場(chǎng)外交易的衍生品中級(jí)5信用

1、Fundamentals of Futures and Options Markets, 7th Ed, Ch 23, Copyright John C. Hull 2010Credit Derivatives信用衍生產(chǎn)品Hull: Chapter 23鄭、陳：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信用衍生證券和信用衍生產(chǎn)品隨著信用風(fēng)險(xiǎn)定價(jià)技術(shù)的發(fā)展，對(duì)于任何損益狀況的分布的估計(jì)已成為可能，從而推動(dòng)了大量信用衍生證券的出現(xiàn)。所謂信用衍生證券就是與信用風(fēng)險(xiǎn)相關(guān)聯(lián)的衍生證券。信用衍生產(chǎn)品是用于分離、轉(zhuǎn)移和交易信用風(fēng)險(xiǎn)的各種工具和技術(shù)的統(tǒng)稱，主要指以

3、貸款或債券的信用狀況為標(biāo)的的衍生金融工具。其實(shí)質(zhì)是對(duì)傳統(tǒng)金融衍生工具的再造，賦予其管理信用風(fēng)險(xiǎn)的新功能。分類信用衍生證券的類型主要有兩類：一類是與違約事件相關(guān)聯(lián)的信用衍生證券；另一類是與信用變化相關(guān)聯(lián)的衍生證券。后者并不需要違約事件的發(fā)生，而前者則需要違約事件的發(fā)生。 信用衍生產(chǎn)品基本可以分為兩大類：無融資的信用衍生產(chǎn)品（Unfunded Credit Derivatives）和融資的信用衍生產(chǎn)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違約觸發(fā)的衍生工具 違約互換 信用違約互換 有限追索權(quán)票據(jù) 資產(chǎn)交換 無融資的信用衍生產(chǎn)品 (Unfunded Credit Derivatives)信用違約互換總收益互換信用溢價(jià)遠(yuǎn)期和期權(quán)Fundamentals of Futures and Options Markets, 7th Ed, Ch 23, Copyright John C. Hull 20106信用溢價(jià)遠(yuǎn)期和期權(quán)定義：以特定風(fēng)險(xiǎn)資產(chǎn)為標(biāo)的

5、、以該資產(chǎn)信用溢價(jià)為標(biāo)的價(jià)格變量的遠(yuǎn)期和期權(quán)合約。信用溢價(jià)遠(yuǎn)期的具體運(yùn)作過程是：交易雙方事先約定未來特定時(shí)刻某資產(chǎn)與基準(zhǔn)利率之間的信用溢價(jià)水平，遠(yuǎn)期到期時(shí)則根據(jù)支付本金修正久期（約定信用溢價(jià)水平實(shí)際信用溢價(jià)水平）進(jìn)行償付。信用溢價(jià)期權(quán)的到期Payoff是非對(duì)稱的。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、信用評(píng)級(jí)變動(dòng)的支付(1) 假如一個(gè)債券發(fā)行者現(xiàn)在的信用評(píng)級(jí)是AAA級(jí)，而合約規(guī)定如果在某一確定日期發(fā)行者的信用評(píng)級(jí)降為AA級(jí)，則合約的持有人將獲得一筆固定金額的支付。假設(shè)利率是不變的。用來解的方程為 2022/9/24CopyrightZhenlong Zheng 2003, Department of Finance, Xiamen University9信用評(píng)級(jí)變動(dòng)的支付(2)合約中如果信用等級(jí)是 AA就必須支付的規(guī)定必須被結(jié)合在邊界條件中。由于除非發(fā)行者被評(píng)級(jí)為AA級(jí),否則不存在支付，故邊界條件是簡(jiǎn)單的 2022/9/24CopyrightZhenlong Zheng 2003, Dep

7、artment of Finance, Xiamen University10信用評(píng)級(jí)變動(dòng)的支付(3)如果將這個(gè)合約看成類似于一個(gè)“敲入”障礙期權(quán)，顯然這將有助于對(duì)其進(jìn)行定價(jià)。在敲入障礙期權(quán)中，支付是由基本標(biāo)的變量達(dá)到某一給定水平而觸發(fā)的。我們的信用衍生證券也具有類似的情況，其中信用等級(jí)水平扮演了基本標(biāo)的變量的角色。同樣，我們必須求解：2022/9/24CopyrightZhenlong Zheng 2003, Department of Finance, Xiamen University11信用評(píng)級(jí)變動(dòng)的支付(4)其邊界條件為但現(xiàn)在我們還有一個(gè)附加條件，它對(duì)應(yīng)于敲入障礙期權(quán)中的邊界條件為：

8、對(duì)于所有的，換句話說，在達(dá)到 AA 等級(jí)的那一刻我們獲得 D的支付。對(duì)于這類合約通常會(huì)對(duì)觸發(fā)有效時(shí)間加以限制。在這類合約中只有當(dāng)觸發(fā)處于有效期內(nèi)對(duì)于的條件才會(huì)生效。 2022/9/24CopyrightZhenlong Zheng 2003, Department of Finance, Xiamen University12收益率差價(jià)衍生工具 違約看漲期權(quán)和違約看跌期權(quán) 信用差價(jià)期權(quán) 2022/9/24CopyrightZhenlong Zheng 2003, Department of Finance, Xiamen University13交換期權(quán)(1) 一個(gè)在時(shí)間T按某一固定的q以零息

9、票風(fēng)險(xiǎn)債券交換零息票無風(fēng)險(xiǎn)債券的期權(quán)所具有的回報(bào)為： 假定在債券到期日TB （TBT）收到的本金為D，則從 可知， 2022/9/24CopyrightZhenlong Zheng 2003, Department of Finance, Xiamen University14交換期權(quán)(2)假定在債券到期日TB （TBT）收到的本金為D，則從 可知，在利率是常數(shù)的情況下，我們可以得到風(fēng)險(xiǎn)債券價(jià)值遵循的偏微分方程是： 2022/9/24CopyrightZhenlong Zheng 2003, Department of Finance, Xiamen University15交換期權(quán)(3)同時(shí)

10、 現(xiàn)在我們的交換期權(quán)回報(bào)的價(jià)值f(p,t)等于：假設(shè)利率和風(fēng)險(xiǎn)率兩者都是隨機(jī)的。V滿足方程： 2022/9/24CopyrightZhenlong Zheng 2003, Department of Finance, Xiamen University16交換期權(quán)(4)首先，我們運(yùn)用下面式子求解基本標(biāo)的債券：然后,求解交換期權(quán),它同樣滿足下式。同時(shí)， 2.融資的信用衍生產(chǎn)品(Funded Credit Derivatives)信用聯(lián)系票據(jù)（CLN）合成型抵押債務(wù)憑證Fundamentals of Futures and Options Markets, 7th Ed, Ch 23, Copyr

11、ight John C. Hull 201017信用衍生產(chǎn)品的作用提供了新的對(duì)沖信用風(fēng)險(xiǎn)的手段有利于信用風(fēng)險(xiǎn)市場(chǎng)定價(jià)的形成為一些投資者提供了進(jìn)入新興市場(chǎng)和貸款市場(chǎng)的便捷渠道對(duì)提高銀行資本的報(bào)酬率很有幫助值得注意的是，信用衍生產(chǎn)品的功能是對(duì)沖信用風(fēng)險(xiǎn)，也就是對(duì)風(fēng)險(xiǎn)在市場(chǎng)中進(jìn)行重新分配與承擔(dān)，但并不意味著信用風(fēng)險(xiǎn)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