场外交易的衍生品中级5信用

1、Fundamentals of Futures and Options Markets, 7th Ed, Ch 23, Copyright John C. Hull 2010Credit Derivatives信用衍生产品Hull: Chapter 23郑、陈：17.3.2（p.327-333)1Fundamentals of Futures and Options Markets, 7th Ed, Ch 23, Copyright John C. Hull 2010Credit DerivativesDerivatives where the payoff depends on the cr

2、edit quality of a company or sovereign entityThe market started to grow fast in the late 1990s22022/9/24CopyrightZhenlong Zheng 2003, Department of Finance, Xiamen University3信用衍生证券和信用衍生产品随着信用风险定价技术的发展，对于任何损益状况的分布的估计已成为可能，从而推动了大量信用衍生证券的出现。所谓信用衍生证券就是与信用风险相关联的衍生证券。信用衍生产品是用于分离、转移和交易信用风险的各种工具和技术的统称，主要指以

3、贷款或债券的信用状况为标的的衍生金融工具。其实质是对传统金融衍生工具的再造，赋予其管理信用风险的新功能。分类信用衍生证券的类型主要有两类：一类是与违约事件相关联的信用衍生证券；另一类是与信用变化相关联的衍生证券。后者并不需要违约事件的发生，而前者则需要违约事件的发生。 信用衍生产品基本可以分为两大类：无融资的信用衍生产品（Unfunded Credit Derivatives）和融资的信用衍生产品（Funded Credit Derivatives）Fundamentals of Futures and Options Markets, 7th Ed, Ch 23, Copyright Joh

4、n C. Hull 201042022/9/24CopyrightZhenlong Zheng 2003, Department of Finance, Xiamen University5违约触发的衍生工具 违约互换 信用违约互换 有限追索权票据 资产交换 无融资的信用衍生产品 (Unfunded Credit Derivatives)信用违约互换总收益互换信用溢价远期和期权Fundamentals of Futures and Options Markets, 7th Ed, Ch 23, Copyright John C. Hull 20106信用溢价远期和期权定义：以特定风险资产为标的

5、、以该资产信用溢价为标的价格变量的远期和期权合约。信用溢价远期的具体运作过程是：交易双方事先约定未来特定时刻某资产与基准利率之间的信用溢价水平，远期到期时则根据支付本金修正久期（约定信用溢价水平实际信用溢价水平）进行偿付。信用溢价期权的到期Payoff是非对称的。Fundamentals of Futures and Options Markets, 7th Ed, Ch 23, Copyright John C. Hull 201072022/9/24CopyrightZhenlong Zheng 2003, Department of Finance, Xiamen University8

6、信用评级变动的支付(1) 假如一个债券发行者现在的信用评级是AAA级，而合约规定如果在某一确定日期发行者的信用评级降为AA级，则合约的持有人将获得一笔固定金额的支付。假设利率是不变的。用来解的方程为 2022/9/24CopyrightZhenlong Zheng 2003, Department of Finance, Xiamen University9信用评级变动的支付(2)合约中如果信用等级是 AA就必须支付的规定必须被结合在边界条件中。由于除非发行者被评级为AA级,否则不存在支付，故边界条件是简单的 2022/9/24CopyrightZhenlong Zheng 2003, Dep

7、artment of Finance, Xiamen University10信用评级变动的支付(3)如果将这个合约看成类似于一个“敲入”障碍期权，显然这将有助于对其进行定价。在敲入障碍期权中，支付是由基本标的变量达到某一给定水平而触发的。我们的信用衍生证券也具有类似的情况，其中信用等级水平扮演了基本标的变量的角色。同样，我们必须求解：2022/9/24CopyrightZhenlong Zheng 2003, Department of Finance, Xiamen University11信用评级变动的支付(4)其边界条件为但现在我们还有一个附加条件，它对应于敲入障碍期权中的边界条件为：

8、对于所有的，换句话说，在达到 AA 等级的那一刻我们获得 D的支付。对于这类合约通常会对触发有效时间加以限制。在这类合约中只有当触发处于有效期内对于的条件才会生效。 2022/9/24CopyrightZhenlong Zheng 2003, Department of Finance, Xiamen University12收益率差价衍生工具 违约看涨期权和违约看跌期权 信用差价期权 2022/9/24CopyrightZhenlong Zheng 2003, Department of Finance, Xiamen University13交换期权(1) 一个在时间T按某一固定的q以零息

9、票风险债券交换零息票无风险债券的期权所具有的回报为： 假定在债券到期日TB （TBT）收到的本金为D，则从 可知， 2022/9/24CopyrightZhenlong Zheng 2003, Department of Finance, Xiamen University14交换期权(2)假定在债券到期日TB （TBT）收到的本金为D，则从 可知，在利率是常数的情况下，我们可以得到风险债券价值遵循的偏微分方程是： 2022/9/24CopyrightZhenlong Zheng 2003, Department of Finance, Xiamen University15交换期权(3)同时

10、 现在我们的交换期权回报的价值f(p,t)等于：假设利率和风险率两者都是随机的。V满足方程： 2022/9/24CopyrightZhenlong Zheng 2003, Department of Finance, Xiamen University16交换期权(4)首先，我们运用下面式子求解基本标的债券：然后,求解交换期权,它同样满足下式。同时， 2.融资的信用衍生产品(Funded Credit Derivatives)信用联系票据（CLN）合成型抵押债务凭证Fundamentals of Futures and Options Markets, 7th Ed, Ch 23, Copyr

11、ight John C. Hull 201017信用衍生产品的作用提供了新的对冲信用风险的手段有利于信用风险市场定价的形成为一些投资者提供了进入新兴市场和贷款市场的便捷渠道对提高银行资本的报酬率很有帮助值得注意的是，信用衍生产品的功能是对冲信用风险，也就是对风险在市场中进行重新分配与承担，但并不意味着信用风险的消除Fundamentals of Futures and Options Markets, 7th Ed, Ch 23, Copyright John C. Hull 201018Fundamentals of Futures and Options Markets, 7th Ed,

12、Ch 23, Copyright John C. Hull 2010Credit Default Swaps (page 501)Buyer of the instrument acquires protection from the seller against a default by a particular company or country (the reference entity)Example: Buyer pays a premium of 90 bps per year for $100 million of 5-year protection against compa

13、ny XPremium is known as the credit default spread. It is paid for life of contract or until defaultIf there is a default, the buyer has the right to sell bonds with a face value of $100 million issued by company X for $100 million (Several bonds may be deliverable)19Fundamentals of Futures and Optio

14、ns Markets, 7th Ed, Ch 23, Copyright John C. Hull 2010CDS Structure Default Protection Buyer, ADefault Protection Seller, B90 bps per yearPayoff if there is a default by reference entity=100(1-R)Recovery rate, R, is the ratio of the value of the bond issued by reference entity immediately after defa

15、ult to the face value of the bond20Fundamentals of Futures and Options Markets, 7th Ed, Ch 23, Copyright John C. Hull 2010Other DetailsPayments are usually made quarterly or semiannually in arrearsIn the event of default there is a final accrual payment by the buyerSettlement can be specified as del

16、ivery of the bonds or a cash equivalent amountSuppose payments are made quarterly in the example just considered. What are the cash flows if there is a default after 3 years and 1 month and recovery rate is 40%?21Moodys Statistics on Recovery Rates (1982-2007) Table 23.1 page 504Fundamentals of Futu

17、res and Options Markets, 7th Ed, Ch 23, Copyright John C. Hull 2010ClassAverage recovery rate (%)Senior secured51.9Senior unsecured36.7Senior subordinated32.4Subordinated31.2Junior subordinated23.922Cheapest-to-deliver bondUsually there are a number of bonds that can be delivered in the event of a d

18、efaultThe protection buyer can choose to deliver the bond with the lowest priceIn the case of cash settlement the calculation agent will base the calculation of the payoff on the cheapest-to-deliver bondFundamentals of Futures and Options Markets, 7th Ed, Ch 23, Copyright John C. Hull 201023Fundamen

19、tals of Futures and Options Markets, 7th Ed, Ch 23, Copyright John C. Hull 2010Attractions of the CDS MarketAllows credit risks to be traded in the same way as market risksCan be used to transfer credit risks to a third partyCan be used to diversify credit risks 24Credit IndicesCDX NA IG tracks the

20、average CDS sppread for a portfolio of 125 investment grade (rated BBB or above) North American companiesiTraxx Europe tracks the average CDS sppread for a portfolio of 125 investment grade European companiesFundamentals of Futures and Options Markets, 7th Ed, Ch 23, Copyright John C. Hull 201025Fun

21、damentals of Futures and Options Markets, 7th Ed, Ch 23, Copyright John C. Hull 2010CDS Spreads and Bond Yields (See page 505)Portfolio consisting of a 5-year par yield corporate bond that provides a yield of 6% and a long position in a 5-year CDS costing 100 basis points per year is (approximately)

22、 a long position in a riskless instrument paying 5% per yearThis shows that CDS spreads should be approximately the same as bond yield spreads 26Fundamentals of Futures and Options Markets, 7th Ed, Ch 23, Copyright John C. Hull 2010ValuationSuppose that conditional on no earlier default a reference

23、entity has a (risk-neutral) probability of default of 2% in each of the next 5 yearsAssume payments are made annually in arrears, that defaults always happen half way through a year, and that the expected recovery rate is 40%Suppose that the breakeven CDS rate is s per dollar of notional principal27

24、Fundamentals of Futures and Options Markets, 7th Ed, Ch 23, Copyright John C. Hull 2010Unconditional Default and Survival Probabilities (Table 23.2)Time (years)Default ProbabilitySurvivalProbability10.02000.980020.01960.960430.01920.941240.01880.922450.01840.903928Fundamentals of Futures and Options

25、 Markets, 7th Ed, Ch 23, Copyright John C. Hull 2010Calculation of PV of PaymentsTable 23.3 (Principal=$1)Time (yrs)Survival ProbExpected PaymtDiscount FactorPV of Exp Pmt10.98000.9800s0.95120.9322s20.96040.9604s0.90480.8690s30.94120.9412s0.86070.8101s40.92240.9224s0.81870.7552s50.90390.9039s0.77880

26、.7040sTotal4.0704s29Fundamentals of Futures and Options Markets, 7th Ed, Ch 23, Copyright John C. Hull 2010Present Value of Expected Payoff Table 23.4 (Principal = $1)Time (yrs)Default Probab.Rec. Rate Expected PayoffDiscount FactorPV of Exp. Payoff0.50.02000.40.01200.97530.01171.50.01960.40.01180.9

27、2770.01092.50.01920.40.01150.88250.01023.50.01880.40.01130.83950.00954.50.01840.40.01110.79850.0088Total0.051130Fundamentals of Futures and Options Markets, 7th Ed, Ch 23, Copyright John C. Hull 2010PV of Accrual Payment made in event of a Default. Table 23.5 (Principal=$1)TimeDefault ProbExpected A

28、ccr PmtDisc FactorPV of Pmt0.50.02000.0100s0.97530.0097s1.50.01960.0098s0.92770.0091s2.50.01920.0096s0.88250.0085s3.50.01880.0094s0.83950.0079s4.50.01840.0092s0.79850.0074sTotal0.0426s31Fundamentals of Futures and Options Markets, 7th Ed, Ch 23, Copyright John C. Hull 2010Putting it all togetherPV o

29、f expected payments is 4.0704s+0.0426s=4.1130sThe breakeven CDS spread is given by4.1130s = 0.0511 or s = 0.0124 (124 bps)The value of a swap with a CDS spread of 150bps would be 4.11300.0150-0.0511 or 0.0106 times the principal.32Fundamentals of Futures and Options Markets, 7th Ed, Ch 23, Copyright

30、 John C. Hull 2010Implying Default Probabilities from CDS SpreadsSuppose that the mid market spread for a 5 year newly issued CDS is 100bps per yearWe can reverse engineer our calculations to conclude that the default probability is 1.61% per year.If probabilities are implied from CDS spreads and th

31、en used to value another CDS the result is not sensitive to the recovery rate providing the same recovery rate is used throughout33Fundamentals of Futures and Options Markets, 7th Ed, Ch 23, Copyright John C. Hull 2010Other Credit DerivativesBinary CDSFirst-to-default Basket CDSTotal return swapCred

32、it default optionCollateralized debt obligation34Fundamentals of Futures and Options Markets, 7th Ed, Ch 23, Copyright John C. Hull 2010Binary CDS (page 510)The payoff in the event of default is a fixed cash amountIn our example the PV of the expected payoff for a binary swap is 0.0852 and the break

33、even binary CDS spread is 207 bps35Fundamentals of Futures and Options Markets, 7th Ed, Ch 23, Copyright John C. Hull 2010First to Default Basket CDS (page 510)Similar to a regular CDS except that several reference entities are specified and there is a payoff when the first one defaultsThis depends

34、on “default correlation”Second, third, and nth to default deals are defined similarly36Fundamentals of Futures and Options Markets, 7th Ed, Ch 23, Copyright John C. Hull 2010Total Return Swap (pages 511-513)Agreement to exchange total return on a corporate bond for LIBOR plus a spreadAt the end ther

35、e is a payment reflecting the change in value of the bondUsually used as financing tools by companies that want an investment in the corporate bond Total ReturnPayerTotal Return ReceiverTotal Return on BondLIBOR plus 25bps37Fundamentals of Futures and Options Markets, 7th Ed, Ch 23, Copyright John C

36、. Hull 2010CDS Options (page 513)Example: European option to buy 5 year protection on Ford for 280 bps starting in one year. If Ford defaults during the one-year life of the option, the option is knocked out Depends on the volatility of CDS spreads38Fundamentals of Futures and Options Markets, 7th E

37、d, Ch 23, Copyright John C. Hull 2010Collateralized Debt Obligation (page 513)A pool of debt issues are put into a special purpose trustTrust issues claims against the debt in a number of tranchesFirst tranche covers x% of notional and absorbs first x% of default lossesSecond tranche covers y% of notional and absorbs next y% of default lossesetcA tranche earns a promised yield on remaining principal in the tranche39Fundamentals of Futures and Options Markets, 7th Ed, Ch 23, Copyright John C. Hull 2010Bond 1Bond 2Bond 3Bond nAverage Yield8.5%TrustTranche 4Loss 2