Asian Economic and Financial Review ANALYSIS OF THE CONTAGION EFFECT TO THE CREDIT DERIVATIVE VALUATION

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1 Asan Economc and Fnancal Revew, 2016, 6(10): Asan Economc and Fnancal Revew ISSN(e): /ISSN(p): URL: ANALYSIS OF HE CONAGION EFFEC O HE CREDI DERIVAIVE VALUAION Hsang Hu Chu Y Fang Chung 2 1 Department of Banng and Fnance Natonal Ch Nan Unversty No.1, Unversty Rd., Pul 545, awan (R.O.C.) 2 Department of Busness Admnstraton De Ln Insttute of echnology, New ape Cty, awan (R.O.C.) ABSRAC hs study explores a credt dervatve prcng model wth counterparty rs and the contagon effect. o compare wth the standard credt dervatve prcng model, we analyze the counterparty rs and the contagon effect to a thto-default Baset Credt Lned Note (BCLN) valuaton by Monte Carlo smulaton. Counterparty rs and the contagon effect show sgnfcant nfluence for th-to-default BCLN valuaton. Especally wth hgh, the BCLN prcng model wth the counterparty rs and the contagon effect captures chan defaultng phenomenon successfully. It ndcates the hgher the th-to-default BCLN or the lower the correlaton degree wthn reference enttes, the more sgnfcant the contagon effect becomes to the th-to-default BCLN valuaton. Parameters senstvty analyses ndcate that the coupon rates of the thrd-to-default BCLN are lower wth the hgher rs-free rate or the shorter maturty. he hgher hazard rate or the lower recovery rate of reference enttes results n the hgher coupon rates of the thrd-to-default BCLN. Lastly, the hazard rate and the recovery rate of the counterparty are less senstve to the BCLN prcng. he major contrbuton n ths study s that the credt dervatve prcng model wth the contagon effect s developed to capture the default chan reacton, and the valuaton performance s sgnfcant numercally AESS Publcatons. All Rghts Reserved. Keywords: Baset credt lned note, Chan default, Contagon effect, Counterparty rs, Credt dervatve valuaton, Credt rs. JEL Classfcaton: C15, C52, C53. Receved: 5 May 2016/ Revsed: 18 June 2016/ Accepted: 14 July 2016/ Publshed: 13 August 2016 Contrbuton/ Orgnalty hs study s one of very few studes whch have nvestgated the credt dervatve valuaton wth the contagon effect to capture the default chan reacton especally durng the global fnancal dstress perod. 1. INRODUCION he credt crunch of n the Unted States s the most substantal fnancal crss to become a global phenomenon, envelopng emergng and developed fnancal and economy marets, wth the collapse of Lehman Brothers worldwde and the strategc move of Goldman Sachs and Morgan Stanley to regulated ban status. Due to the collapses of large fnancal nsttutons n the subprme mortgage crss, more and more researchers and practtoners have found that the ssuer default rs and the correlatons between the ssuer and the underlyng assets Correspondng author DOI: /journal.aefr/ / ISSN(e): /ISSN(p): AESS Publcatons. All Rghts Reserved. 571

2 Asan Economc and Fnancal Revew, 2016, 6(10): sgnfcantly affect the credt dervatves prcng. And an approprate modelng and forecastng of nherent credt rs s emphaszed by rs managers and analytcs. Generally, default probablty of the borrower, losses gven default, exposures, and correlaton between borrowers are consdered n credt rs models. Hull and Whte (2001); Hu and Lo (2002) and Mashal and Nald (2005) analyze how the possblty of ssuer default affects the credt dervatves prcng. Relatvely, the counterparty rs model proposed by Wu (2010) s consdered drectly the default correlaton between the reference enttes and the ssuer. her numercal results obvously ndcate that ssuer default rs mpacts the Baset Credt Lned Note (BCLN) valuaton. Dungey (2008) measures the contagon effect n the money maret and equty maret n US, UK, Europe, Japan and Australa from July 2007 to September Results show that there exsts a domnant effect from contagon at 78 percent of observed volatlty n the US equty maret. In all other marets the contagon effects le between 4 and 10 percent, wth the greatest contrbuton n the Australan money maret and the smallest n the Japanese equty maret. In addton, the asymmetrc dependency between borrowers n a portfolo has been gradually emphaszed and s frst appled to the credt prcng model va credt nfecton or neural networ connecton between borrowers by Davs and Lo (2001) and Egloff et al. (2007). hey fnd that even a small nfecton probablty has a dramatc mpact on the tals of the loss dstrbuton. Neu and Kühn (2004) observe the collectve phenomena such as bursts and avalanches by extendng the standard Credt Metrcs credt rs model based on heterogeneous, and non-symmetrc nter frm dependences. hen Rösch and Wnterfeldt (2008) develop a contagon extenson of factor models whch are used n most credt rs models. her fndngs ndcate that contagon effects are sgnfcant and can serously affect loss dstrbuton. he structure form approach (Merton, 1974; Blac and Cox, 1976; Gese, 1977; Leland, 1994; Longstaff and Schwartz, 1995) and the reduced form approach (Jarrow and urnbull, 1995; Duffe and Sngleton, 1997; Jarrow et al., 1997) are two man credt rs models. hey consder default events as the reflecton n a company s captal structure and beng exogenous, separately. he copula approach for specfyng the dependence among default tmes was developed and extended by L (2000) and Andersen et al. (2003); Hull and Whte (2004); Laurent and Gregory (2005) and Slar (1959). he concepton s that default events are ndependent condtonal on the common factor. hs study explores a credt dervatve prcng model wth counterparty rs and the contagon effect. o compare wth the naïve credt dervatve prcng model, we analyze counterparty rs and the contagon effect to the BCLN valuaton by Monte Carlo smulaton. A BCLN s a note wth coupon lned to credt events of reference enttes. he buyer of a BCLN durng the contract perod receves constant coupon payments perodcally untl the default event occurs. For example, the default event of a frst to default BCLN means that one of the reference enttes goes nto default before the contract maturty. he buyer wll receve the nomnal at the maturty date f the default event does not occur, but be redeemed at the recovery value determned ntally. he structure of ths paper s organzed as follows: In Secton 2, we ntroduce the factor Gaussan copula model, the BCLN valuaton model wthout counterparty rs, the BCLN prcng model wth counterparty rs. he smulaton results of the BCLN prcng wth the ssuer default probablty and the contagon effect analyss are presented n Secton 3. Secton 4 s the concluson. 2. HE MODEL 2.1. he Factor Gaussan Copula Consder a portfolo wth N reference enttes, and the default tmes and hazard rates are, 2,, N, 2,, 1 N and 1, respectvely. Accordng to the reduced form model, the cumulatve default probablty of each reference entty before tme t s 2016 AESS Publcatons. All Rghts Reserved. 572

3 Asan Economc and Fnancal Revew, 2016, 6(10): t F ( t) P( t) 1 e, 1, 2,, N. (1) Applyng the Gaussan copula, the multvarate jont dstrbuton of default tmes s F(,,, ) ( ( F ( )), ( F ( )),, ( ( ))). (2) 1 2 N F N N and denote the multvarate and one dmensonal cumulatve normal dstrbuton, respectvely. In the sngle factor model, assume random varable,, correspondng to the reference entty defaultng tme assocates a common factor Y and a specfc rs factor,, Y 1, 1,2,..., N. (3) Y 2, Y Y and are ndependent standard normal varables and the correlaton coeffcent between hen the reference entty defaultng tme s ln(1 ( )) 1 F ( ( )), 1, 2,, N. (4) and Y s Y Prcng BCLN wthout Counterparty Rs Assume a th-to-default BCLN nvolvng N reference enttes whch the notonal prncpal of each reference entty s one dollar. he coupon rate s c, the payment dates are t, 1, 2,,, and the maturty date of the BCLN s t. Moreover, Q denotes the rs-neutral probablty measure, I () s an ndcator functon, r s the rsfree rate, s the th default tme, and s the recovery rate of the th default reference entty. he value of a th-to-default BCLN s E Q [ c 1 e rt I( t ) e r I( t ) e rt I( t )] (5) hen the far coupon rate c at the ntal tme can be yeld, c E Q [1 e E r I( t Q [ 1 e rt ) e rt I( t )] I( t )] (6) 2.3. Prcng BCLN wth Counterparty Rs o base on the standard stuaton that nether counterparty rs nor the contagon effect s consdered, Wu (2010) consders the counterparty rs drectly the default correlaton between the reference enttes and the ssuer. he random varable correspondng to the default tme of the underlyng reference entty s, Z 1, 1,2,..., N. (7) Z 2, Z he common factor n the standard model s replaced by the ssuer varable, Z AESS Publcatons. All Rghts Reserved. 573

4 Asan Economc and Fnancal Revew, 2016, 6(10): he ssuer default rs s nvolved n the BCLN prcng by Wu (2010) he consders whether the ssuer defaults before or after the th default tme. Assume,, and be the ssuer default tme, the ssuer recovery rate, and hazard rate, respectvely. he value of a th-to-default BCLN s E Q rt r [ c e I( t mn(, )) e I( mn(, t )) 1 (8) r rt e I( mn(, t )) e I( t mn(, ))] hen the far coupon rate c at the ntal tme can be yeld, r 1 ( mn(, )) e I t Q E ˆ r ˆ rt e I( ˆ mn(, t )) e I( t mn(, ˆ)) c Q rt E [ e I( t mn(, ˆ))] 1 (9) 2.4. Prcng BCLN wth Counterparty Rs and the Contagon Effect In ths study, the major pont s to examne the BCLN valuaton wth ncorporatng the counterparty rs and the contagon effect. he prcng model of the former sector s extended wth the contagon dea proposed by Rösch and Wnterfeldt (2008). he credt contagon effect n Rösch and Wnterfeldt (2008) s estmated by the default rate of the nfectng frms. Assume there are two groups of frms, the nfectng frms and the nfected frms. Default events n the nfectng frms group nfect the nfected frms group by the contagon effect parameter, 0, and the default rate of the nfectng frms, D / I. D and I denote the defaulted number of the nfectng frms and the number of the nfectng frms, separately. he random varable correspondng to the nfectng frms and the nfected frms are, Y, Y Y Y , Y, Y f nfectng D I frm f nfected frm Besdes, the far coupon rate c of the nth to default BCLN at the ntal tme s the same as equaton (9). (10) 3. NUMERICAL RESULS 3.1. Valuaton of a BCLN wth Dfferent Models Suppose that 10 reference enttes have notonal prncpal one dollar, hazard rate 5%, recovery rate 30%, the coupon s pad annually, and the rs free rate s 2%. 100,000 runs of Monte Carlo smulaton are mplemented, and the far coupon rates of a 5 year th-to-default BCLN wth standard BCLN prcng model are shown n Fgure 1. As expected, the far coupon rate of a BCLN decreases wth and the absolute value of correlaton degree wthn reference enttes as a result of the low jont default event possblty. Furthermore, let the hazard rate and recovery rate of the ssuer be 1% and 30%, respectvely. he far coupon rates of a 5 year th-to-default BCLN wth the counterparty rs are shown n Fgure 2. he BCLN coupon rate curve becomes flat, especally when =3. It s because that the BCLN coupon rate vares less wth the low degree correlaton wthn reference enttes, but rses apparently when the reference enttes are hgh postve or negatve correlated. Moreover, assume the contagon effect parameter to be -2 and Fgure 3 dsplays the far coupon rates of a 5 year th-to-default BCLN both wth counterparty rs and the contagon effect. No matter what the correlaton degrees wthn the reference enttes are, the far coupon rate of a th-to-default BCLN gets hgher. And the hgher s, the hgher far coupon rate of a BCLN s. When the 2016 AESS Publcatons. All Rghts Reserved. 574

5 Asan Economc and Fnancal Revew, 2016, 6(10): thrd-to-default BCLN s evaluated, the coupon rate s nearly accordng to the correlaton among reference enttes, but t vares sgnfcantly f the contagon effect s consdered. Fgure-1. Coupon rates of the th-to-default BCLN n the standard BCLN prcng model wth =1, =2, and =3. Source: Smulaton results. Fgure-2. Coupon rates of the th-to-default BCLN n the BCLN prcng model wth the counterparty rs (=1, =2, and =3). Source: Smulaton results. Fgure-3. Coupon rates of the th-to-default BCLN n the BCLN prcng model wth counterparty rs and the contagon effect (=1, =2, and =3). Source: Smulaton results AESS Publcatons. All Rghts Reserved. 575

6 Asan Economc and Fnancal Revew, 2016, 6(10): Fgure 4, Fgure 5, and Fgure 6 dsplay the smulaton results of the frst-, the second-, and the thrd-to-default BCLN wth dfferent prcng models. Relatve to the standard BCLN prcng model, the coupon rate of a BCLN prcng model wth counterparty rs s always hgher due to the exstence of the ssuer default possblty. Moreover, f the contagon effect s consdered, the far coupon rate of the BCLN prcng model wth counterparty rs and the contagon effect becomes much hgher except for the frst-to-default BCLN. It lely because that when the frst-todefault BCLN s consdered, the contract termnated smultaneously wth the ssuer default or the frst reference entty default occurs, so the contagon nteracton among the reference enttes has less mpact on the BCLN prcng. On the other hand, the contagon effect plays actvely for the second- or the thrd-to-default BCLN prcng. Fgure-4. Coupon rates of the frst-to-default BCLN n three models, the standard BCLN prcng model, the BCLN prcng model wth counterparty rs, and the BCLN prcng model wth counterparty rs and the contagon effect. Source: Smulaton results. Fgure-5. Coupon rates of the second-to-default BCLN n three models, the standard BCLN prcng model, the BCLN prcng model wth counterparty rs, and the BCLN prcng model wth counterparty rs and the contagon effect. Source: Smulaton results AESS Publcatons. All Rghts Reserved. 576

7 Asan Economc and Fnancal Revew, 2016, 6(10): Fgure-6. Coupon rates of the thrd-to-default BCLN n three models, the standard BCLN prcng model, the BCLN prcng model wth counterparty rs, and the BCLN prcng model wth counterparty rs and the contagon effect. Source: Smulaton results he Contagon Parameter Analyss o understand the contagon parameter nfluence on the BCLN valuaton, ths study let the contagon parameters, β, be -1, -2, -3, -4, -5, -6, -7, -8, and -9. he hgher the absolute value of the contagon parameter s, the larger the contagon effect ntensty s. he frst-, the second-, and the thrd-to-default BCLN valuatons wth dfferent contagon effect ntenstes are shown n Fgure 7, Fgure 8, and Fgure 9. When the frst-to-default BCLN s consdered, Fgure 7 shows that the coupon rate of a BCLN decreases wth the contagon effect ntensty ncreases gradually. But the frst-to-default BCLN valuaton vares less for the extreme contagon effect parameter (β=-7, -8, and -9). When the second- or the thrd-to-default BCLN s consdered, the contagon effect dsplays an nterestng phenomenon wth dfferent correlaton degrees of reference enttes. he relatve results are dsplayed n Fgure 8 and Fgure 9. Obvously, wth the hghly postve or negatve correlated reference enttes, the coupon rate of a BCLN decreases wth the contagon effect ntensty ncreases. Otherwse, the contagon effect s not explct wth the low degree correlaton of reference enttes. It ndcates that f the hgher th-to-default BCLN or a low correlaton degree wthn reference enttes s consdered, the contagon effect s more remarable to the th-to-default BCLN valuaton. Fgure-7. Coupon rates of the frst-to-default BCLN wth dfferent contagon effect parameters (β=-1, -2, -3, -4, -5, -6, -7, -8, and -9). Source: Smulaton results AESS Publcatons. All Rghts Reserved. 577

8 Asan Economc and Fnancal Revew, 2016, 6(10): Fgure-8. Coupon rates of the second-to-default BCLN wth dfferent contagon effect parameters (β=-1, -2, -3, -4, -5, -6, -7, -8, and -9). Source: Smulaton results. Fgure-9. Coupon rates of the thrd-to-default BCLN wth dfferent contagon effect parameters (β=-1, -2, -3, -4, -5, -6, -7, -8, and -9). Source: Smulaton results Other Parameters Senstvty Analyses Both counterparty rs and the contagon effect are consdered, we study the senstvty analyses wth the rsfree rate, the maturty, the hazard rate of reference enttes, the recovery rate of reference enttes, the hazard rate of the counterparty, and the recovery rate of the counterparty. Fgure 10 shows that the coupon rates of the thrd-todefault BCLN are lower wth the hgher rs-free rate, snce the hgher dscount effect. Due to the hgher occurrence possblty of default event, the coupon rates are hgher wth the longer maturty as the results of Fgure AESS Publcatons. All Rghts Reserved. 578

9 Asan Economc and Fnancal Revew, 2016, 6(10): Fgure-10. Coupon rates of the thrd-to-default BCLN wth dfferent rs-free rate (r=1%, 1.2%, 1.4%, 1.6%, 1.8%, 2%, 2.2%, 2.4%, 2.6%, and 2.8%). Source: Smulaton results. Fgure-11. Coupon rates of the thrd-to-default BCLN wth dfferent maturty ( t =5y, 5.5y, 6y, 6.5y, 7y, 7.5y, 8y, 8.5y, 9y, and 9.5y). Source: Smulaton results. Fgure 12 and Fgure 13 show that the hgher hazard rate or the lower recovery rate of reference enttes results n the hgher coupon rates of the thrd-to-default BCLN. Explctly, the hgher hazard rate or the lower recovery rate causes the severer expected loss. Lastly, the hazard rate and the recovery rate of the counterparty are less senstve to the BCLN prcng. Fgure 14 and Fgure 15 dsplay that the BCLN valuaton converges especally wth the postve correlaton coeffcent AESS Publcatons. All Rghts Reserved. 579

10 Asan Economc and Fnancal Revew, 2016, 6(10): Fgure-12. Coupon rates of the thrd-to-default BCLN wth dfferent hazard rate of reference enttes ( =5%, 5.2%, 5.4%, 5.6%, 5.8%, 6%, 6.2%, 6.4%, 6.6%, and 6.8%, 1,..., 10 ). Source: Smulaton results. Fgure-13. Coupon rates of the thrd-to-default BCLN wth dfferent recovery rate of reference enttes ( =0%, 10%, 20%, 30%, 40%, 50%, 60%, 70%, 80%, and 90%, 1,..., 10 ). Source: Smulaton results. Fgure-14. Coupon rates of the thrd-to-default BCLN wth dfferent hazard rate of the counterparty ( =1%, 1.2%, 1.4%, 1.6%, 1.8%, 2%, 2.2%, 2.4%, 2.6%, and 2.8%). Source: Smulaton results AESS Publcatons. All Rghts Reserved. 580

11 Asan Economc and Fnancal Revew, 2016, 6(10): Fgure-15. Coupon rates of the thrd-to-default BCLN wth dfferent recovery rate of the counterparty ( =0%, 10%, 20%, 30%, 40%, 50%, 60%, 70%, 80%, and 90%). Source: Smulaton results. 4. CONCLUSIONS hs study explores a credt dervatve prcng model wth counterparty rs and the contagon effect. o compare wth the standard credt dervatve prcng model, we analyze counterparty rs and the contagon effect to a th-todefault BCLN valuaton by Monte Carlo smulaton. If the counterparty rs or the contagon effect s consdered, the far coupon rate of a th-to-default BCLN decreases wth the absolute value of the correlaton coeffcent wthn the reference enttes. Relatve to the standard BCLN prcng model, when the counterparty rs and the contagon effect are both consdered, the far coupon rate of the BCLN becomes much hgher except for the frst-to-default BCLN. Especally wth the hghly postve or negatve correlated reference enttes, the coupon rate of a BCLN decreases wth the contagon effect ntensty ncreases. he counterparty rs and the contagon effect have substantal nfluence on th-to-default BCLN valuaton. Especally wth hgh, the BCLN prcng model wth counterparty rs and the contagon effect captures the chan defaultng phenomenon successfully. It ndcates that f the hgher th-to-default BCLN or a low correlaton degree wthn reference enttes s consdered, the contagon effect s more sgnfcant to the th-to-default BCLN valuaton. Other parameters senstvty analyses are also studed. he coupon rates of the thrd-to-default BCLN are lower wth the hgher rs-free rate or the shorter maturty. he hgher hazard rate or the lower recovery rate of reference enttes results n the hgher coupon rates of the thrd-to-default BCLN. Lastly, the hazard rate and the recovery rate of the counterparty are less senstve to the BCLN prcng. he major contrbuton n ths study s that the credt dervatve prcng model wth the contagon effect s developed to capture the default chan reacton, and the valuaton performance s sgnfcant numercally. Fundng: hs study receved no specfc fnancal support. Competng Interests: he authors declare that they have no competng nterests. Contrbutors/Acnowledgement: All authors contrbuted equally to the concepton and desgn of the study. he authors than the anonymous referees for help comments. Furthermore, we are grateful to the Mnstry of Scence and echnology for support. REFERENCES Andersen, L., J. Sdenus and S. Basu, All your hedges n one baset. Rs, 16(11): Blac, F. and J. Cox, Valung corporate securtes: Some effects of bond ndenture provsons. Journal of Fnance, 31(2): AESS Publcatons. All Rghts Reserved. 581

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