default, default-able bond changes to risk-free bond. From another view, this assumption can be comprehended easily: after default, investor put all the money to risk-free bond, which matures at time T.然后，Jarrow and Turnbull(1995)就得到了他们的结果，其中为计算出来的信用风险修正因子。结果看上去很漂亮，利率r和信用风险参数d 和l 都很好的分离开了，也就是说，只要无风险债券价格乘上一个信用风险修正因子就可以得到可违约债券的价格了。其实由于假定利率过程与违约过程的相关系数为零，模型的结果是显而易见的，这一点后来在