信用风险和资产证券化衍生品定价方法介绍_第一部分-信用风险理论介绍

1、金融工程案例分析课程， George Yuan， 2013-14第一部分第一部分: 信用风险理论介绍（信用风险理论介绍（2014年年7日上午日上午9:00am-11:30am）第1章信用风险介绍和信用评级第2章Merton 信用风险模型, KMV等模型应用介绍第3章信用风险价值量调整(CVA)介绍第4章单因子结构模型和信用风险价值量(Credit VaR) 介绍 第二部分第二部分: 信用和资产证券化衍生品定价信用和资产证券化衍生品定价( (2014年年7日下午日下午2:30pm-5:00pm) 第1章: 信用互换衍生品（CDS）定价介绍第2章: 资产证券化衍生品介绍第3章: 资产证券化衍生品定

2、价中南财大讲学大纲中南财大讲学大纲 信用风险和资产证券化衍生品定价方法介绍信用风险和资产证券化衍生品定价方法介绍袁先智（同济大学风险管理研究所）第3页2012-7-253一一信用风险管理的基本理论信用风险管理的基本理论目录目录金融工程案例分析课程， George Yuan， 2013-14第一部分第一部分: 信用风险理论介绍（信用风险理论介绍（2014年年7日上午日上午9:00am-11:30am）第第1章信用风险介绍和信用评级章信用风险介绍和信用评级第2章Merton 信用风险模型, KMV等模型应用介绍第3章信用风险价值量调整(CVA)介绍第4章单因子结构模型和信用风险价值量(Credit

3、 VaR) 介绍 中南财大讲学大纲中南财大讲学大纲 信用风险和资产证券化衍生品定价方法介绍信用风险和资产证券化衍生品定价方法介绍袁先智（同济大学风险管理研究所）4第3页2012-7-253一一Altman Score in Altman Score in 信用信用风险风险分析分析目录目录Altman Z-score IntroductionOriginal z-score component definitions variable definition weighting factorZ-score estimated for private firmsZ-score estimated f

4、or non-manufacturers & emerging markets金融工程案例分析课程， George Yuan， 2013-14目录目录- -Altman Score in Altman Score in 信用信用风险风险分析分析1. Altman Z-score Introduction1 Estimation of the formulaThe Z-score is a linear combination of four or five common business ratios, weighted by coefficients. The coefficients we

5、re estimated by identifying a set of firms which had declared bankruptcy and then collecting a matched sample of firms which had survived, with matching by industry and approximate size (assets).Altman applied the statistical method of discriminant analysis to a dataset of publicly held manufacturer

6、s. The estimation was originally based on data from publicly held manufacturers, but has since been re-estimated based on other datasets for private manufacturing, non-manufacturing and service companies.The original data sample consisted of 66 firms, half of which had filed for bankruptcy under Cha

7、pter 7. All businesses in the database were manufacturers, and small firms with assets of 2.99 -“Safe” Zones1.81 Z 2.99 -“Grey” ZonesZ 2.9 -“Safe” Zone1.23 Z 2.9 -“Grey” ZoneZ 3.75 -“Safe” Zone1.1 Z 3.75 -“Grey” ZoneZ化层化校正校正信用风险预期损失计量：客户信用评级信用风险预期损失计量：客户信用评级评级评级风险成本风险成本( (预期损失预期损失) )(EL

8、)(EL)=违约概率违约概率(PD)(PD)违约损失率违约损失率(LGD)(LGD)违约风险敞口违约风险敞口(EAD)(EAD)思想成就未来，专业创造价值第8页50%50%75%75%100%100%有担保有担保清收清收损失损失无担保收回无担保收回比例比例担保收回比担保收回比例例债项评级对应债项评级对应LGDLGD0%0%25%25%信用风险预期损失计算：债项评级信用风险预期损失计算：债项评级客户违约后不同业务的损失相同吗？客户违约后不同业务的损失相同吗？风险暴露头寸风险暴露头寸无担保无担保风险成本风险成本( (预期损失预期损失) )(EL)(EL)=违约概率违约概率(PD)(PD)违约损失率

9、违约损失率(LGD)(LGD)违约风险敞口违约风险敞口(EAD)(EAD)违约损失率违约损失率(LGD)(LGD) 授信风险暴露授信风险暴露(EAD)(EAD)思想成就未来，专业创造价值第9页信用风险预期损失计算示例信用风险预期损失计算示例举例举例1:1:评级为评级为A A的客户、房产抵押的的客户、房产抵押的1 1亿元流动资金贷款，风险成本是多少？亿元流动资金贷款，风险成本是多少？=客户信用评级客户信用评级0.30%0.30%违约概率违约概率(PD)(PD)债项评级债项评级贷款余额贷款余额50%50%1 1亿元亿元违约损失率违约损失率(LGD)(LGD) 授信风险暴露授信风险暴露(EAD)(E

10、AD)1515万万! !风险成本风险成本( (预期损失预期损失EL)EL)举例举例2:2:0.15%0.15%评级为评级为B B的客户、房产抵押的的客户、房产抵押的1 1亿元流动资金贷款，风险成本是多少？亿元流动资金贷款，风险成本是多少？=LGD)LGD) EAD)EAD)客户信用评级客户信用评级3.5%3.5%违约概率违约概率(PD)(PD)50%50%债项评级债项评级贷款余额贷款余额1 1亿元亿元175175万万! !风险成本风险成本( (预期损失预期损失EL)EL)1.75%!1.75%!第10页信用风险的预期损失和非预期损失信用风险的预期损失和非预期损失预期损失预期损失nnnnn预期损

11、失(EL)是预计损失的平均值如果要获得精确的预期损失，银行必须对违约暴露、违约概率和违约损失率三项风险因素进行准确的计量可以被银行所预见并成为需要考虑的经营成本，实质上并不构成真正的风险在银行提供的产品价格（如贷款利率）中得到了补偿通过提取坏账准备来抵补预期损失，影响银行的当期损益非预期损失非预期损失nnnn非预期损失是损失围绕预期损失的波动，它表现为风险损失标准差的若干倍违约事件可能存在相关性，当违约同时发生时，资产的价值将发生剧烈的变化非预期损失不能被预见，是真正意义上的风险，需要使用资本金来抵补非预期损失。通常，稳健的银行都应至少保证资本金足以抵御概率99.9%以上的非预期损失非预期损失

12、也是PD、LGD和EAD的函数，只不过更加复杂信用风险非预期损失与经济资本信用风险非预期损失与经济资本概率经济资本量是在一定置信水平下（用评级表示）应对非预期损失的能力。第11页损失非预期损失是在一定置信水平上的风险损失衡量。置信水平越高，损失越大，但概率越低。经济资本不是真正的银行资本，它是一个虚拟的资本数字，在数额上与非预期损失相等。某国际活跃银行信用评级目标评级置信度A AA AAA99.9% 99.95% 99.99%第12页信用风险的贷款定价信用风险的贷款定价贷款定价需要考虑的要素贷款定价需要考虑的要素目标利润目标利润总成本总成本根据盈利要求确定根据盈利要求确定考虑经济资本考虑经济资

13、本资金转移价格资金转移价格根据预期损失提根据预期损失提取减值准备取减值准备运营费用运营费用适用税率适用税率资本成本资本成本资金成本资金成本风险成本风险成本运营成本运营成本税务成本税务成本第3页2012-7-253一一信用评级信用评级-Credit Ratings-Credit Ratings目录目录第13页贷款收入贷款收入 - - 资金成本资金成本 - - 运营成本运营成本税务成本税务成本 - - 风险成本风险成本经济资本经济资本RAROCRAROC = =RAROCRAROC是考虑了风险的贷款盈利性指标，即是考虑了风险的贷款盈利性指标，即“风险调整后资本收益率风险调整后资本收益率”。风险调整

14、后资本收益率和经济增加值风险调整后资本收益率和经济增加值评价用评价用RAROCRAROC，兑现绩效用，兑现绩效用EVAEVA经济增加值经济增加值EVAEVA= = 贷款收入贷款收入 - - 资金成本资金成本 - - 运营成本运营成本 - - 税务成本税务成本 - - 风险成本风险成本资本成本资本成本即：即：EVAEVA = = 经风险调整收益经风险调整收益 资本成本资本成本第14页信用风险限额管理信用风险限额管理复复杂杂程程度度风风险险反反映映程程度度基于风险等级限额基于风险等级限额基于经济资本的限额管理基于经济资本的限额管理低低高高高高基于贷款规模基于贷款规模经济资本配置管理是指以限额形式将

15、经济资本分解到各分支机构、部门、产品、客户等对象，并进行动态监控和适时调整信用风险限额管理的发展：控制风险 风险收益的平衡第15页信用风险集中度限额管理信用风险集中度限额管理信用集中度风险控制信用集中度风险限额管理信用集中度风险缓释措施客户集中度风险限额信用集中度风险组合限额监管要求单一客户单一集团客户最大十家单一客户行业维度资产组合再平衡地区维度产品维度机构维度信贷资产组合管理信贷资产组合管理n银行要管理单笔信贷资产风险，更要管理信贷资产组合风险。贷款组合风险绝不是单笔贷款风险银行要管理单笔信贷资产风险，更要管理信贷资产组合风险。贷款组合风险绝不是单笔贷款风险的算术加总，需要考虑单笔贷款之间

16、的相关性。的算术加总，需要考虑单笔贷款之间的相关性。 资产组合管理的核心是掌握不同资产之间的风险资产组合管理的核心是掌握不同资产之间的风险相关性（叠加、消减和对冲等），从而利用资产相关性实现风险的对冲和消减，通过风险相关性配相关性（叠加、消减和对冲等），从而利用资产相关性实现风险的对冲和消减，通过风险相关性配置实现风险的对冲和消减，降低资产同时损失的可能性。置实现风险的对冲和消减，降低资产同时损失的可能性。n银行整个资产组合可以按照不同方式划分成不同的细分组合，比如不同地区组合、不同客户对象组银行整个资产组合可以按照不同方式划分成不同的细分组合，比如不同地区组合、不同客户对象组合、不同行业组合

17、等，最低层次是单个客户的信贷资产组合。不同层次组合之间不但有两两相关性，合、不同行业组合等，最低层次是单个客户的信贷资产组合。不同层次组合之间不但有两两相关性，还有多重相关性；不但有一个层次上的相关，还有不同层次之间的相关性。因此，信贷资产组合的还有多重相关性；不但有一个层次上的相关，还有不同层次之间的相关性。因此，信贷资产组合的研研究是一个庞大的体系。究是一个庞大的体系。第16页+1+1-1-10 0相关系数相关系数风险消减效应风险消减效应最小最小最大最大第17页银监会对公司客户内部评级体系的要求银监会对公司客户内部评级体系的要求总体要求评级维度评级结构债务人评级方法论和评级时间跨度评级标准

18、nnn至少包括债务人评级和债项评级两个维度债务人不同债项的债务人评级一致初级法、高级法对债项评级要求不同nn使用IRB确定非零售债务人、债项评级，债务人包括所有债务人与保证人计量模型法、专家判断法或综合应用nnn债务人评级至少7个非违约级别，1个违约级别有效区分风险，较高级别的风险小于较低级别的风险风险在不同级别合理分布nnnnnnnn可以采取时点评级法、跨周期评级法以及介于两者之间评级方法估计债务人的PD同时考虑影响债务人违约风险的非系统性和系统性因素应估计债务人未来一年违约概率。监管部门鼓励商业银行采用长于一年的时间跨度。模型使用计量模型应在评估违约特征和损失特征中发挥重要作用。应通过必要

19、的专家判断保证内部评级考虑了所有相关信息。可以根据业务的复杂程度以及风险管理水平建立多种评级体系监测模型的预测能力，验证模型，持续改进模型表现。对模型数据和假设的要求。nnnnn评级定义和标准应合理、直观，且能够有意义地区分风险考虑与债务人和债项评级相关的所有重要信息。银行拥有的信息越少，评级应越保守。内部评级可参考外部评级结果，但应考虑其他信息。文档化管理书面记录非零售风险暴露内部评级的设计。书面记录内部评级的重要过程、评级标准及各级别定义就模型的方法论、使用范围等建立完整文档技术要求第18页内部评级和五级分类内部评级和五级分类nnnn内部评级法是以历史数据为基础，通过数理统计分析方法，计算

20、借款人和债项的数据指标，并籍此开展信用风险管理。而五级分类则以主观判断为特征，缺乏必要的定量分析，而且没有反映出贷款的实际损失。内部评级法体现了对贷款发放的事先管理，强调通过分析既有的数据来预测和防范未来的风险，因而能有效地决定贷款发放与否、贷款发放额度、贷款利息水平以及抵押担保的要求等。而五级分类则只能在事后的检查或补救中发挥作用。内部评级是二维评级系统。一维是针对借款人；另一维是针对债项的评级。五级分类则是一维评级系统，不区分借款人与债项这两类不同性质的风险以及影响这两类风险的重要因素，因此，它并不能准确地反映风险，更不利于有效地防范风险。借助内部评级，可以把五级分类细化为十二级分类。第1

21、9页信用风险内部评级法的风险暴露分类信用风险内部评级法的风险暴露分类银行账户的风险暴露分类金融机构银行类机构风险暴露非银行类机构风险暴露两类公司一般公司风险暴露专业贷款项目融资物品融资商品融资产生收入的房地产中小企业风险暴露零售个人住房抵押贷款风险暴露合格循环零售贷款风险暴露其它零售风险暴露股权银行直接或间接持有的股东权益主权主权国家或经济实体区域及其中央银行非中央政府公共部门实体适用零风险权重的多边开发银行、国际清算银行、国际货币基金组织等的债权第20页nnn信用风险管理应与业务发展相协调，并满足监管要求。信用风险管理工具应支持信贷业务人员，而不是取代或影响他们的权威。信用风险管理工具只有在

22、被理解的情况下，才能充分发挥作用。信用风险管理的基本理念信用风险管理的基本理念信用风险管理是银行的核心能力信用风险管理是银行的核心能力nn虽然定量工具对于信用风险管理的科学性具有重要作用，但是信用风险管理不是一门工程学科。对于历史数据的分析相当有用，但其在信用风险管理中并不起决定性作用。业务专家的主观判断对于信用风险管理非常重要。n西方国家的模型开发技术，需要根据国内中小银行的数据质量状况和市场背景，有选择地采用。信用风险管理是一门实用艺术信用风险管理是一门实用艺术nn为了达到风险管理的最佳效果，需要风险管理体系的各方面协调统一，包括：治理架构、政策制度、工具方法、人员培训和信息科技等。只有在

23、一个完善的风险管理体系中，复杂的信用风险管理工具和方法，才能充分发挥其作用。风险管理是一个复杂的体系风险管理是一个复杂的体系金融工程案例分析课程， George Yuan， 2013-14Historical Data Historical data provided by rating agencies are also used to estimate the probability of default金融工程案例分析课程， George Yuan， 2013-14Cumulative Ave Default Rates (%) (1970-2009, Moodys, Table 23.

24、1, page 522) 1 2 3 4 5 7 10 Aaa 0.000 0.012 0.012 0.037 0.105 0.245 0.497 Aa 0.022 0.059 0.091 0.159 0.234 0.384 0.542 A 0.051 0.165 0.341 0.520 0.717 1.179 2.046 Baa 0.176 0.494 0.912 1.404 1.926 2.996 4.851 Ba 1.166 3.186 5.583 8.123 10.397 14.318 19.964 B 4.546 10.426 16.188 21.256 25.895 34.473

25、44.377 Caa-C 17.723 29.384 38.682 46.094 52.286 59.771 71.376 金融工程案例分析课程， George Yuan， 2013-14Interpretation The table shows the probability of default for companies starting with a particular credit rating A company with an initial credit rating of Baa has a probability of 0.176% of defaulting by t

26、he end of the first year, 0.494% by the end of the second year, and so on金融工程案例分析课程， George Yuan， 2013-14Do Default Probabilities Increase with Time? For a company that starts with a good credit rating default probabilities tend to increase with time For a company that starts with a poor credit rati

27、ng default probabilities tend to decrease with time 金融工程案例分析课程， George Yuan， 2013-14Hazard Rates vs Unconditional Default Probabilities (page 522-523) The hazard rate (also called default intensity) is the probability of default for a certain time period conditional on no earlier default The uncondi

28、tional default probability is the probability of default for a certain time period as seen at time zero What are the default intensities and unconditional default probabilities for a Caa rated company in the third year?金融工程案例分析课程， George Yuan， 2013-14Hazard Rate The hazard rate that is usually quote

29、d is an instantaneous rate If V(t) is the probability of a company surviving to time tttdttetQtetVtVttVttVt)()(1)()()()()()(0is timeby default ofy probabilit cumulative Theto leads This金融工程案例分析课程， George Yuan， 2013-1437Recovery Rate The recovery rate for a bond is usually defined as the price of the

30、 bond immediately after default as a percent of its face value Recovery rates tend to decrease as default rates increase金融工程案例分析课程， George Yuan， 2013-14Recovery Rates; Moodys: 1982 to 2009Class Mean(%) 1st lien bank loan 65.6 2nd lien bank loan 32.8 Sen Unsec. bank loan 48.7 Senior Secured 49.8 Seni

31、or Unsecured 36.6 Senior Subordinated 30.7 Subordinated 31.3 Junior Subordinated 24.7 金融工程案例分析课程， George Yuan， 2013-14Estimating Default Probabilities Alternatives:Use Bond PricesUse CDS spreadsUse Historical DataUse Mertons Model金融工程案例分析课程， George Yuan， 2013-14Using Bond Prices (Equation 23.2, page

32、 524)Average default intensity over life of bond is approximatelywhere s is the spread of the bonds yield over the risk-free rate and R is the recovery rateRs1金融工程案例分析课程， George Yuan， 2013-14More Exact Calculation Assume that a five year corporate bond pays a coupon of 6% per annum (semiannually). T

33、he yield is 7% with continuous compounding and the yield on a similar risk-free bond is 5% (with continuous compounding) Price of risk-free bond is 104.09; price of corporate bond is 95.34; expected loss from defaults is 8.75 Suppose that the probability of default is Q per year and that defaults al

34、ways happen half way through a year (immediately before a coupon payment. 金融工程案例分析课程， George Yuan， 2013-14Calculations (Table 23.3, page 525)金融工程案例分析课程， George Yuan， 2013-14Calculations continued We set 288.48Q = 8.75 to get Q = 3.03% This analysis can be extended to allow defaults to take place mor

35、e frequently With several bonds we can use more parameters to describe the default probability distribution金融工程案例分析课程， George Yuan， 2013-14The Risk-Free Rate The risk-free rate when default probabilities are estimated is usually assumed to be the LIBOR/swap zero rate (or sometimes 10 bps below the L

36、IBOR/swap rate) Asset swaps provide a direct estimates of the spread of bond yields over swap rates金融工程案例分析课程， George Yuan， 2013-14Real World vs Risk-Neutral Default Probabilities The default probabilities backed out of bond prices or credit default swap spreads are risk-neutral default probabilitie

37、s The default probabilities backed out of historical data are real-world default probabilities金融工程案例分析课程， George Yuan， 2013-14A Comparison Calculate 7-year default intensities from the Moodys data, 1970-2009, (These are real world default probabilities) Use Merrill Lynch data to estimate average 7-y

38、ear default intensities from bond prices, 1996 to 2007 (these are risk-neutral default intensities) Assume a risk-free rate equal to the 7-year swap rate minus 10 basis points金融工程案例分析课程， George Yuan， 2013-14Data from Moodys and Merrill Lynch Cumulative 7-year default probability (Moodys: 1970-2009)

39、Average bond yield spread in bps* (Merrill Lynch: 1996 to June 2007) Aaa 0.245% 35.74 Aa 0.384% 43.67 A 1.179% 58.68 Baa 2.996% 127.53 Ba 14.318% 280.28 B 34.473% 481.04 Caa 59.771% 1103.70 *The benchmark risk-free rate for calculating spreads is assumed to be the swap rate minus 10 basis points. Bo

40、nds are corporate bonds with a life of approximately 7 years. 金融工程案例分析课程， George Yuan， 2013-1448Real World vs Risk Neutral Hazard Rates (Table 23.4, page 527) 1 Calculated asln(1-d)/7 where d is the Moodys 7 yr default rate. For example, in the case of Aaa companies, d=0.00245 and -ln(0.99755)/7=0.0

41、004 or 4bps. For investment grade companies the historical hazard rate is approximately d/7. 2 Calculated as s/(1-R) where s is the bond yield spread and R is the recovery rate (assumed to be 40%). 金融工程案例分析课程， George Yuan， 2013-1449Average Risk Premiums Earned By Bond Traders 1 Equals average spread

42、 of our benchmark risk-free rate over Treasuries.2 Equals historical hazard rate times (1-R) where R is the recovery rate. For example, in the case of Baa, 26bps is 0.6 times 43bps.金融工程案例分析课程， George Yuan， 2013-1450Possible Reasons for These Results (The third reason is the most important) Corporate

43、 bonds are relatively illiquid The subjective default probabilities of bond traders may be much higher than the estimates from Moodys historical data Bonds do not default independently of each other. This leads to systematic risk that cannot be diversified away. Bond returns are highly skewed with l

44、imited upside. The non-systematic risk is difficult to diversify away and may be priced by the market金融工程案例分析课程， George Yuan， 2013-14Which World Should We Use? We should use risk-neutral estimates for valuing credit derivatives and estimating the present value of the cost of default We should use re

45、al world estimates for calculating credit VaR and scenario analysis金融工程案例分析课程， George Yuan， 2013-14第一部分第一部分: 信用风险理论介绍（信用风险理论介绍（2014年年7日上午日上午9:00am-11:30am）第1章信用风险介绍和信用评级第第2 2章章Merton Merton 信用风险模型信用风险模型, KMV, KMV等模型应用介绍等模型应用介绍第3章信用风险价值量调整(CVA)介绍第4章单因子结构模型和信用风险价值量(Credit VaR) 介绍 中南财大讲学大纲中南财大讲学大纲 信用风险

46、和资产证券化衍生品定价方法介绍信用风险和资产证券化衍生品定价方法介绍袁先智（同济大学风险管理研究所）金融工程案例分析课程， George Yuan， 2013-14Using Equity Prices: Mertons Model (page 530-531) Mertons model regards the equity as an option on the assets of the firm In a simple situation the equity value ismax(VT D, 0)where VT is the value of the firm and D is

47、the debt repayment required金融工程案例分析课程， George Yuan， 2013-14Equity vs. Assets The Black-Scholes-Merton option pricing model enables the value of the firms equity today, E0, to be related to the value of its assets today, V0, and the volatility of its assets, sVEV N dDeN ddVDrTTddTrTVVV0012102212()()l

48、n()();wheresss金融工程案例分析课程， George Yuan， 2013-14Volatilities sssEVVEEVVN dV0010()This equation together with the option pricing relationship enables V0 and sV to be determined from E0 and sE金融工程案例分析课程， George Yuan， 2013-14Example A companys equity is $3 million and the volatility of the equity is 80%

49、The risk-free rate is 5%, the debt is $10 million and time to debt maturity is 1 year Solving the two equations yields V0=12.40 and sv=21.23% The probability of default is N(d2) or 12.7%金融工程案例分析课程， George Yuan， 2013-14The Implementation of Mertons Model Choose time horizon Calculate cumulative oblig

50、ations to time horizon. This is termed by KMV the “default point”. We denote it by D Use Mertons model to calculate a theoretical probability of default Use historical data or bond data to develop a one-to-one mapping of theoretical probability into either real-world or risk-neutral probability of d

51、efault.金融工程案例分析课程， George Yuan， 2013-14The Implementation of Mertons Model KMV 模型介绍模型介绍Asset (A(t) = Debt (D(t) + Equity (E(t)Three main approaches to modeling credit risk in the finance literatureStructural approach: Assumptions are made about the dynamics of a firms assets, its capital structure,

52、and its debt and share holders. A firm defaults if the assets are insufficient according to some measure. A liability is characterized as an option on the firms assets.Reduced form approach: No assumptions are made concerning why a default occurs. Rather, the dynamics of default are exogenously give

53、n by the default rate (or intensity). Prices of credit sensitive securities can be calculated as if they were default free using the risk free rate adjusted by the level of intensity. Incomplete information approach: Combines the structural and reduced form approaches.Structural approach: default in

54、 the classical Merton model (1974).1. We want to use the structural approach to incorporate bond default risk in bond valuationThe value of the firms assets are assumed to follow the process, where is the instantaneous expected rate of return on assets, and is the standard deviation of the return on

55、 assets.Let D(t,T) be the date t market value of debt with promised payment B at date t. The second line in (18.2) says that the payoff to the creditors equals the promised payment (B) minus the payoff on a European put option written on the firms assets with exercise price B.Market value of firm de

56、bt, D(t)Let P(t,T) represent the current date t price of a default-free, zero-coupon bond that pays $1 at date T, where the bond conforms with the Vasicek model in Ch. 9. Pennacchi asserts that using results for pricing options (Ch. 9.3) when interest rates are random (as in 9.58), we can write Mark

57、et value of firm equity, E(t)Shareholder equity is similar to a call option on the firms assets, since at maturity the payoff to equity holders is max A(t) B, 0. However, shareholder equity is different from a European option if the firm pay dividends to shareholders prior to maturity as reflected i

58、n the first term of the last line in (18.4) where denotes the dividend rate.Critique of the Merton modelThe Merton model assumption is that the firm has a single issue of zero-coupon debt. That is unrealistic. Modeling multiple issues with different maturities and seniorities complicates default. In

59、 response some models have suggested that default occurs when the firms assets hit a lower boundary. That boundary has a monotonic relation to the firms total outstanding debt. The first passage time is when the value of the firms assets crosses through the lower boundary. First passage model - - bo

60、nd indenture provisions often include safety covenants that give bond holders the right to reorganize the firm if the value falls below a given barrier.The first passage model defines the survival probability as p(t,T) that the distance to default does not reach zero at any date between t and T. The